BAKVEC


      SUBROUTINE BAKVEC(NM,N,T,E,M,Z,IERR)

    This subroutine forms the eigenvectors of a NONSYMMETRIC
    TRIDIAGONAL matrix by back transforming those of the
    corresponding symmetric matrix determined by  FIGI.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       T contains the nonsymmetric matrix.  Its subdiagonal is
         stored in the last N-1 positions of the first column,
         its diagonal in the N positions of the second column,
         and its superdiagonal in the first N-1 positions of
         the third column.  T(1,1) and T(N,3) are arbitrary.
       E contains the subdiagonal elements of the symmetric
         matrix in its last N-1 positions.  E(1) is arbitrary.
       M is the number of eigenvectors to be back transformed.
       Z contains the eigenvectors to be back transformed
         in its first M columns.
    On OUTPUT
       T is unaltered.
       E is  Destroyed.
       Z contains the transformed eigenvectors
         in its first M columns.
       IERR is set to
         Zero       for normal return,
         2*N+I      if E(I) is zero with T(I,1) or T(I-1,3) non-zero.
                    In this case, the symmetric matrix is not similar
                    to the original matrix, and the eigenvectors
                    cannot be found by this program.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



BALANC


      SUBROUTINE BALANC(NM,N,A,LOW,IGH,SCALE)

    This subroutine is a translation of the ALGOL procedure BALANCE,
    NUM. MATH. 13, 293-304(1969) by Parlett and Reinsch.
    HANDBOOk FOR AUTO. COMP., Vol.II-LINEAR ALGEBRA, 315-326(1971).
    This subroutine balances a REAL matrix and isolates
    eigenvalues whenever possible.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       A contains the input matrix to be balanced.
    On OUTPUT
       A contains the balanced matrix.
       LOW and IGH are two integers such that A(I,J)
         is equal to zero if
          (1) I is greater than J and
          (2) J=1,...,LOW-1 or I=IGH+1,...,N.
       SCALE contains information determining the
          permutations and scaling factors used.
    Suppose that the principal submatrix in rows LOW through IGH
    has been balanced, that P(J) denotes the index interchanged
    with J during the permutation step, and that the elements
    of the diagonal matrix used are denoted by D(I,J).  Then
       SCALE(J) = P(J),    for J = 1,...,LOW-1
                = D(J,J),      J = LOW,...,IGH
                = P(J)         J = IGH+1,...,N.
    The order in which the interchanges are made is N to IGH+1,
    then 1 TO LOW-1.
    Note that 1 is returned for IGH if IGH is zero formally.
    The ALGOL procedure EXC contained in BALANCE appears in
    BALANC  in line.  (Note that the ALGOL roles of identifiers
    K,L have been reversed.)
    Questions and comments should be directed to B. S. Garbow,
    Applied Mathematics Division, ARGONNE NATIONAL LABORATORY



BALBAK


      SUBROUTINE BALBAK(NM,N,LOW,IGH,SCALE,M,Z)

    This subroutine is a translation of the ALGOL procedure BALBAK,
    NUM. MATH. 13, 293-304(1969) by Parlett and Reinsch.
    HANDBOOK FOR AUTO. COMP., Vol.II-LINEAR ALGEBRA, 315-326(1971).
    This subroutine forms the eigenvectors of a REAL GENERAL
    matrix by back transforming those of the corresponding
    balanced matrix determined by  BALANC.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by  BALANC.
       SCALE contains information determining the permutations
         and scaling factors used by  BALANC.
       M is the number of columns of Z to be back transformed.
       Z contains the real and imaginary parts of the eigen-
         vectors to be back transformed in its first M columns.
    On OUTPUT
       Z contains the real and imaginary parts of the
         transformed eigenvectors in its first M columns.
    Questions and comments should be directed to B. S. Garbow,
    Applied Mathematics Division, ARGONNE NATIONAL LABORATORY



BANDR


      SUBROUTINE BANDR(NM,N,MB,A,D,E,E2,MATZ,Z)

    This subroutine is a translation of the ALGOL procedure BANDRD,
    NUM. MATH. 12, 231-241(1968) by Schwarz.
    HANDBOOK FOR AUTO. COMP., VOL.II-lINEAR ALGEBRA, 273-283(1971).
    This subroutine reduces a REAL SYMMETRIC BAND matrix
    to a symmetric tridiagonal matrix using and optionally
    accumulating orthogonal similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       MB is the (half) band width of the matrix, defined as the
         number of adjacent diagonals, including the principal
         diagonal, required to specify the non-zero portion of the
         lower triangle of the matrix.
       A contains the lower triangle of the symmetric band input
         matrix stored as an N by MB array.  Its lowest subdiagonal
         is stored in the last N+1-MB positions of the first column,
         its next subdiagonal in the last N+2-MB positions of the
         second column, further subdiagonals similarly, and finally
         its principal diagonal in the N positions of the last column.
         Contents of storages not part of the matrix are arbitrary.
       MATZ should be set to .TRUE. if the transformation matrix is
         to be accumulated, and to .FALSE. otherwise.
    On OUTPUT
       A has been destroyed, except for its last two columns which
         contain a copy of the tridiagonal matrix.
       D contains the diagonal elements of the tridiagonal matrix.
       E contains the subdiagonal elements of the tridiagonal
         matrix in its last N-1 positions.  E(1) is set to zero.
       E2 contains the squares of the corresponding elements of E.
         E2 may coincide with E if the squares are not needed.
       Z contains the orthogonal transformation matrix produced in
         the reduction if MATZ has been set to .TRUE.  Otherwise, Z
         is not referenced.
    Questions and comments should be directed to B. S. Garbow,
    Applied Mathematics Division, ARGONNE NATIONAL LABORATORY



BANDV


      SUBROUTINE BANDV(NM,N,MBW,A,E21,M,W,Z,IERR,NV,RV,RV6)

    This subroutine finds those eigenvectors of a REAL SYMMETRIC
    BAND matrix corresponding to specified eigenvalues, using inverse
    iteration.  The subroutine may also be used to solve systems
    of linear equations with a symmetric or non-symmetric band
    coefficient matrix.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       MBW is the number of columns of the array A used to store the
         band matrix.  If the matrix is symmetric, MBW is its (half)
         band width, denoted MB and defined as the number of adjacent
         diagonals, including the principal diagonal, required to
         specify the non-zero portion of the lower triangle of the
         matrix.  If the subroutine is being used to solve systems
         of linear equations and the coefficient matrix is not
         symmetric, it must however have the same number of adjacent
         diagonals above the main diagonal as below, and in this
         case, MBW=2*MB-1.
       A contains the lower triangle of the symmetric band input
         matrix stored as an N by MB array.  Its lowest subdiagonal
         is stored in the last N+1-MB positions of the first column,
         its next subdiagonal in the last N+2-MB positions of the
         second column, further subdiagonals similarly, and finally
         its principal diagonal in the N positions of column MB.
         If the subroutine is being used to solve systems of linear
         equations and the coefficient matrix is not symmetric, A is
         N by 2*MB-1 instead with lower triangle as above and with
         its first superdiagonal stored in the first N-1 positions of
         column MB+1, its second superdiagonal in the first N-2
         positions of column MB+2, further superdiagonals similarly,
         and finally its highest superdiagonal in the first N+1-MB
         positions of the last column.
         Contents of storages not part of the matrix are arbitrary.
       E21 specifies the ordering of the eigenvalues and contains
           0.0E0 if the eigenvalues are in ascending order, or
           2.0E0 if the eigenvalues are in descending order.
         If the subroutine is being used to solve systems of linear
         equations, E21 should be set to 1.0E0 if the coefficient
         matrix is symmetric and to -1.0E0 if not.
       M is the number of specified eigenvalues or the number of
         systems of linear equations.
       W contains the M eigenvalues in ascending or descending order.
         If the subroutine is being used to solve systems of linear
         equations (A-W(R)*I)*X(R)=B(R), where I is the identity
         matrix, W(R) should be set accordingly, for R=1,2,...,M.
       Z contains the constant matrix columns (B(R),R=1,2,...,M), if
         the subroutine is used to solve systems of linear equations.
       NV must be set to the dimension of the array parameter RV
         as declared in the calling program dimension statement.
    On OUTPUT
       A and W are unaltered.
       Z contains the associated set of orthogonal eigenvectors.
         Any vector which fails to converge is set to zero.  If the
         subroutine is used to solve systems of linear equations,
         Z contains the solution matrix columns (X(R),R=1,2,...,M).
       IERR is set to
         Zero       for normal return,
         -R         if the eigenvector corresponding to the R-th
                    eigenvalue fails to converge, or if the R-th
                    system of linear equations is nearly singular.
       RV and RV6 are temporary storage arrays.  Note that RV is
         of dimension at least N*(2*MB-1).  If the subroutine
         is being used to solve systems of linear equations, the
         determinant (up to sign) of A-W(M)*I is available, upon
         return, as the product of the first N elements of RV.
    Questions and comments should be directed to B. S. Garbow,
    Applied Mathematics Division, ARGONNE NATIONAL LABORATORY



BCRH


      FUNCTION BCRH(XLL,XRR,IZ,C,A,BH,F,SGN)



BDIFF


      SUBROUTINE BDIFF(L,V)

    BDIFF computes the sum of B(L,K)*V(K)*(-1)**K where B(L,K)
    are the binomial coefficients.  Truncated sums are computed by
    setting last part of the V vector to zero. On return, the binomial
    sum is in V(L).


BISECT


      SUBROUTINE BISECT(N,EPS1,D,E,E2,LB,UB,MM,M,W,IND,IERR,RV4,RV5)

    This subroutine is a translation of the bisection technique
    in the ALGOL procedure TRISTURM by Peters and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 418-439(1971).
    This subroutine finds those eigenvalues of a TRIDIAGONAL
    SYMMETRIC matrix which lie in a specified interval,
    using bisection.
    On INPUT
       N is the order of the matrix.
       EPS1 is an absolute error tolerance for the computed
         eigenvalues.  If the input EPS1 is non-positive,
         it is reset for each submatrix to a default value,
         namely, minus the product of the relative machine
         precision and the 1-norm of the submatrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
       E2 contains the squares of the corresponding elements of E.
         E2(1) is arbitrary.
       LB and UB define the interval to be searched for eigenvalues.
         If LB is not less than UB, no eigenvalues will be found.
       MM should be set to an upper bound for the number of
         eigenvalues in the interval.  WARNING. If more than
         MM eigenvalues are determined to lie in the interval,
         an error return is made with no eigenvalues found.
    On OUTPUT
       EPS1 is unaltered unless it has been reset to its
         (last) default value.
       D and E are unaltered.
       Elements of E2, corresponding to elements of E regarded
         as negligible, have been replaced by zero causing the
         matrix to split into a direct sum of submatrices.
         E2(1) is also set to zero.
       M is the number of eigenvalues determined to lie in (LB,UB).
       W contains the M eigenvalues in ascending order.
       IND contains in its first M positions the submatrix indices
         associated with the corresponding eigenvalues in W --
         1 for eigenvalues belonging to the first submatrix from
         the top, 2 for those belonging to the second submatrix, etc.
       IERR is set to
         Zero       for normal return,
         3*N+1      if M exceeds MM.
       RV4 and RV5 are temporary storage arrays.
    The ALGOL procedure STURMCNT contained in TRISTURM
    appears in BISECT in-line.
    Note that subroutine TQL1 or IMTQL1 is generally faster than
    BISECT, if more than N/4 eigenvalues are to be found.
    Questions and comments should be directed to B. S. Garbow,
    Applied Mathematics Division, ARGONNE NATIONAL LABORATORY



BKIAS


      SUBROUTINE BKIAS(X,N,KTRMS,T,ANS,IND,MS,GMRN,H,IERR)

    BKIAS computes repeated integrals of the K0 Bessel function
    by the asymptotic expansion


BKISR


      SUBROUTINE BKISR(X,N,SUM,IERR)

    BKISR computes repeated integrals of the K0 Bessel function
    by the series for N=0,1, and 2.


BKSOL


      SUBROUTINE BKSOL(N,A,X)



BLKTR1


      SUBROUTINE BLKTR1(N,AN,BN,CN,M,AM,BM,CM,IDIMY,Y,B,W1,W2,W3,WD,WW,



BNDACC


      SUBROUTINE BNDACC(G,MDG,NB,IP,IR,MT,JT)

    These subroutines solve the least squares problem Ax = b for
    banded matrices A using sequential accumulation of rows of the
    data matrix.  Exactly one right-hand side vector is permitted.
    These subroutines are intended for the type of least squares
    systems that arise in applications such as curve or surface
    fitting of data.  The least squares equations are accumulated and
    processed using only part of the data.  This requires a certain
    user interaction during the solution of Ax = b.
    Specifically, suppose the data matrix (A B) is row partitioned
    into Q submatrices.  Let (E F) be the T-th one of these
    submatrices where E = (0 C 0).  Here the dimension of E is MT by N
    and the dimension of C is MT by NB.  The value of NB is the
    bandwidth of A.  The dimensions of the leading block of zeros in E
    are MT by JT-1.
    The user of the subroutine BNDACC provides MT,JT,C and F for
    T=1,...,Q.  Not all of this data must be supplied at once.
    Following the processing of the various blocks (E F), the matrix
    (A B) has been transformed to the form (R D) where R is upper
    triangular and banded with bandwidth NB.  The least squares
    system Rx = d is then easily solved using back substitution by
    executing the statement CALL BNDSOL(1,...). The sequence of
    values for JT must be nondecreasing.  This may require some
    preliminary interchanges of rows and columns of the matrix A.
    The primary reason for these subroutines is that the total
    processing can take place in a working array of dimension MU by
    NB+1.  An acceptable value for MU is
                      MU = MAX(MT + N + 1),
    where N is the number of unknowns.
    Here the maximum is taken over all values of MT for T=1,...,Q.
    Notice that MT can be taken to be a small as one, showing that
    MU can be as small as N+2.  The subprogram BNDACC processes the
    rows more efficiently if MU is large enough so that each new
    block (C F) has a distinct value of JT.
    The four principle parts of these algorithms are obtained by the
    following call statements
    CALL BNDACC(...)  Introduce new blocks of data.
    CALL BNDSOL(1,...)Compute solution vector and length of
                      residual vector.
    CALL BNDSOL(2,...)Given any row vector H solve YR = H for the
                      row vector Y.
    CALL BNDSOL(3,...)Given any column vector W solve RZ = W for
                      the column vector Z.
    The dots in the above call statements indicate additional
    arguments that will be specified in the following paragraphs.
    The user must dimension the array appearing in the call list..
    G(MDG,NB+1)
    Description of calling sequence for BNDACC..
    The entire set of parameters for BNDACC are
    Input..
    G(*,*)            The working array into which the user will
                      place the MT by NB+1 block (C F) in rows IR
                      through IR+MT-1, columns 1 through NB+1.
                      See descriptions of IR and MT below.
    MDG               The number of rows in the working array
                      G(*,*).  The value of MDG should be .GE. MU.
                      The value of MU is defined in the abstract
                      of these subprograms.
    NB                The bandwidth of the data matrix A.
    IP                Set by the user to the value 1 before the
                      first call to BNDACC.  Its subsequent value
                      is controlled by BNDACC to set up for the
                      next call to BNDACC.
    IR                Index of the row of G(*,*) where the user is
                      to place the new block of data (C F).  Set by
                      the user to the value 1 before the first call
                      to BNDACC.  Its subsequent value is controlled
                      by BNDACC. A value of IR .GT. MDG is considered
                      an error.
    MT,JT             Set by the user to indicate respectively the
                      number of new rows of data in the block and
                      the index of the first nonzero column in that
                      set of rows (E F) = (0 C 0 F) being processed.
    Output..
    G(*,*)            The working array which will contain the
                      processed rows of that part of the data
                      matrix which has been passed to BNDACC.
    IP,IR             The values of these arguments are advanced by
                      BNDACC to be ready for storing and processing
                      a new block of data in G(*,*).
    Description of calling sequence for BNDSOL..
    The user must dimension the arrays appearing in the call list..
    G(MDG,NB+1), X(N)
    The entire set of parameters for BNDSOL are
    Input..
    MODE              Set by the user to one of the values 1, 2, or
                      3.  These values respectively indicate that
                      the solution of AX = B, YR = H or RZ = W is
                      required.
    G(*,*),MDG,       These arguments all have the same meaning and
     NB,IP,IR         contents as following the last call to BNDACC.
    X(*)              With mode=2 or 3 this array contains,
                      respectively, the right-side vectors H or W of
                      the systems YR = H or RZ = W.
    N                 The number of variables in the solution
                      vector.  If any of the N diagonal terms are
                      zero the subroutine BNDSOL prints an
                      appropriate message.  This condition is
                      considered an error.
    Output..
    X(*)              This array contains the solution vectors X,
                      Y or Z of the systems AX = B, YR = H or
                      RZ = W depending on the value of MODE=1,
                      2 or 3.
    RNORM             If MODE=1 RNORM is the Euclidean length of the
                      residual vector AX-B.  When MODE=2 or 3 RNORM
                      is set to zero.
    Remarks..
    To obtain the upper triangular matrix and transformed right-hand
    side vector D so that the super diagonals of R form the columns
    of G(*,*), execute the following Fortran statements.
    NBP1=NB+1
    DO 10 J=1, NBP1
 10 G(IR,J) = 0.E0
    MT=1
    JT=N+1
    CALL BNDACC(G,MDG,NB,IP,IR,MT,JT)


BNDSOL


      SUBROUTINE BNDSOL(MODE,G,MDG,NB,IP,IR,X,N,RNORM)

    These subroutines solve the least squares problem Ax = b for
    banded matrices A using sequential accumulation of rows of the
    data matrix.  Exactly one right-hand side vector is permitted.
    These subroutines are intended for the type of least squares
    systems that arise in applications such as curve or surface
    fitting of data.  The least squares equations are accumulated and
    processed using only part of the data.  This requires a certain
    user interaction during the solution of Ax = b.
    Specifically, suppose the data matrix (A B) is row partitioned
    into Q submatrices.  Let (E F) be the T-th one of these
    submatrices where E = (0 C 0).  Here the dimension of E is MT by N
    and the dimension of C is MT by NB.  The value of NB is the
    bandwidth of A.  The dimensions of the leading block of zeros in E
    are MT by JT-1.
    The user of the subroutine BNDACC provides MT,JT,C and F for
    T=1,...,Q.  Not all of this data must be supplied at once.
    Following the processing of the various blocks (E F), the matrix
    (A B) has been transformed to the form (R D) where R is upper
    triangular and banded with bandwidth NB.  The least squares
    system Rx = d is then easily solved using back substitution by
    executing the statement CALL BNDSOL(1,...). The sequence of
    values for JT must be nondecreasing.  This may require some
    preliminary interchanges of rows and columns of the matrix A.
    The primary reason for these subroutines is that the total
    processing can take place in a working array of dimension MU by
    NB+1.  An acceptable value for MU is
                      MU = MAX(MT + N + 1),
    where N is the number of unknowns.
    Here the maximum is taken over all values of MT for T=1,...,Q.
    Notice that MT can be taken to be a small as one, showing that
    MU can be as small as N+2.  The subprogram BNDACC processes the
    rows more efficiently if MU is large enough so that each new
    block (C F) has a distinct value of JT.
    The four principle parts of these algorithms are obtained by the
    following call statements
    CALL BNDACC(...)  Introduce new blocks of data.
    CALL BNDSOL(1,...)Compute solution vector and length of
                      residual vector.
    CALL BNDSOL(2,...)Given any row vector H solve YR = H for the
                      row vector Y.
    CALL BNDSOL(3,...)Given any column vector W solve RZ = W for
                      the column vector Z.
    The dots in the above call statements indicate additional
    arguments that will be specified in the following paragraphs.
    The user must dimension the array appearing in the call list..
    G(MDG,NB+1)
    Description of calling sequence for BNDACC..
    The entire set of parameters for BNDACC are
    Input..
    G(*,*)            The working array into which the user will
                      place the MT by NB+1 block (C F) in rows IR
                      through IR+MT-1, columns 1 through NB+1.
                      See descriptions of IR and MT below.
    MDG               The number of rows in the working array
                      G(*,*).  The value of MDG should be .GE. MU.
                      The value of MU is defined in the abstract
                      of these subprograms.
    NB                The bandwidth of the data matrix A.
    IP                Set by the user to the value 1 before the
                      first call to BNDACC.  Its subsequent value
                      is controlled by BNDACC to set up for the
                      next call to BNDACC.
    IR                Index of the row of G(*,*) where the user is
                      the user to the value 1 before the first call
                      to BNDACC.  Its subsequent value is controlled
                      by BNDACC. A value of IR .GT. MDG is considered
                      an error.
    MT,JT             Set by the user to indicate respectively the
                      number of new rows of data in the block and
                      the index of the first nonzero column in that
                      set of rows (E F) = (0 C 0 F) being processed.
    Output..
    G(*,*)            The working array which will contain the
                      processed rows of that part of the data
                      matrix which has been passed to BNDACC.
    IP,IR             The values of these arguments are advanced by
                      BNDACC to be ready for storing and processing
                      a new block of data in G(*,*).
    Description of calling sequence for BNDSOL..
    The user must dimension the arrays appearing in the call list..
    G(MDG,NB+1), X(N)
    The entire set of parameters for BNDSOL are
    Input..
    MODE              Set by the user to one of the values 1, 2, or
                      3.  These values respectively indicate that
                      the solution of AX = B, YR = H or RZ = W is
                      required.
    G(*,*),MDG,       These arguments all have the same meaning and
     NB,IP,IR         contents as following the last call to BNDACC.
    X(*)              With mode=2 or 3 this array contains,
                      respectively, the right-side vectors H or W of
                      the systems YR = H or RZ = W.
    N                 The number of variables in the solution
                      vector.  If any of the N diagonal terms are
                      zero the subroutine BNDSOL prints an
                      appropriate message.  This condition is
                      considered an error.
    Output..
    X(*)              This array contains the solution vectors X,
                      Y or Z of the systems AX = B, YR = H or
                      RZ = W depending on the value of MODE=1,
                      2 or 3.
    RNORM             If MODE=1 RNORM is the Euclidean length of the
                      residual vector AX-B.  When MODE=2 or 3 RNORM
                      is set to zero.
    Remarks..
    To obtain the upper triangular matrix and transformed right-hand
    side vector D so that the super diagonals of R form the columns
    of G(*,*), execute the following Fortran statements.
    NBP1=NB+1
    DO 10 J=1, NBP1
 10 G(IR,J) = 0.E0
    MT=1
    JT=N+1
    CALL BNDACC(G,MDG,NB,IP,IR,MT,JT)


BNFAC


      SUBROUTINE BNFAC(W,NROWW,NROW,NBANDL,NBANDU,IFLAG)



BNSLV


      SUBROUTINE BNSLV(W,NROWW,NROW,NBANDL,NBANDU,B)



BQR


      SUBROUTINE BQR(NM,N,MB,A,T,R,IERR,NV,RV)

    This subroutine is a translation of the ALGOL procedure BQR,
    NUM. MATH. 16, 85-92(1970) by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL II-LINEAR ALGEBRA, 266-272(1971).
    This subroutine finds the eigenvalue of smallest (usually)
    magnitude of a REAL SYMMETRIC BAND matrix using the
    QR algorithm with shifts of origin.  Consecutive calls
    can be made to find further eigenvalues.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       MB is the (half) band width of the matrix, defined as the
         number of adjacent diagonals, including the principal
         diagonal, required to specify the non-zero portion of the
         lower triangle of the matrix.
       A contains the lower triangle of the symmetric band input
         matrix stored as an N by MB array.  Its lowest subdiagonal
         is stored in the last N+1-MB positions of the first column,
         its next subdiagonal in the last N+2-MB positions of the
         second column, further subdiagonals similarly, and finally
         its principal diagonal in the N positions of the last column.
         Contents of storages not part of the matrix are arbitrary.
         On a subsequent call, its output contents from the previous
         call should be passed.
       T specifies the shift (of eigenvalues) applied to the diagonal
         of A in forming the input matrix. What is actually determined
         is the eigenvalue of A+TI (I is the identity matrix) nearest
         to T.  On a subsequent call, the output value of T from the
         previous call should be passed if the next nearest eigenvalue
         is sought.
       R should be specified as zero on the first call, and as its
         output value from the previous call on a subsequent call.
         It is used to determine when the last row and column of
         the transformed band matrix can be regarded as negligible.
       NV must be set to the dimension of the array parameter RV
         as declared in the calling program dimension statement.
    On OUTPUT
       A contains the transformed band matrix.  The matrix A+TI
         derived from the output parameters is similar to the
         input A+TI to within rounding errors.  Its last row and
         column are null (if IERR is zero).
       T contains the computed eigenvalue of A+TI (if IERR is zero).
       R contains the maximum of its input value and the norm of the
         last column of the input matrix A.
       IERR is set to
         Zero       for normal return,
         N          if the eigenvalue has not been
                    determined after 30 iterations.
       RV is a temporary storage array of dimension at least
         (2*MB**2+4*MB-3).  The first (3*MB-2) locations correspond
         to the ALGOL array B, the next (2*MB-1) locations correspond
         to the ALGOL array H, and the final (2*MB**2-MB) locations
         correspond to the MB by (2*MB-1) ALGOL array U.
    NOTE. For a subsequent call, N should be replaced by N-1, but
    MB should not be altered even when it exceeds the current N.
    Calls PYTHAG(A,B) for SQRT(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    Applied Mathematics Division, ARGONNE NATIONAL LABORATORY



BSGQ8


      SUBROUTINE BSGQ8(FUN,XT,BC,N,KK,ID,A,B,INBV,ERR,ANS,IERR,WORK)



CAXPY


      SUBROUTINE CAXPY(N,CA,CX,INCX,CY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      CA  complex scalar multiplier
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
      CY  complex vector with N elements
    INCY  storage spacing between elements of CY
    --Output--
      CY  complex result (unchanged if N .LE. 0)
    Overwrite complex CY with complex  CA*CX + CY.
    For I = 0 to N-1, replace  CY(LY+I*INCY) with CA*CX(LX+I*INCX) +
      CY(LY+I*INCY), where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N
      and LY is defined in a similar way using INCY.


CBABK2


      SUBROUTINE CBABK2(NM,N,LOW,IGH,SCALE,M,ZR,ZI)

    This subroutine is a translation of the ALGOL procedure
    CBABK2, which is a complex version of BALBAK,
    NUM. MATH. 13, 293-304(1969) by Parlett and Reinsch.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
    This subroutine forms the eigenvectors of a COMPLEX GENERAL
    matrix by back transforming those of the corresponding
    balanced matrix determined by  CBAL.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by  CBAL.
       SCALE contains information determining the permutations
         and scaling factors used by  CBAL.
       M is the number of eigenvectors to be back transformed.
       ZR and ZI contain the real and imaginary parts,
         respectively, of the eigenvectors to be
         back transformed in their first M columns.
    On OUTPUT
       ZR and ZI contain the real and imaginary parts,
         respectively, of the transformed eigenvectors
         in their first M columns.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



CBAL


      SUBROUTINE CBAL(NM,N,AR,AI,LOW,IGH,SCALE)

    This subroutine is a translation of the ALGOL procedure
    CBALANCE, which is a complex version of BALANCE,
    NUM. MATH. 13, 293-304(1969) by Parlett and Reinsch.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
    This subroutine balances a COMPLEX matrix and isolates
    eigenvalues whenever possible.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       AR and AI contain the real and imaginary parts,
         respectively, of the complex matrix to be balanced.
    On OUTPUT
       AR and AI contain the real and imaginary parts,
         respectively, of the balanced matrix.
       LOW and IGH are two integers such that AR(I,J) and AI(I,J)
         are equal to zero if
          (1) I is greater than J and
          (2) J=1,...,LOW-1 or I=IGH+1,...,N.
       SCALE contains information determining the
          permutations and scaling factors used.
    Suppose that the principal submatrix in rows LOW through IGH
    has been balanced, that P(J) denotes the index interchanged
    with J during the permutation step, and that the elements
    of the diagonal matrix used are denoted by D(I,J).  Then
       SCALE(J) = P(J),    for J = 1,...,LOW-1
                = D(J,J)       J = LOW,...,IGH
                = P(J)         J = IGH+1,...,N.
    The order in which the interchanges are made is N to IGH+1,
    then 1 to LOW-1.
    Note that 1 is returned for IGH if IGH is zero formally.
    The ALGOL procedure EXC contained in CBALANCE appears in
    CBAL  in line.  (Note that the ALGOL roles of identifiers
    K,L have been reversed.)
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



CCHDC


      SUBROUTINE CCHDC(A,LDA,P,WORK,JPVT,JOB,INFO)

    CCHDC computes the Cholesky decomposition of a positive definite
    matrix.  A pivoting option allows the user to estimate the
    condition of a positive definite matrix or determine the rank
    of a positive semidefinite matrix.
    On Entry
        A      COMPLEX(LDA,P).
               A contains the matrix whose decomposition is to
               be computed.  Only the upper half of A need be stored.
               The lower part of The array A is not referenced.
        LDA    INTEGER.
               LDA is the leading dimension of the array A.
        P      INTEGER.
               P is the order of the matrix.
        WORK   COMPLEX.
               WORK is a work array.
        JPVT   INTEGER(P).
               JPVT contains integers that control the selection
               of the pivot elements, if pivoting has been requested.
               Each diagonal element A(K,K)
               is placed in one of three classes according to the
               value of JPVT(K)).
                  If JPVT(K)) .GT. 0, then X(K) is an initial
                                     element.
                  If JPVT(K)) .EQ. 0, then X(K) is a free element.
                  If JPVT(K)) .LT. 0, then X(K) is a final element.
               Before the decomposition is computed, initial elements
               are moved by symmetric row and column interchanges to
               the beginning of the array A and final
               elements to the end.  Both initial and final elements
               are frozen in place during the computation and only
               free elements are moved.  At the K-th stage of the
               reduction, if A(K,K) is occupied by a free element
               it is interchanged with the largest free element
               A(L,L) with L .GE. K.  JPVT is not referenced if
               JOB .EQ. 0.
       JOB     INTEGER.
               JOB is an integer that initiates column pivoting.
               IF JOB .EQ. 0, no pivoting is done.
               IF JOB .NE. 0, pivoting is done.
    On Return
        A      A contains in its upper half the Cholesky factor
               of the matrix A as it has been permuted by pivoting.
        JPVT   JPVT(J) contains the index of the diagonal element
               of A that was moved into the J-th position,
               provided pivoting was requested.
        INFO   contains the index of the last positive diagonal
               element of the Cholesky factor.
    For positive definite matrices INFO = P is the normal return.
    For pivoting with positive semidefinite matrices INFO will
    in general be less than P.  However, INFO may be greater than
    the rank of A, since rounding error can cause an otherwise zero
    element to be positive.  Indefinite systems will always cause
    INFO to be less than P.
    LINPACK.  This version dated 03/19/79 .
    J. J. Dongarra and G. W. Stewart, Argonne National Laboratory and
    University of Maryland.
    BLAS CAXPY,CSWAP
    Fortran SQRT,REAL,CONJG


CCHDD


      SUBROUTINE CCHDD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S,INFO)

    CCHDD downdates an augmented Cholesky decomposition or the
    triangular factor of an augmented QR decomposition.
    Specifically, given an upper triangular matrix R of order P,  a
    row vector X, a column vector Z, and a scalar Y, CCHDD
    determines a unitary matrix U and a scalar ZETA such that
                       (R   Z )     (RR  ZZ)
                   U * (      )  =  (      ) ,
                       (0 ZETA)     ( X   Y)
    where RR is upper triangular.  If R and Z have been obtained
    from the factorization of a least squares problem, then
    RR and ZZ are the factors corresponding to the problem
    with the observation (X,Y) removed.  In this case, if RHO
    is the norm of the residual vector, then the norm of
    the residual vector of the downdated problem is
    SQRT(RHO**2 - ZETA**2).  CCHDD will simultaneously downdate
    several triplets (Z,Y,RHO) along with R.
    For a less terse description of what CCHDD does and how
    it may be applied, see the LINPACK Guide.
    The matrix U is determined as the product U(1)*...*U(P)
    where U(I) is a rotation in the (P+1,I)-plane of the
    form
                      ( C(I)  -CONJG(S(I)) )
                      (                    ) .
                      ( S(I)       C(I)    )
    the rotations are chosen so that C(I) is real.
    The user is warned that a given downdating problem may
    be impossible to accomplish or may produce
    inaccurate results.  For example, this can happen
    if X is near a vector whose removal will reduce the
    rank of R.  Beware.
    On Entry
        R      COMPLEX(LDR,P), where LDR .GE. P.
               R contains the upper triangular matrix
               that is to be downdated.  The part of R
               below the diagonal is not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        p      INTEGER.
               P is the order of the matrix R.
        X      COMPLEX(P).
               X contains the row vector that is to
               be removed from R.  X is not altered by CCHDD.
        Z      COMPLEX(LDZ,NZ), where LDZ .GE. P.
               Z is an array of NZ P-vectors which
               are to be downdated along with R.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of vectors to be downdated
               NZ may be zero, in which case Z, Y, and RHO
               are not referenced.
        Y      COMPLEX(NZ).
               Y contains the scalars for the downdating
               of the vectors Z.  Y is not altered bY CCHDD.
        RHO    REAL(NZ).
               RHO contains the norms of the residual
               vectors that are to be downdated.
    On Return
        R
        Z      contain the downdated quantities.
        RHO
        C      REAL(P).
               C contains the cosines of the transforming
               rotations.
        S      COMPLEX(P).
               S contains the sines of the transforming
               rotations.
        INFO   INTEGER.
               INFO is set as follows.
                  INFO = 0  if the entire downdating
                            was successful.
                  INFO =-1  if R could not be downdated.
                            in this case, all quantities
                            are left unaltered.
                  INFO = 1  if some RHO could not be
                            downdated.  The offending RHO's are
                            set to -1.
    LINPACK.  This version dated 08/14/78 .
    Stewart, G. W., University of Maryland, Argonne National Lab.
    CCHDD uses the following functions and subprograms.
    Fortran AIMAG,CABS,CONJG
    BLAS CDOTC, SCNRM2


CCHEX


      SUBROUTINE CCHEX(R,LDR,P,K,L,Z,LDZ,NZ,C,S,JOB)

    CCHEX updates the Cholesky factorization
                  A = CTRANS(R)*R
    of a positive definite matrix A of order P under diagonal
    permutations of the form
                  TRANS(E)*A*E
    where E is a permutation matrix.  Specifically, given
    an upper triangular matrix R and a permutation matrix
    E (which is specified by K, L, and JOB), CCHEX determines
    a unitary matrix U such that
                          U*R*E = RR,
    where RR is upper triangular.  At the users option, the
    transformation U will be multiplied into the array Z.
    If A = CTRANS(X)*X, so that R is the triangular part of the
    QR factorization of X, then RR is the triangular part of the
    QR factorization of X*E, i.e. X with its columns permuted.
    For a less terse description of what CCHEX does and how
    it may be applied, see the LINPACK Guide.
    The matrix Q is determined as the product U(L-K)*...*U(1)
    of plane rotations of the form
                          (    C(I)       S(I) )
                          (                    ) ,
                          ( -CONJG(S(I))  C(I) )
    where C(I) is real.  The rows these rotations operate on
    are described below.
    There are two types of permutations, which are determined
    by the value of JOB.
    1. Right circular shift (JOB = 1).
        The columns are rearranged in the following order.
               1,...,K-1,L,K,K+1,...,L-1,L+1,...,P.
        U is the product of L-K rotations U(I), where U(I)
        acts in the (L-I,L-I+1)-plane.
    2. Left circular shift (JOB = 2).
        The columns are rearranged in the following order
               1,...,K-1,K+1,K+2,...,L,K,L+1,...,P.
        U is the product of L-K rotations U(I), where U(I)
        acts in the (K+I-1,K+I)-plane.
    On Entry
        R      COMPLEX(LDR,P), where LDR .GE. P.
               R contains the upper triangular factor
               that is to be updated.  Elements of R
               below the diagonal are not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        P      INTEGER.
               P is the order of the matrix R.
        K      INTEGER.
               K is the first column to be permuted.
        L      INTEGER.
               L is the last column to be permuted.
               L must be strictly greater than K.
        Z      COMPLEX(LDZ,NZ), where LDZ .GE. P.
               Z is an array of NZ P-vectors into which the
               transformation U is multiplied.  Z is
               not referenced if NZ = 0.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of columns of the matrix Z.
        JOB    INTEGER.
               JOB determines the type of permutation.
                      JOB = 1  right circular shift.
                      JOB = 2  left circular shift.
    On Return
        R      contains the updated factor.
        Z      contains the updated matrix Z.
        C      REAL(P).
               C contains the cosines of the transforming rotations.
        S      COMPLEX(P).
               S contains the sines of the transforming rotations.
    LINPACK.  This version dated 08/14/78 .
    Stewart, G. W., University of Maryland, Argonne National Lab.
    CCHEX uses the following functions and subroutines.
    Extended BLAS CROTG
    Fortran MIN0


CCHUD


      SUBROUTINE CCHUD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S)

    CCHUD updates an augmented Cholesky decomposition of the
    triangular part of an augmented QR decomposition.  Specifically,
    given an upper triangular matrix R of order P, a row vector
    X, a column vector Z, and a scalar Y, CCHUD determines a
    unitary matrix U and a scalar ZETA such that
                             (R  Z)     (RR   ZZ )
                        U  * (    )  =  (        ) ,
                             (X  Y)     ( 0  ZETA)
    where RR is upper triangular.  If R and Z have been
    obtained from the factorization of a least squares
    problem, then RR and ZZ are the factors corresponding to
    the problem with the observation (X,Y) appended.  In this
    case, if RHO is the norm of the residual vector, then the
    norm of the residual vector of the updated problem is
    SQRT(RHO**2 + ZETA**2).  CCHUD will simultaneously update
    several triplets (Z,Y,RHO).
    For a less terse description of what CCHUD does and how
    it may be applied see the LINPACK Guide.
    The matrix U is determined as the product U(P)*...*U(1),
    where U(I) is a rotation in the (I,P+1) plane of the
    form
                      (     (CI)      S(I) )
                      (                    ) .
                      ( -CONJG(S(I))  (CI) )
    The rotations are chosen so that C(I) is real.
    On Entry
        R      COMPLEX(LDR,P), where LDR .GE. P.
               R contains the upper triangular matrix
               that is to be updated.  The part of R
               below the diagonal is not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        P      INTEGER.
               P is the order of the matrix R.
        X      COMPLEX(P).
               X contains the row to be added to R.  X is
               not altered by CCHUD.
        Z      COMPLEX(LDZ,NZ), where LDZ .GE. P.
               Z is an array containing NZ P-vectors to
               be updated with R.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of vectors to be updated
               NZ may be zero, in which case Z, Y, and RHO
               are not referenced.
        Y      COMPLEX(NZ).
               Y contains the scalars for updating the vectors
               Z.  Y is not altered bY CCHUD.
        RHO    REAL(NZ).
               RHO contains the norms of the residual
               vectors that are to be updated.  If RHO(J)
               is negative, it is left unaltered.
    On Return
        RC
        RHO    contain the updated quantities.
        Z
        C      REAL(P).
               C contains the cosines of the transforming
               rotations.
        S      COMPLEX(P).
               S contains the sines of the transforming
               rotations.
    LINPACK.  This version dated 08/14/78 .
    Stewart, G. W., University of Maryland, Argonne National Lab.
    CCHUD uses the following functions and subroutines.
    Extended BLAS CROTG
    Fortran CONJG,SQRT


CCMPB


      SUBROUTINE CCMPB(N,IERROR,AN,BN,CN,B,AH,BH)



CCOPY


      SUBROUTINE CCOPY(N,CX,INCX,CY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
      CY  complex vector with N elements
    INCY  storage spacing between elements of CY
    --Output--
      CY  copy of vector CX (unchanged if N .LE. 0)
    Copy complex CX to complex CY.
    For I = 0 to N-1, copy CX(LX+I*INCX) to CY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


CDCDOT


      COMPLEX FUNCTION CDCDOT(N,CB,CX,INCX,CY,INCY)

                 B L A S Subprogram
 Description of Parameters
 --INPUT--
         N  number of elements in input vector(s)
        CB  complex scalar to be added to inner product
        CX  complex vector with N elements
      INCX  storage spacing between elements of CX
        CY  complex vector with N elements
      INCY  storage spacing between elements of CY
 --OUTPUT--
    CDCDOT  complex dot product (zero if N .LE. 0)
  Returns result with dot product accumulated in D.P.
    CDCDOT = CB + sum for I = 0 to N-1 of CX(LX+I*INCY)*CY(LY+I*INCY)
    where LX + 1 if INCX .GE. 0, else LX =(-INCX)*N, AND LY IS
    defined in a similar way using INCY.


CDIV


      SUBROUTINE CDIV(AR,AI,BR,BI,CR,CI)



CDOTC


      COMPLEX FUNCTION CDOTC(N,CX,INCX,CY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
      CY  complex vector with N elements
    INCY  storage spacing between elements of CY
    --Output--
   CDOTC  complex result (zero if N .LE. 0)
    Returns the dot product for complex CX and CY, uses CONJUGATE(CX)
    CDOTC = SUM for I = 0 to N-1 of CONJ(CX(LX+I*INCX))*CY(LY+I*INCY)
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


CDOTU


      COMPLEX FUNCTION CDOTU(N,CX,INCX,CY,INCY)

               B L A S  Subprogram
   Description of parameters
    --Input--
       N  number of elements in input vector(s)
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
      CY  complex vector with N elements
    INCY  storage spacing between elements of CY
    --Output--
   CDOTU  complex result (zero if N .LE. 0)
    Returns the dot product for complex CX and CY, no conjugation
    CDOTU = SUM for I = 0 to N-1 of  CX(LX+I*INCX) * CY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


CG


      SUBROUTINE CG(NM,N,AR,AI,WR,WI,MATZ,ZR,ZI,FV1,FV2,FV3,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    of a COMPLEX GENERAL matrix.
    On INPUT
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrix  A=(AR,AI).
       AR  and  AI  contain the real and imaginary parts,
       respectively, of the complex general matrix.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On OUTPUT
       WR  and  WI  contain the real and imaginary parts,
       respectively, of the eigenvalues.
       ZR  and  ZI  contain the real and imaginary parts,
       respectively, of the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       FV1, FV2, and  FV3  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



CGBCO


      SUBROUTINE CGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)

    CGBCO factors a complex band matrix by Gaussian
    elimination and estimates the condition of the matrix.
    If  RCOND  is not needed, CGBFA is slightly faster.
    To solve  A*X = B , follow CGBCO by CGBSL.
    To compute  INVERSE(A)*C , follow CGBCO by CGBSL.
    To compute  DETERMINANT(A) , follow CGBCO by CGBDI.
    On Entry
       ABD     COMPLEX(LDA, N)
               contains the matrix in band storage.  The columns
               of the matrix are stored in the columns of  ABD  and
               the diagonals of the matrix are stored in rows
               ML+1 through 2*ML+MU+1 of  ABD .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. 2*ML + MU + 1 .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT. N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT. N .
               More efficient if  ML .LE. MU .
    On Return
       ABD     an upper triangular matrix in band storage and
               the multipliers which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  And  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Band Storage
          if  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  M = ML + MU + 1
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-MU)
                     I2 = MIN0(N, J+Ml)
                     DO 10 I = I1, I2
                        K = I - J + M
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses rows  ML+1  through  2*ML+MU+1  of  ABD .
          In addition, the first  ML  rows in  ABD  are used for
          elements generated during the triangularization.
          The total number of rows needed in  ABD  is  2*ML+MU+1 .
          The  ML+MU by ML+MU  upper left triangle and the
          ML by ML  lower right triangle are not referenced.
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABD should contain
           *  *  *  +  +  +  , * = not used
           *  * 13 24 35 46  , + = used for pivoting
           * 12 23 34 45 56
          11 22 33 44 55 66
          21 32 43 54 65  *
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CGBFA
    BLAS CAXPY,CDOTC,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,MAX0,MIN0,REAL


CGBDI


      SUBROUTINE CGBDI(ABD,LDA,N,ML,MU,IPVT,DET)

    CGBDI computes the determinant of a band matrix
    using the factors computed by CGBCO or CGBFA.
    If the inverse is needed, use CGBSL  N  times.
    On Entry
       ABD     COMPLEX(LDA, N)
               the output from CGBCo or CGBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from CGBCO or CGBFA.
    On Return
       DET     COMPLEX(2)
               determinant of original matrix.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. CABS1(DET(1)) .LT. 10.0
               or  DET(1) = 0.0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    Fortran ABS,AIMAG,CMPLX,REAL


CGBFA


      SUBROUTINE CGBFA(ABD,LDA,N,ML,MU,IPVT,INFO)

    CGBFA factors a complex band matrix by elimination.
    CGBFA is usually called by CGBCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    On Entry
       ABD     COMPLEX(LDA, N)
               contains the matrix in band storage.  The columns
               of the matrix are stored in the columns of  ABD  and
               the diagonals of the matrix are stored in rows
               ML+1 through 2*ML+MU+1 of  ABD .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. 2*ML + MU + 1 .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT. N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT. N .
               More efficient if  ML .LE. MU .
    On Return
       ABD     an upper triangular matrix in band storage and
               the multipliers which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if  U(K,K) .EQ. 0.0 .  This is not an error
                    condition for this subroutine, but it does
                    indicate that CGBSL will divide by zero if
                    called.  Use  RCOND  in CGBCO for a reliable
                    indication of singularity.
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  m = ML + MU + 1
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-MU)
                     I2 = MIN0(n, J+ML)
                     DO 10 I = I1, I2
                        K = I - J + M
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses rows  ML+1  through  2*ML+MU+1  of  ABD .
          In addition, the first  ML  rows in  ABD  are used for
          elements generated during the triangularization.
          The total number of rows needed in  ABD  is  2*ML+MU+1 .
          The  ML+MU by ML+MU  upper left triangle and the
          ML by ML  lower right triangle are not referenced.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CSCAL,ICAMAX
    Fortran ABS,AIMAG,MAX0,MIN0,REAL


CGBSL


      SUBROUTINE CGBSL(ABD,LDA,N,ML,MU,IPVT,B,JOB)

    CGBSL solves the complex band system
    A * X = B  or  CTRANS(A) * X = B
    using the factors computed by CGBCO or CGBFA.
    On Entry
       ABD     COMPLEX(LDA, N)
               the output from CGBCo or CGBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from CGBCO or CGBFA.
       B       COMPLEX(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B ,
               = nonzero   to solve  CTRANS(A)*X = B , where
                           CTRANS(A)  is the conjugate transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically this indicates singularity
       but it is often caused by improper arguments or improper
       setting of LDA .  It will not occur if the subroutines are
       called correctly and if CGBCO has set RCOND .GT. 0.0
       or CGBFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)
          IF (RCOND is too small) GO TO ...
          DO 10 J = 1, P
             CALL CGBSL(ABD,LDA,N,ML,MU,IPVT,C(1,J),0)
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTC
    Fortran CONJG,MIN0


CGECO


      SUBROUTINE CGECO(A,LDA,N,IPVT,RCOND,Z)

    CGECO factors a complex matrix by Gaussian elimination
    and estimates the condition of the matrix.
    If  RCOND  is not needed, CGEFA is slightly faster.
    To solve  A*X = B , follow CGECO By CGESL.
    To Compute  INVERSE(A)*C , follow CGECO by CGESL.
    To compute  DETERMINANT(A) , follow CGECO by CGEDI.
    To compute  INVERSE(A) , follow CGECO by CGEDI.
    On Entry
       A       COMPLEX(LDA, N)
               the matrix to be factored.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix and the multipliers
               which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CGEFA
    BLAS CAXPY,CDOTC,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,REAL


CGEDI


      SUBROUTINE CGEDI(A,LDA,N,IPVT,DET,WORK,JOB)

    CGEDI computes the determinant and inverse of a matrix
    using the factors computed by CGECO or CGEFA.
    On Entry
       A       COMPLEX(LDA, N)
               the output from CGECO or CGEFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       IPVT    INTEGER(N)
               the pivot vector from CGECO or CGEFA.
       WORK    COMPLEX(N)
               work vector.  Contents destroyed.
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       A       inverse of original matrix if requested.
               Otherwise unchanged.
       DET     COMPLEX(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. CABS1(DET(1)) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if CGECO has set RCOND .GT. 0.0 or CGEFA has set
       INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CSCAL,CSWAP
    Fortran ABS,AIMAG,CMPLX,MOD,REAL


CGEEV


      SUBROUTINE CGEEV(A,LDA,N,E,V,LDV,WORK,JOB,INFO)

    LICEPACK.    This version dated 08/08/80.
    David Kahner, Cleve Moler, G. W. Stewart
      N.B.S.         U.N.M.      N.B.S./U.MD.
    Abstract
     CGEEV computes the eigenvalues and, optionally,
     the eigenvectors of a general complex matrix.
    Call Sequence Parameters-
      (The values of parameters marked with * (star) will be changed
        by CGEEV.)
       A*      COMPLEX(LDA,N)
               complex nonsymmetric input matrix.
       LDA     INTEGER
               set by the user to
               the leading dimension of the complex array A.
       N       INTEGER
               set by the user to
               the order of the matrices A and V, and
               the number of elements in E.
       E*      COMPLEX(N)
               on return from CGEEV E contains the eigenvalues of A.
               See also INFO below.
       V*      COMPLEX(LDV,N)
               on return from CGEEV if the user has set JOB
               = 0        V is not referenced.
               = nonzero  the N eigenvectors of A are stored in the
               first N columns of V.  See also INFO below.
               (If the input matrix A is nearly degenerate, V
                will be badly conditioned, i.e. have nearly
                dependent columns.)
       LDV     INTEGER
               set by the user to
               the leading dimension of the array V if JOB is also
               set nonzero.  In that case N must be .LE. LDV.
               If JOB is set to zero LDV is not referenced.
       WORK*   REAL(3N)
               temporary storage vector.  Contents changed by CGEEV.
       JOB     INTEGER
               set by the user to
               = 0        eigenvalues only to be calculated by CGEEV.
                          neither V nor LDV are referenced.
               = nonzero  eigenvalues and vectors to be calculated.
                          In this case A & V must be distinct arrays.
                          Also,  if LDA > LDV,  CGEEV changes all the
                          elements of A thru column N.  If LDA < LDV,
                          CGEEV changes all the elements of V through
                          column N.  If LDA = LDV only A(I,J) and V(I,
                          J) for I,J = 1,...,N are changed by CGEEV.
       INFO*   INTEGER
               on return from CGEEV the value of INFO is
               = 0  normal return, calculation successful.
               = K  if the eigenvalue iteration fails to converge,
                    eigenvalues K+1 through N are correct, but
                    no eigenvectors were computed even if they were
                    requested (JOB nonzero).
     Error Messages
          No. 1  recoverable  N is greater than LDA
          No. 2  recoverable  N is less than one.
          No. 3  recoverable  JOB is nonzero and N is greater than LDV
          No. 4  warning      LDA > LDV,  elements of A other than the
                              N by N input elements have been changed
          No. 5  warning      LDA < LDV,  elements of V other than the
                              N by N output elements have been changed
    Subroutines Used
    EISPACK-  CBABK2, CBAL, COMQR, COMQR2, CORTH
    BLAS-  SCOPY
    SLATEC- XERROR


CGEFA


      SUBROUTINE CGEFA(A,LDA,N,IPVT,INFO)

    CGEFA factors a complex matrix by Gaussian elimination.
    CGEFA is usually called by CGECO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (Time for CGECO) = (1 + 9/N)*(Time for CGEFA) .
    On Entry
       A       COMPLEX(LDA, N)
               the matrix to be factored.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix and the multipliers
               which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if  U(K,K) .EQ. 0.0 .  This is not an error
                    condition for this subroutine, but it does
                    indicate that CGESL or CGEDI will divide by zero
                    if called.  Use  RCOND  in CGECO for a reliable
                    indication of singularity.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CSCAL,ICAMAX
    Fortran ABS,AIMAG,REAL


CGEFS


      SUBROUTINE CGEFS(A,LDA,N,V,ITASK,IND,WORK,IWORK)

   Subroutine CGEFS solves A general NxN system of complex
   linear equations using LINPACK subroutines CGECO
   and CGESL.  That is, if A is an NxN complex matrix
   and if X  and B are complex  N-vectors, then CGEFS
   solves the equation
                         A*X=B.
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to find the
   solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeedIng solutions.  In this case, the contents of A,
   LDA, N and IWORK must not have been altered by the user follow-
   ing factorization (ITASK=1).  IND will not be changed by CGEFS
   in this case.
 Argument Description ***
   A      COMPLEX(LDA,N)
            on entry, the doubly subscripted array with dimension
              (LDA,N) which contains the coefficient matrix.
            on return, an upper triangular matrix U and the
              multipliers necessary to construct a matrix L
              so that A=L*U.
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (Terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  The first N elements of
            the array A are the elements of the first column of
            the matrix A.  N must be greater than or equal to 1.
            (Terminal error message IND=-2)
   V      COMPLEX(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            if ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            if ITASK .GT. 1, the equation is solved using the existing
              factored matrix A and IWORK.
            if ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT.0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT.0  see error message corresponding to IND below.
   WORK   COMPLEX(N)
            a singly subscripted array of dimension at least N.
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   The matrix A is computationally singular.
                        A solution has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              NOTE-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


CGEIR


      SUBROUTINE CGEIR(A,LDA,N,V,ITASK,IND,WORK,IWORK)

   Subroutine CGEIR solves a general NxN system of complex
   linear equations using LINPACK subroutines CGEFA and CGESL.
   One pass of iterative refinement is used only to obtain an
   estimate of the accuracy.  That is, if A is an NxN complex
   matrix and if X and B are complex N-vectors, then CGEIR solves
   the equation
                         A*X=B.
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to calculate
   the solution, X.  Then the residual vector is found and
   used to calculate an estimate of the relative error, IND.
   IND estimates the accuracy of the solution only when the
   input matrix and the right hand side are represented
   exactly in the computer and does not take into
   account any errors in the input data.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeedIng solutions.  In this case, the contents of A,
   LDA, N, WORK, and IWORK must not have been altered by the
   user following factorization (ITASK=1).  IND will not be
   changed by CGEIR in this case.
 Argument Description ***
   A      COMPLEX(LDA,N)
            the doubly subscripted array with dimension (LDA,N)
            which contains the coefficient matrix.  A is not
            altered by the routine.
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (Terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  The first N elements of
            the array A are the elements of the first column of
            matrix A.  N must be greater than or equal to 1.
            (Terminal error message IND=-2)
   V      COMPLEX(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            if ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            if ITASK .GT. 1, the equation is solved using the existing
              factored matrix A (stored in work).
            if ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT.0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.  IND=75 means
                    that the solution vector X is zero.
            LT.0  see error message corresponding to IND below.
   WORK   COMPLEX(N*(N+1))
            a singly subscripted array of dimension at least N*(N+1).
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than one.
   IND=-3  terminal   ITASK is less than one.
   IND=-4  terminal   The matrix A is computationally singular.
                        A solution has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              NOTE-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


CGESL


      SUBROUTINE CGESL(A,LDA,N,IPVT,B,JOB)

    CGESL solves the complex system
    A * X = B  or  CTRANS(A) * X = B
    using the factors computed by CGECO or CGEFA.
    On Entry
       A       COMPLEX(LDA, N)
               the output from CGECO or CGEFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       IPVT    INTEGER(N)
               the pivot vector from CGECO or CGEFA.
       B       COMPLEX(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B ,
               = nonzero   to solve  CTRANS(A)*X = B  where
                           CTRANS(A)  is the conjugate transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically this indicates singularity
       but it is often caused by improper arguments or improper
       setting of LDA .  It will not occur if the subroutines are
       called correctly and if CGECO has set RCOND .GT. 0.0
       or CGEFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CGECO(A,LDA,N,IPVT,RCOND,Z)
          IF (RCOND is too small) GO TO ...
          DO 10 J = 1, P
             CALL CGESL(A,LDA,N,IPVT,C(1,J),0)
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTC
    Fortran CONJG


CGTSL


      SUBROUTINE CGTSL(N,C,D,E,B,INFO)

    CGTSL given a general tridiagonal matrix and a right hand
    side will find the solution.
    On Entry
       N       INTEGER
               is the order of the tridiagonal matrix.
       C       COMPLEX(N)
               is the subdiagonal of the tridiagonal matrix.
               C(2) through C(N) should contain the subdiagonal.
               On output C is destroyed.
       D       COMPLEX(N)
               is the diagonal of the tridiagonal matrix.
               On output D is destroyed.
       E       COMPLEX(N)
               is the superdiagonal of the tridiagonal matrix.
               E(1) through E(N-1) should contain the superdiagonal.
               On output E is destroyed.
       B       COMPLEX(N)
               is the right hand side vector.
    On Return
       B       is the solution vector.
       INFO    INTEGER
               = 0 normal value.
               = K if the K-th element of the diagonal becomes
                   exactly zero.  The subroutine returns when
                   this is detected.
    LINPACK.  This version dated 08/14/78 .
    Jack Dongarra, Argonne National Laboratory.
    No Externals
    Fortran ABS,AIMAG,REAL


CH


      SUBROUTINE CH(NM,N,AR,AI,W,MATZ,ZR,ZI,FV1,FV2,FM1,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    of a COMPLEX HERMITIAN matrix.
    On INPUT
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrix  A=(AR,AI).
       AR  and  AI  contain the real and imaginary parts,
       respectively, of the complex hermitian matrix.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On OUTPUT
       W  contains the eigenvalues in ascending order.
       ZR  and  ZI  contain the real and imaginary parts,
       respectively, of the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       FV1, FV2, and  FM1  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



CHICO


      SUBROUTINE CHICO(A,LDA,N,KPVT,RCOND,Z)

    CHICO factors a complex Hermitian matrix by elimination with
    symmetric pivoting and estimates the condition of the matrix.
    If  RCOND  is not needed, CHIFA is slightly faster.
    To solve  A*X = B , follow CHICO by CHISL.
    To compute  INVERSE(A)*C , follow CHICO by CHISL.
    To compute  INVERSE(A) , follow CHICO by CHIDI.
    To compute  DETERMINANT(A) , follow CHICO by CHIDI.
    To compute  INERTIA(A), follow CHICO by CHIDI.
    On Entry
       A       COMPLEX(LDA, N)
               the Hermitian matrix to be factored.
               Only the diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    Output
       A       a block diagonal matrix and the multipliers which
               were used to obtain it.
               The factorization can be written  A = U*D*CTRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , CTRANS(U) is the
               conjugate transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KVPT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CHIFA
    BLAS CAXPY,CDOTC,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,IABS,REAL


CHIDI


      SUBROUTINE CHIDI(A,LDA,N,KPVT,DET,INERT,WORK,JOB)

    CHIDI computes the determinant, inertia and inverse
    of a complex Hermitian matrix using the factors from CHIFA.
    On Entry
       A       COMPLEX(LDA,N)
               the output from CHIFA.
       LDA     INTEGER
               the leading dimension of the array A.
       N       INTEGER
               the order of the matrix A.
       KVPT    INTEGER(N)
               the pivot vector from CHIFA.
       WORK    COMPLEX(N)
               work vector.  Contents destroyed.
       JOB     INTEGER
               JOB has the decimal expansion  ABC  where
                  if  C .NE. 0, the inverse is computed,
                  if  B .NE. 0, the determinant is computed,
                  if  A .NE. 0, the inertia is computed.
               For example, JOB = 111  gives all three.
    On Return
       Variables not requested by JOB are not used.
       A      contains the upper triangle of the inverse of
              the original matrix.  The strict lower triangle
              is never referenced.
       DET    REAL(2)
              determinant of original matrix.
              Determinant = DET(1) * 10.0**DET(2)
              with 1.0 .LE. ABS(DET(1)) .LT. 10.0
              or DET(1) = 0.0.
       INERT  INTEGER(3)
              the inertia of the original matrix.
              INERT(1)  =  number of positive eigenvalues.
              INERT(2)  =  number of negative eigenvalues.
              INERT(3)  =  number of zero eigenvalues.
    Error Condition
       A division by zero may occur if the inverse is requested
       and  CHICO  has set RCOND .EQ. 0.0
       or  CHIFA  has set  INFO .NE. 0 .
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CCOPY,CDOTC,CSWAP
    Fortran ABS,CABS,CMPLX,CONJG,IABS,MOD,REAL


CHIEV


      SUBROUTINE CHIEV(A,LDA,N,E,V,LDV,WORK,JOB,INFO)

    David Kahaner, Cleve Moler, G. W. Stewart,
      N.B.S.         U.N.M.      N.B.S./U.MD.
    Abstract
     CHIEV computes the eigenvalues and, optionally,
     the eigenvectors of a complex Hermitian matrix.
    Call Sequence Parameters-
      (the values of parameters marked with * (star) will be changed
        by CHIEV.)
       A*      COMPLEX(LDA,N)
               complex Hermitian input matrix.
               Only the upper triangle of A need be
               filled in.  Elements on diagonal must be real.
       LDA     INTEGER
               set by the user to
               the leading dimension of the complex array A.
       N       INTEGER
               set by the user to
               the order of the matrices A and V, and
               the number of elements in E.
       E*      REAL(N)
               on return from CHIEV E contains the eigenvalues of A.
               See also INFO below.
       V*      COMPLEX(LDV,N)
               on return from CHIEV if the user has set JOB
               = 0        V is not referenced.
               = nonzero  the N eigenvectors of A are stored in the
               first N columns of V.  See also INFO below.
       LDV     INTEGER
               set by the user to
               the leading dimension of the array V if JOB is also
               set nonzero.  In that case N must be .LE. LDV.
               If JOB is set to zero LDV is not referenced.
       WORK*   REAL(4N)
               temporary storage vector.  Contents changed by CHIEV.
       JOB     INTEGER
               set by the user to
               = 0        eigenvalues only to be calculated by CHIEV.
                          Neither V nor LDV are referenced.
               = nonzero  eigenvalues and vectors to be calculated.
                          In this case A and V must be distinct arrays
                          also if LDA .GT. LDV CHIEV changes all the
                          elements of A thru column N.  If LDA < LDV
                          CHIEV changes all the elements of V through
                          column N.  If LDA = LDV only A(I,J) and V(I,
                          J) for I,J = 1,...,N are changed by CHIEV.
       INFO*   INTEGER
               on return from CHIEV the value of INFO is
               = 0  normal return, calculation successful.
               = K  if the eigenvalue iteration fails to converge,
                    eigenvalues (and eigenvectors if requested)
                    1 through K-1 are correct.
     Error Messages
          No. 1  recoverable  N is greater than LDA
          No. 2  recoverable  N is less than one.
          No. 3  recoverable  JOB is nonzero and N is greater than LDV
          No. 4  warning      LDA > LDV,  elements of A other than the
                              N by N input elements have been changed
          No. 5  warning      LDA < LDV,  elements of V other than the
                              N by N output elements have been changed
          No. 6  recoverable  nonreal element on diagonal of A.
    Subroutines Used
    EISPACK-  HTRIBK, HTRIDI, IMTQL2, TQLRAT
    BLAS-  SCOPY, SCOPYM
    SLATEC- XERROR


CHIFA


      SUBROUTINE CHIFA(A,LDA,N,KPVT,INFO)

    CHIFA factors a complex Hermitian matrix by elimination
    with symmetric pivoting.
    To solve  A*X = B , follow CHIFA by CHISL.
    To compute  INVERSE(A)*C , follow CHIFA by CHISL.
    To compute  DETERMINANT(A) , follow CHIFA by CHIDI.
    To compute  INERTIA(A) , follow CHIFA by CHIDI.
    To compute  INVERSE(A) , follow CHIFA by CHIDI.
    On Entry
       A       COMPLEX(LDA,N)
               the Hermitian matrix to be factored.
               Only the diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       a block diagonal matrix and the multipliers which
               were used to obtain it.
               The factorization can be written  A = U*D*CTRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , CTRANS(U) is the
               conjugate transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KVPT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if the K-th pivot block is singular.  This is
                    not an error condition for this subroutine,
                    but it does indicate that CHISL or CHIDI may
                    divide by zero if called.
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CSWAP,ICAMAX
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,REAL,SQRT


CHISL


      SUBROUTINE CHISL(A,LDA,N,KPVT,B)

    CHISL solves the complex Hermitian system
    A * X = B
    using the factors computed by CHIFA.
    On Entry
       A       COMPLEX(LDA,N)
               the output from CHIFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       KVPT    INTEGER(N)
               the pivot vector from CHIFA.
       B       COMPLEX(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero may occur if  CHICO  has set RCOND .EQ. 0.0
       or  CHIFA  has set INFO .NE. 0  .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CHIFA(A,LDA,N,KVPT,INFO)
          IF (INFO .NE. 0) GO TO ...
          DO 10 J = 1, p
             CALL CHISL(A,LDA,N,KVPT,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTC
    Fortran CONJG,IABS


CHPCO


      SUBROUTINE CHPCO(AP,N,KPVT,RCOND,Z)

    CHPCO factors a complex Hermitian matrix stored in packed
    form by elimination with symmetric pivoting and estimates
    the condition of the matrix.
    if  RCOND  is not needed, CHPFA is slightly faster.
    To solve  A*X = B , follow CHPCO by CHPSL.
    To compute  INVERSE(A)*C , follow CHPCO by CHPSL.
    To compute  INVERSE(A) , follow CHPCO by CHPDI.
    To compute  DETERMINANT(A) , follow CHPCO by CHPDI.
    To compute  INERTIA(A), follow CHPCO by CHPDI.
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the packed form of a Hermitian matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    Output
       AP      a block diagonal matrix and the multipliers which
               were used to obtain it stored in packed form.
               The factorization can be written  A = U*D*CTRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , CTRANS(U) is the
               conjugate transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KVPT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Packed Storage
         The following program segment will pack the upper
         triangle of a Hermitian matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CHPFA
    BLAS CAXPY,CDOTC,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,IABS,REAL


CHPDI


      SUBROUTINE CHPDI(AP,N,KPVT,DET,INERT,WORK,JOB)

    CHPDI computes the determinant, inertia and inverse
    of a complex Hermitian matrix using the factors from CHPFA,
    where the matrix is stored in packed form.
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the output from CHPFA.
       N       INTEGER
               the order of the matrix A.
       KVPT    INTEGER(N)
               the pivot vector from CHPFA.
       WORK    COMPLEX(N)
               work vector.  Contents ignored.
       JOB     INTEGER
               JOB has the decimal expansion  ABC  where
                  if  C .NE. 0, the inverse is computed,
                  if  B .NE. 0, the determinant is computed,
                  if  A .NE. 0, the inertia is computed.
               For example, JOB = 111  gives all three.
    On Return
       Variables not requested by JOB are not used.
       AP     contains the upper triangle of the inverse of
              the original matrix, stored in packed form.
              The columns of the upper triangle are stored
              sequentially in a one-dimensional array.
       DET    REAL(2)
              determinant of original matrix.
              Determinant = DET(1) * 10.0**DET(2)
              with 1.0 .LE. ABS(DET(1)) .LT. 10.0
              or DET(1) = 0.0.
       INERT  INTEGER(3)
              the inertia of the original matrix.
              INERT(1)  =  number of positive eigenvalues.
              INERT(2)  =  number of negative eigenvalues.
              INERT(3)  =  number of zero eigenvalues.
    Error Condition
       A division by zero will occur if the inverse is requested
       and  CHPCO  has set RCOND .EQ. 0.0
       or  CHPFA  has set  INFO .NE. 0 .
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CCOPY,CDOTC,CSWAP
    Fortran ABS,CABS,CMPLX,CONJG,IABS,MOD,REAL


CHPFA


      SUBROUTINE CHPFA(AP,N,KPVT,INFO)

    CHPFA factors a complex Hermitian matrix stored in
    packed form by elimination with symmetric pivoting.
    To solve  A*X = B , follow CHPFA by CHPSL.
    To compute  INVERSE(A)*C , follow CHPFA by CHPSL.
    To compute  DETERMINANT(A) , follow CHPFA by CHPDI.
    To compute  INERTIA(A) , follow CHPFA by CHPDI.
    To compute  INVERSE(A) , follow CHPFA by CHPDI.
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the packed form of a Hermitian matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    Output
       AP      A block diagonal matrix and the multipliers which
               were used to obtain it stored in packed form.
               The factorization can be written  A = U*D*CTRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , CTRANS(U) is the
               conjugate transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KVPT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if the K-th pivot block is singular.  This is
                    not an error condition for this subroutine,
                    but it does indicate that CHPSL or CHPDI may
                    divide by zero if called.
    Packed Storage
         The following program segment will pack the upper
         triangle of a Hermitian matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K)  = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CSWAP,ICAMAX
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,REAL,SQRT


CHPSL


      SUBROUTINE CHPSL(AP,N,KPVT,B)

    CHISL solves the complex Hermitian system
    A * X = B
    using the factors computed by CHPFA.
    On Entry
       AP      COMPLEX(N*(N+1)/2)
               the output from CHPFA.
       N       INTEGER
               the order of the matrix  A .
       KVPT    INTEGER(N)
               the pivot vector from CHPFA.
       B       COMPLEX(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero may occur if  CHPCO  has set RCOND .EQ. 0.0
       or  CHPFA  has set INFO .NE. 0  .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CHPFA(AP,N,KVPT,INFO)
          IF (INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL CHPSL(AP,N,KVPT,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTC
    Fortran CONJG,IABS


CINVIT


      SUBROUTINE CINVIT(NM,N,AR,AI,WR,WI,SELECT,MM,M,ZR,ZI,IERR,RM1,

    This subroutine is a translation of the ALGOL procedure CXINVIT
    by Peters and Wilkinson.
    HANDBOOK FOR AUTO. COMP. VOL.II-LINEAR ALGEBRA, 418-439(1971).
    This subroutine finds those eigenvectors of A COMPLEX UPPER
    Hessenberg matrix corresponding to specified eigenvalues,
    using inverse iteration.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       AR and AI contain the real and imaginary parts,
         respectively, of the Hessenberg matrix.
       WR and WI contain the real and imaginary parts, respectively,
         of the eigenvalues of the matrix.  The eigenvalues must be
         stored in a manner identical to that of subroutine  COMLR,
         which recognizes possible splitting of the matrix.
       SELECT specifies the eigenvectors to be found.  The
         eigenvector corresponding to the J-th eigenvalue is
         specified by setting SELECT(J) to .TRUE.
       MM should be set to an upper bound for the number of
         eigenvectors to be found.
    On OUTPUT
       AR, AI, WI, and SELECT are unaltered.
       WR may have been altered since close eigenvalues are perturbed
         slightly in searching for independent eigenvectors.
       M is the number of eigenvectors actually found.
       ZR and ZI contain the real and imaginary parts, respectively,
         of the eigenvectors.  The eigenvectors are normalized
         so that the component of largest magnitude is 1.
         Any vector which fails the acceptance test is set to zero.
       IERR is set to
         Zero       for normal return,
         -(2*N+1)   if more than MM eigenvectors have been specified,
         -K         if the iteration corresponding to the K-th
                    value fails,
         -(N+K)     if both error situations occur.
       RM1, RM2, RV1, and RV2 are temporary storage arrays.
    The ALGOL procedure GUESSVEC appears in CINVIT in line.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Calls CDIV for complex division.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



CNBCO


      SUBROUTINE CNBCO(ABE,LDA,N,ML,MU,IPVT,RCOND,Z)

    CNBCO factors a complex band matrix by Gaussian
    elimination and estimates the condition of the matrix.
    If RCOND is not needed, CNBFA is slightly faster.
    To solve  A*X = B , follow CNBCO by CNBSL.
    To compute  INVERSE(A)*C , follow CNBCO by CNBSL.
    To compute  DETERMINANT(A) , follow CNBCO by CNBDI.
    On Entry
       ABE     COMPLEX(LDA, NC)
               contains the matrix in band storage.  The rows
               of the original matrix are stored in the rows
               of ABE and the diagonals of the original matrix
               are stored in columns 1 through ML+MU+1 of ABE.
               NC must be .GE. 2*ML+MU+1 .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array ABE.
               LDA must be .GE. N .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT. N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT. N .
               More efficient if ML .LE. MU .
    On Return
       ABE     an upper triangular matrix in band storage
               and the multipliers which were used to obtain it.
               The factorization can be written  A = L*U  where
               L is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                        1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  DO 20 I = 1, N
                     J1 = MAX0(1, I-ML)
                     J2 = MIN0(N, I+MU)
                     DO 10 J = J1, J2
                        K = J - I + ML + 1
                        ABE(I,K) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses columns  1  through  ML+MU+1  of ABE .
          Furthermore,  ML  additional columns are needed in
          ABE  starting with column  ML+MU+2  for elements
          generated during the triangularization.  The total
          number of columns needed in  ABE  is  2*ML+MU+1 .
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
           * 11 12 13  +     , * = not used
          21 22 23 24  +     , + = used for pivoting
          32 33 34 35  +
          43 44 45 46  +
          54 55 56  *  +
          65 66  *  *  +
    SLATEC.  This version dated 07/30/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
    SLATEC CNBFA
     CDOTC,CAXPY,CSSCAL,SCASUM
    Fortran  ABS,AIMAG,AMAX1,CMPLX,CONJG,MAX0,MIN0,SIGN


CNBDI


      SUBROUTINE CNBDI(ABE,LDA,N,ML,MU,IPVT,DET)

    CNBDI computes the determinant of a band matrix
    using the factors computed by CNBCO or CNBFA.
    If the inverse is needed, use CNBSL  N  times.
    On Entry
       ABE     COMPLEX(LDA, NC)
               the output from CNBCO or CNBFA.
               NC must be .GE. 2*ML+MU+1 .
       LDA     INTEGER
               the leading dimension of the array  ABE .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from CNBCO or CNBFA.
    On Return
       DET     COMPLEX(2)
               determinant of original matrix.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. CABS1(DET(1)) .LT. 10.0
               or  DET(1) = 0.0 .
    SLATEC.  This version dated 07/30/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
    Fortran ABS,AIMAG,CMPLX,REAL


CNBFA


      SUBROUTINE CNBFA(ABE,LDA,N,ML,MU,IPVT,INFO)

   CNBFA factors a complex band matrix by elimination.
    CNBFA is usually called by CNBCO, but it can be called
    directly with a saving in time if RCOND is not needed.
    On Entry
       ABE     COMPLEX(LDA, NC)
               contains the matrix in band storage.  The rows
               of the original matrix are stored in the rows
               of ABE and the diagonals of the original matrix
               are stored in columns 1 through ML+MU+1 of ABE.
               NC must be .GE. 2*ML+MU+1 .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array ABE.
               LDA must be .GE. N .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT. N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT. N .
               More efficient if ML .LE. MU .
    On Return
       ABE     an upper triangular matrix in band storage
               and the multipliers which were used to obtain it.
               the factorization can be written  A = L*U  where
               L is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               =0  normal value
               =K  if  U(K,K) .EQ. 0.0 .  This is not an error
               condition for this subroutine, but it does
               indicate that CNBSL will divide by zero if
               called.  Use RCOND in CNBCO for a reliable
               indication of singularity.
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  DO 20 I = 1, N
                     J1 = MAX0(1, I-ML)
                     J2 = MIN0(N, I+MU)
                     DO 10 J = J1, J2
                        K = J - I + ML + 1
                        ABE(I,K) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses columns  1  through  ML+MU+1  of ABE .
          Furthermore,  ML  additional columns are needed in
          ABE  starting with column  ML+MU+2  for elements
          generated during the triangularization.  The total
          number of columns needed in  ABE  is  2*ML+MU+1 .
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
           * 11 12 13  +     , * = not used
          21 22 23 24  +     , + = used for pivoting
          32 33 34 35  +
          43 44 45 46  +
          54 55 56  *  +
          65 66  *  *  +
    SLATEC.  This version dated 07/30/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
    BLAS ICAMAX,CAXPY,CSCAL,CSWAP
    Fortran  ABS,AIMAG,MIN0,REAL


CNBFS


      SUBROUTINE CNBFS(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK)

   Subroutine CNBFS solves a general nonsymmetric banded NxN
   system of single precision complex linear equations using
   SLATEC subroutines CNBCO and CNBSL.  These are adaptations
   of the LINPACK subroutines CGBCO and CGBSL which require
   a different format for storing the matrix elements.  If
   A  is an NxN complex matrix and if  X  and  B  are complex
   N-vectors, then CNBFS solves the equation
                         A*X=B.
   A band matrix is a matrix whose nonzero elements are all
   fairly near the main diagonal, specifically  A(I,J) = 0
   if  I-J is greater than  ML  or  J-I  is greater than
   MU .  The integers ML and MU are called the lower and upper
   band widths and  M = ML+MU+1  is the total band width.
   CNBFS uses less time and storage than the corresponding
   program for general matrices (CGEFS) if 2*ML+MU .LT. N .
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to find the
   solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, N and IWORK must not have been altered by the user follow-
   ing factorization (ITASK=1).  IND will not be changed by CNBFS
   in this case.
   Band Storage
         If  A  is a band matrix, the following program segment
         will set up the input.
                 ML = (band width below the diagonal)
                 MU = (band width above the diagonal)
                 DO 20 I = 1, N
                    J1 = MAX0(1, I-ML)
                    J2 = MIN0(N, I+MU)
                    DO 10 J = J1, J2
                       K = J - I + ML + 1
                       ABE(I,K) = A(I,J)
              10    CONTINUE
              20 CONTINUE
         This uses columns  1  through  ML+MU+1  of ABE .
         Furthermore,  ML  additional columns are needed in
         ABE  starting with column  ML+MU+2  for elements
         generated during the triangularization.  The total
         number of columns needed in  ABE  is  2*ML+MU+1 .
   Example:  If the original matrix is
         11 12 13  0  0  0
         21 22 23 24  0  0
          0 32 33 34 35  0
          0  0 43 44 45 46
          0  0  0 54 55 56
          0  0  0  0 65 66
    then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
          * 11 12 13  +     , * = not used
         21 22 23 24  +     , + = used for pivoting
         32 33 34 35  +
         43 44 45 46  +
         54 55 56  *  +
         65 66  *  *  +
 Argument Description ***
   ABE    COMPLEX(LDA,NC)
            on entry, contains the matrix in band storage as
              described above.  NC  must not be less than
              2*ML+MU+1 .  The user is cautioned to specify  NC
              with care since it is not an argument and cannot
              be checked by CNBFS.  The rows of the original
              matrix are stored in the rows of  ABE  and the
              diagonals of the original matrix are stored in
              columns  1  through  ML+MU+1  of  ABE .
            on return, contains an upper triangular matrix U and
              the multipliers necessary to construct a matrix L
              so that A=L*U.
   LDA    INTEGER
            the leading dimension of array ABE.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater
            than or equal to 1 .  (terminal error message IND=-2)
   ML     INTEGER
            the number of diagonals below the main diagonal.
            ML  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-5)
   MU     INTEGER
            the number of diagonals above the main diagonal.
            MU  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-6)
   V      COMPLEX(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            if ITASK = 1, the matrix A is factored and then the
              linear equation is solved.
            if ITASK .GT. 1, the equation is solved using the existing
              factored matrix A and IWORK.
            if ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT. 0  see error message corresponding to IND below.
   WORK   COMPLEX(N)
            a singly subscripted array of dimension at least N.
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   The matrix A is computationally singular.
                        A solution has not been computed.
   IND=-5  terminal   ML is less than zero or is greater than
                        or equal to N .
   IND=-6  terminal   MU is less than zero or is greater than
                        or equal to N .
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              NOTE-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


CNBIR


      SUBROUTINE CNBIR(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK)

   Subroutine CNBIR solves a general nonsymmetric banded NxN
   system of single precision complex linear equations using
   SLATEC subroutines CNBFA and CNBSL.  These are adaptations
   of the LINPACK subroutines CGBFA and CGBSL which require
   a different format for storing the matrix elements.
   One pass of iterative refinement is used only to obtain an
   estimate of the accuracy.  If  A  is an NxN complex banded
   matrix and if  X  and  B  are complex N-vectors, then CNBIR
   solves the equation
                         A*X=B.
   A band matrix is a matrix whose nonzero elements are all
   fairly near the main diagonal, specifically  A(I,J) = 0
   if  I-J is greater than  ML  or  J-I  is greater than
   MU .  The integers ML and MU are called the lower and upper
   band widths and  M = ML+MU+1  is the total band width.
   CNBIR uses less time and storage than the corresponding
   program for general matrices (CGEIR) if 2*ML+MU .LT. N .
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to find the
   solution vector X .  Then the residual vector is found and used
   to calculate an estimate of the relative error, IND .  IND esti-
   mates the accuracy of the solution only when the input matrix
   and the right hand side are represented exactly in the computer
   and does not take into account any errors in the input data.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A, LDA,
   N, WORK and IWORK must not have been altered by the user follow-
   ing factorization (ITASK=1).  IND will not be changed by CNBIR
   in this case.
   Band Storage
         If  A  is a band matrix, the following program segment
         will set up the input.
                 ML = (band width below the diagonal)
                 MU = (band width above the diagonal)
                 DO 20 I = 1, N
                    J1 = MAX0(1, I-ML)
                    J2 = MIN0(N, I+MU)
                    DO 10 J = J1, J2
                       K = J - I + ML + 1
                       ABE(I,K) = A(I,J)
              10    CONTINUE
              20 CONTINUE
         This uses columns  1  through  ML+MU+1  of ABE .
   Example:  If the original matrix is
         11 12 13  0  0  0
         21 22 23 24  0  0
          0 32 33 34 35  0
          0  0 43 44 45 46
          0  0  0 54 55 56
          0  0  0  0 65 66
    then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
          * 11 12 13        , * = not used
         21 22 23 24
         32 33 34 35
         43 44 45 46
         54 55 56  *
         65 66  *  *
 Argument Description ***
   ABE    COMPLEX(LDA,MM)
            on entry, contains the matrix in band storage as
              described above.  MM  must not be less than  M =
              ML+MU+1 .  The user is cautioned to dimension  ABE
              with care since MM is not an argument and cannot
              be checked by CNBIR.  The rows of the original
              matrix are stored in the rows of  ABE  and the
              diagonals of the original matrix are stored in
              columns  1  through  ML+MU+1  of  ABE .  ABE  is
              not altered by the program.
   LDA    INTEGER
            the leading dimension of array ABE.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater
            than or equal to 1 .  (terminal error message IND=-2)
   ML     INTEGER
            the number of diagonals below the main diagonal.
            ML  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-5)
   MU     INTEGER
            the number of diagonals above the main diagonal.
            MU  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-6)
   V      COMPLEX(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            if ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            if ITASK .GT. 1, the equation is solved using the existing
              factored matrix A and IWORK.
            if ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X .  IND=75 means
                    that the solution vector  X  is zero.
            LT. 0  see error message corresponding to IND below.
   WORK   COMPLEX(N*(NC+1))
            a singly subscripted array of dimension at least
            N*(NC+1)  where  NC = 2*ML+MU+1 .
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   The matrix A is computationally singular.
                        A solution has not been computed.
   IND=-5  terminal   ML is less than zero or is greater than
                        or equal to N .
   IND=-6  terminal   MU is less than zero or is greater than
                        or equal to N .
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              NOTE-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


CNBSL


      SUBROUTINE CNBSL(ABE,LDA,N,ML,MU,IPVT,B,JOB)

    CNBSL solves the complex band system
    A * X = B  or  CTRANS(A) * X = B
    using the factors computed by CNBCO or CNBFA.
    On Entry
       ABE     COMPLEX(LDA, NC)
               the output from CNBCO or CNBFA.
               NC must be .GE. 2*ML+MU+1 .
       LDA     INTEGER
               the leading dimension of the array  ABE .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from CNBCO or CNBFA.
       B       COMPLEX(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B .
               = nonzero   to solve  CTRANS(A)*X = B , where
                           CTRANS(A)  is the conjugate transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically this indicates singularity
       but it is often caused by improper arguments or improper
       setting of LDA.  It will not occur if the subroutines are
       called correctly and if CNBCO has set RCOND .GT. 0.0
       or CNBFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CNBCO(ABE,LDA,N,ML,MU,IPVT,RCOND,Z)
          IF (RCOND is too small) GO TO ...
          DO 10 J = 1, P
            CALL CNBSL(ABE,LDA,N,ML,MU,IPVT,C(1,J),0)
       10 CONTINUE
    SLATEC.  This version dated 07/30/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
    BLAS CAXPY,CDOTC
    Fortran  CONJG,MIN0


COMBAK


      SUBROUTINE COMBAK(NM,LOW,IGH,AR,AI,INT,M,ZR,ZI)

    This subroutine is a translation of the ALGOL procedure COMBAK,
    NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    This subroutine forms the eigenvectors of a COMPLEX GENERAL
    matrix by back transforming those of the corresponding
    upper Hessenberg matrix determined by  COMHES.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       LOW and IGH are integers determined by the balancing
         subroutine  CBAL.  If  CBAL  has not been used,
         set LOW=1 and IGH equal to the order of the matrix.
       AR and AI contain the multipliers which were used in the
         reduction by  COMHES  in their lower triangles
         below the subdiagonal.
       INT contains information on the rows and columns
         interchanged in the reduction by  COMHES.
         Only elements LOW through IGH are used.
       M is the number of eigenvectors to be back transformed.
       ZR and ZI contain the real and imaginary parts,
         respectively, of the eigenvectors to be
         back transformed in their first M columns.
    On OUTPUT
       ZR and ZI contain the real and imaginary parts,
         respectively, of the transformed eigenvectors
         in their first M columns.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



COMHES


      SUBROUTINE COMHES(NM,N,LOW,IGH,AR,AI,INT)

    This subroutine is a translation of the ALGOL procedure COMHES,
    NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    Given a COMPLEX GENERAL matrix, this subroutine
    reduces a submatrix situated in rows and columns
    LOW through IGH to upper Hessenberg form by
    stabilized elementary similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH ARE integers determined by the balancing
         subroutine  CBAL.  If  CBAL  has not been used,
         set LOW=1, IGH=N.
       AR and AI contain the real and imaginary parts,
         respectively, of the complex input matrix.
    On OUTPUT
       AR and AI contain the real and imaginary parts,
         respectively, of the Hessenberg matrix.  The
         multipliers which were used in the reduction
         are stored in the remaining triangles under the
         Hessenberg matrix.
       INT contains information on the rows and columns
         interchanged in the reduction.
         Only elements LOW through IGH are used.
    Calls CDIV for complex division.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



COMLR


      SUBROUTINE COMLR(NM,N,LOW,IGH,HR,HI,WR,WI,IERR)

    This subroutine is a translation of the ALGOL procedure COMLR,
    NUM. MATH. 12, 369-376(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 396-403(1971).
    This subroutine finds the eigenvalues of a COMPLEX
    UPPER Hessenberg matrix by the modified LR method.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  CBAL.  If  CBAL  has not been used,
         set LOW=1, IGH=N.
       HR and HI contain the real and imaginary parts,
         respectively, of the complex upper Hessenberg matrix.
         Their lower triangles below the subdiagonal contain the
         multipliers which were used in the reduction by  COMHES,
         if performed.
    On OUTPUT
       The upper Hessenberg portions of HR and HI have been
         destroyed.  Therefore, they must be saved before
         calling  COMLR  if subsequent calculation of
         eigenvectors is to be performed.
       WR and WI contain the real and imaginary parts,
         respectively, of the eigenvalues.  If an error
         exit is made, the eigenvalues should be correct
         for indices IERR+1,...,N.
       IERR is set to
         Zero       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after a total of 30*N iterations.
    Calls CSROOT for complex square root.
    Calls CDIV for complex division.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



COMLR2


      SUBROUTINE COMLR2(NM,N,LOW,IGH,INT,HR,HI,WR,WI,ZR,ZI,IERR)

    This subroutine is a translation of the ALGOL procedure COMLR2,
    NUM. MATH. 16, 181-204(1970) by Peters and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
    This subroutine finds the eigenvalues and eigenvectors
    of a COMPLEX UPPER Hessenberg matrix by the modified LR
    method.  The eigenvectors of a COMPLEX GENERAL matrix
    can also be found if  COMHES  has been used to reduce
    this general matrix to Hessenberg form.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  CBAL.  If  CBAL  has not been used,
         set LOW=1, IGH=N.
       INT contains information on the rows and columns interchanged
         in the reduction by  COMHES, if performed.  Only elements
         LOW through IGH are used.  If the eigenvectors of the HESSEN-
         BERG matrix are desired, set INT(J)=J for these elements.
       HR and HI contain the real and imaginary parts,
         respectively, of the complex upper Hessenberg matrix.
         Their lower triangles below the subdiagonal contain the
         multipliers which were used in the reduction by  COMHES,
         if performed.  If the eigenvectors of the Hessenberg
         matrix are desired, these elements must be set to zero.
    On OUTPUT
       The upper Hessenberg portions of HR and HI have been
         destroyed, but the location HR(1,1) contains the norm
         of the triangularized matrix.
       WR and WI contain the real and imaginary parts,
         respectively, of the eigenvalues.  If an error
         exit is made, the eigenvalues should be correct
         for indices IERR+1,...,N.
       ZR and ZI contain the real and imaginary parts,
         respectively, of the eigenvectors.  The eigenvectors
         are unnormalized.  If an error exit is made, none of
         the eigenvectors has been found.
       IERR is set to
         Zero       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after a total of 30*N iterations.
    Calls CSROOT for complex square root.
    Calls CDIV for complex division.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



COMPB


      SUBROUTINE COMPB(N,IERROR,AN,BN,CN,B,AH,BH)



COMQR


      SUBROUTINE COMQR(NM,N,LOW,IGH,HR,HI,WR,WI,IERR)

    This subroutine is a translation of a unitary analogue of the
    ALGOL procedure  COMLR, NUM. MATH. 12, 369-376(1968) by Martin
    and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 396-403(1971).
    The unitary analogue substitutes the QR algorithm of Francis
    (COMP. JOUR. 4, 332-345(1962)) for the LR algorithm.
    This subroutine finds the eigenvalues of a COMPLEX
    upper Hessenberg matrix by the QR method.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  CBAL.  If  CBAL  has not been used,
         set LOW=1, IGH=N.
       HR and HI contain the real and imaginary parts,
         respectively, of the complex upper Hessenberg matrix.
         Their lower triangles below the subdiagonal contain
         information about the unitary transformations used in
         the reduction by  CORTH, if performed.
    On OUTPUT
       The upper Hessenberg portions of HR and HI have been
         destroyed.  Therefore, they must be saved before
         calling  COMQR  if subsequent calculation of
         eigenvectors is to be performed.
       WR and WI contain the real and imaginary parts,
         respectively, of the eigenvalues.  If an error
         exit is made, the eigenvalues should be correct
         for indices IERR+1,...,N.
       IERR is set to
         ZERO       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after a total of 30*N iterations.
    Calls CSROOT for complex square root.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Calls CDIV for complex division.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



COMQR2


      SUBROUTINE COMQR2(NM,N,LOW,IGH,ORTR,ORTI,HR,HI,WR,WI,ZR,ZI,IERR)

    This subroutine is a translation of a unitary analogue of the
    ALGOL procedure  COMLR2, NUM. MATH. 16, 181-204(1970) by Peters
    and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
    The unitary analogue substitutes the QR algorithm of Francis
    (COMP. JOUR. 4, 332-345(1962)) for the LR algorithm.
    This subroutine finds the eigenvalues and eigenvectors
    of a COMPLEX UPPER Hessenberg matrix by the QR
    method.  The eigenvectors of a COMPLEX GENERAL matrix
    can also be found if  CORTH  has been used to reduce
    this general matrix to Hessenberg form.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  CBAL.  If  CBAL  has not been used,
         set LOW=1, IGH=N.
       ORTR and ORTI contain information about the unitary trans-
         formations used in the reduction by  CORTH, if performed.
         Only elements LOW through IGH are used.  If the eigenvectors
         of the Hessenberg matrix are desired, set ORTR(J) and
         ORTI(J) to 0.0E0 for these elements.
       HR and HI contain the real and imaginary parts,
         respectively, of the complex upper Hessenberg matrix.
         Their lower triangles below the subdiagonal contain further
         information about the transformations which were used in the
         reduction by  CORTH, if performed.  If the eigenvectors of
         the Hessenberg matrix are desired, these elements may be
         arbitrary.
    On OUTPUT
       ORTR, ORTI, and the upper Hessenberg portions of HR and HI
         have been destroyed.
       WR and WI contain the real and imaginary parts,
         respectively, of the eigenvalues.  If an error
         exit is made, the eigenvalues should be correct
         for indices IERR+1,...,N.
       ZR and ZI contain the real and imaginary parts,
         respectively, of the eigenvectors.  The eigenvectors
         are unnormalized.  If an error exit is made, none of
         the eigenvectors has been found.
       IERR is set to
         Zero       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after a total of 30*N iterations.
    Calls CSROOT for complex square root.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Calls CDIV for complex division.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



CORTB


      SUBROUTINE CORTB(NM,LOW,IGH,AR,AI,ORTR,ORTI,M,ZR,ZI)

    This subroutine is a translation of a complex analogue of
    the ALGOL procedure ORTBAK, NUM. MATH. 12, 349-368(1968)
    by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    This subroutine forms the eigenvectors of a COMPLEX GENERAL
    matrix by back transforming those of the corresponding
    upper Hessenberg matrix determined by  CORTH.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       LOW and IGH are integers determined by the balancing
         subroutine  CBAL.  If  CBAL  has not been used,
         set LOW=1 and IGH equal to the order of the matrix.
       AR and AI contain information about the unitary
         transformations used in the reduction by  CORTH
         in their strict lower triangles.
       ORTR and ORTI contain further information about the
         transformations used in the reduction by  CORTH.
         Only elements LOW through IGH are used.
       M is the number of columns of ZR and ZI to be back transformed.
       ZR and ZI contain the real and imaginary parts,
         respectively, of the eigenvectors to be
         back transformed in their first M columns.
    On OUTPUT
       ZR and ZI contain the real and imaginary parts,
         respectively, of the transformed eigenvectors
         in their first M columns.
       ORTR and ORTI have been altered.
    Note that CORTB preserves vector Euclidean norms.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



CORTH


      SUBROUTINE CORTH(NM,N,LOW,IGH,AR,AI,ORTR,ORTI)

    This subroutine is a translation of a complex analogue of
    the ALGOL procedure ORTHES, NUM. MATH. 12, 349-368(1968)
    by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    Given a COMPLEX GENERAL matrix, this subroutine
    reduces a submatrix situated in rows and columns
    LOW through IGH to upper Hessenberg form by
    unitary similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  CBAL.  If  CBAL  has not been used,
         set LOW=1, IGH=N.
       AR and AI contain the real and imaginary parts,
         respectively, of the complex input matrix.
    On OUTPUT
       AR and AI contain the real and imaginary parts,
         respectively, of the Hessenberg matrix.  Information
         about the unitary transformations used in the reduction
         is stored in the remaining triangles under the
         Hessenberg matrix.
       ORTR and ORTI contain further information about the
         transformations.  Only elements LOW through IGH are used.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



CPBCO


      SUBROUTINE CPBCO(ABD,LDA,N,M,RCOND,Z,INFO)

    CPBCO factors a complex Hermitian positive definite matrix
    stored in band form and estimates the condition of the matrix.
    If  RCOND  is not needed, CPBFA is slightly faster.
    To solve  A*X = B , follow CPBCO by CPBSL.
    To compute  INVERSE(A)*C , follow CPBCO by CPBSL.
    To compute  DETERMINANT(A) , follow CPBCO by CPBDI.
    On Entry
       ABD     COMPLEX(LDA, N)
               the matrix to be factored.  The columns of the upper
               triangle are stored in the columns of ABD and the
               diagonals of the upper triangle are stored in the
               rows of ABD .  See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. M + 1 .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
               0 .LE. M .LT. N .
    On Return
       ABD     an upper triangular matrix  R , stored in band
               form, so that  A = CTRANS(R)*R .
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is singular to working precision, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    Band Storage
          If  A  is a Hermitian positive definite band matrix,
          the following program segment will set up the input.
                  M = (band width above diagonal)
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-M)
                     DO 10 I = I1, J
                        K = I-J+M+1
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses  M + 1  rows of  A , except for the  M by M
          upper left triangle, which is ignored.
    Example:  If the original matrix is
          11 12 13  0  0  0
          12 22 23 24  0  0
          13 23 33 34 35  0
           0 24 34 44 45 46
           0  0 35 45 55 56
           0  0  0 46 56 66
    then  N = 6 , M = 2  and  ABD  should contain
           *  * 13 24 35 46
           * 12 23 34 45 56
          11 22 33 44 55 66
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CPBFA
    BLAS CAXPY,CDOTC,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,MAX0,MIN0,REAL


CPBDI


      SUBROUTINE CPBDI(ABD,LDA,N,M,DET)

    CPBDI computes the determinant
    of a complex Hermitian positive definite band matrix
    using the factors computed by CPBCO or CPBFA.
    If the inverse is needed, use CPBSL  N  times.
    On Entry
       ABD     COMPLEX(LDA, N)
               the output from CPBCO or CPBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
    On Return
       DET     REAL(2)
               determinant of original matrix in the form
               determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    Fortran REAL


CPBFA


      SUBROUTINE CPBFA(ABD,LDA,N,M,INFO)

    CPBFA factors a complex Hermitian positive definite matrix
    stored in band form.
    CPBFA is usually called by CPBCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    On Entry
       ABD     COMPLEX(LDA, N)
               the matrix to be factored.  The columns of the upper
               triangle are stored in the columns of ABD and the
               diagonals of the upper triangle are stored in the
               rows of ABD .  See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. M + 1 .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
               0 .LE. M .LT. N .
    On Return
       ABD     an upper triangular matrix  R , stored in band
               form, so that  A = CTRANS(R)*R .
       INFO    INTEGER
               = 0  for normal return.
               = K  if the leading minor of order  K  is not
                    positive definite.
    Band Storage
          If  A  is a Hermitian positive definite band matrix,
          the following program segment will set up the input.
                  M = (band width above diagonal)
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-M)
                     DO 10 I = I1, J
                        K = I-J+M+1
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CDOTC
    Fortran AIMAG,CMPLX,CONJG,MAX0,REAL,SQRT


CPBSL


      SUBROUTINE CPBSL(ABD,LDA,N,M,B)

    CPBSL solves the complex Hermitian positive definite band
    system  A*X = B
    using the factors computed by CPBCO or CPBFA.
    On Entry
       ABD     COMPLEX(LDA, N)
               the output from CPBCO or CPBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
       B       COMPLEX(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically this indicates
       singularity but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CPBCO(ABD,LDA,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL CPBSL(ABD,LDA,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTC
    Fortran MIN0


CPEVL


      SUBROUTINE CPEVL(N,M,A,Z,C,B,KBD)



CPEVLR


      SUBROUTINE CPEVLR(N,M,A,X,C)



CPOCO


      SUBROUTINE CPOCO(A,LDA,N,RCOND,Z,INFO)

    CPOCO factors a complex Hermitian positive definite matrix
    and estimates the condition of the matrix.
    If  RCOND  is not needed, CPOFA is slightly faster.
    To solve  A*X = B , follow CPOCO by CPOSL.
    To compute  INVERSE(A)*C , follow CPOCO by CPOSL.
    To compute  DETERMINANT(A) , follow CPOCO by CPODI.
    To compute  INVERSE(A) , follow CPOCO by CPODI.
    On Entry
       A       COMPLEX(LDA, N)
               the Hermitian matrix to be factored.  Only the
               diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix  R  so that  A =
               CTRANS(R)*R where  CTRANS(R)  is the conjugate
               transpose.  The strict lower triangle is unaltered.
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CPOFA
    BLAS CAXPY,CDOTC,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,REAL


CPODI


      SUBROUTINE CPODI(A,LDA,N,DET,JOB)

    CPODI computes the determinant and inverse of a certain
    complex Hermitian positive definite matrix (see below)
    using the factors computed by CPOCO, CPOFA or CQRDC.
    On Entry
       A       COMPLEX(LDA, N)
               the output  A  from CPOCO or CPOFA
               or the output  X  from CQRDC.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       A       If CPOCO or CPOFA was used to factor  A  then
               CPODI produces the upper half of INVERSE(A) .
               If CQRDC was used to decompose  X  then
               CPODI produces the upper half of INVERSE(CTRANS(X)*X)
               where CTRANS(X) is the conjugate transpose.
               Elements of  A  below the diagonal are unchanged.
               If the units digit of JOB is zero,  A  is unchanged.
       DET     REAL(2)
               determinant of  A  or of  CTRANS(X)*X  if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       a division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if CPOCO or CPOFA has set INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CSCAL
    Fortran CONJG,MOD,REAL


CPOFA


      SUBROUTINE CPOFA(A,LDA,N,INFO)

    CPOFA factors a complex Hermitian positive definite matrix.
    CPOFA is usually called by CPOCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (Time for CPOCO) = (1 + 18/N)*(Time for CPOFA) .
    On Entry
       A       COMPLEX(LDA, N)
               the Hermitian matrix to be factored.  Only the
               diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix  R  so that  A =
               CTRANS(R)*R where  CTRANS(R)  is the conjugate
               transpose.  The strict lower triangle is unaltered.
               If  INFO .NE. 0 , the factorization is not complete.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CDOTC
    Fortran AIMAG,CMPLX,CONJG,REAL,SQRT


CPOFS


      SUBROUTINE CPOFS(A,LDA,N,V,ITASK,IND,WORK)

   Subroutine CPOFS solves a  positive definite symmetric
   NxN system of complex linear equations using LINPACK
   subroutines CPOCO and CPOSL.  That is, if A is an NxN
   complex positive definite symmetric matrix and if X and B
   are complex N-vectors, then CPOFS solves the equation
                         A*X=B.
   Care should be taken not to use CPOFS with a non-Hermitian
   matrix.
   The matrix A is first factored into upper and lower tri-
   angular matrices R and R-TRANSPOSE.  These factors are used to
   find the solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of a does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeedIng solutions.  In this case, the contents of A,
   LDA, and N must not have been altered by the user following
   factorization (ITASK=1).  IND will not be changed by CPOFS
   in this case.
 Argument Description ***
   A      COMPLEX(LDA,N)
            on entry, the doubly subscripted array with dimension
              (LDA,N) which contains the coefficient matrix.  Only
              the upper triangle, including the diagonal, of the
              coefficient matrix need be entered and will subse-
              quently be referenced and changed by the routine.
            on return, contains in its upper triangle an upper
              triangular matrix R such that  A = (R-TRANSPOSE) * R .
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater
            than or equal to 1.  (terminal error message IND=-2)
   V      COMPLEX(N)
            on entry the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            if ITASK = 1, the matrix A is factored and then the
              linear equation is solved.
            if ITASK .GT. 1, the equation is solved using the existing
              factored matrix A.
            if ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT. 0  see error message corresponding to IND below.
   WORK   COMPLEX(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   The matrix A is computationally singular or
                        is not positive definite.  A solution
                        has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the
                        matrix A may be poorly scaled.
              NOTE-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


CPOIR


      SUBROUTINE CPOIR(A,LDA,N,V,ITASK,IND,WORK)

   Subroutine CPOIR solves a complex positive definite Hermitian
   NxN system of single precision linear equations using LINPACK
   subroutines CPOFA and CPOSL.  One pass of iterative refine-
   ment is used only to obtain an estimate of the accuracy.  That
   is, if A is an NxN complex positive definite Hermitian matrix
   and if X and B are complex N-vectors, then CPOIR solves the
   equation
                         A*X=B.
   Care should be taken not to use CPOIR with a non-Hermitian
   matrix.
   The matrix A is first factored into upper and lower
   triangular matrices R and R-TRANSPOSE.  These
   factors are used to calculate the solution, X.
   Then the residual vector is found and used
   to calculate an estimate of the relative error, IND.
   IND estimates the accuracy of the solution only when the
   input matrix and the right hand side are represented
   exactly in the computer and does not take into account
   any errors in the input data.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, N, and WORK must not have been altered by the user
   following factorization (ITASK=1).  IND will not be changed
   by CPOIR in this case.
 Argument Description ***
   A       COMPLEX(LDA,N)
            the doubly subscripted array with dimension (LDA,N)
            which contains the coefficient matrix.  Only the
            upper triangle, including the diagonal, of the
            coefficient matrix need be entered.  A is not
            altered by the routine.
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater than
            or equal to one.  (terminal error message IND=-2)
   V      COMPLEX(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            if ITASK = 1, the matrix A is factored and then the
              linear equation is solved.
            if ITASK .GT. 1, the equation is solved using the existing
              factored matrix A (stored in WORK).
            if ITASK .LT. 1, then terminal terminal error IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.  IND=75 means
                    that the solution vector X is zero.
            LT. 0  see error message corresponding to IND below.
   WORK   COMPLEX(N*(N+1))
            a singly subscripted array of dimension at least N*(N+1).
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than one.
   IND=-3  terminal   ITASK is less than one.
   IND=-4  terminal   The matrix A is computationally singular
                        or is not positive definite.
                        A solution has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        the solution may be inaccurate or the matrix
                        a may be poorly scaled.
              NOTE-  the above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


CPOSL


      SUBROUTINE CPOSL(A,LDA,N,B)

    CPOSL solves the COMPLEX Hermitian positive definite system
    A * X = B
    using the factors computed by CPOCO or CPOFA.
    On Entry
       A       COMPLEX(LDA, N)
               the output from CPOCO or CPOFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       B       COMPLEX(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically this indicates
       singularity but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CPOCO(A,LDA,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL CPOSL(A,LDA,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTC


CPPCO


      SUBROUTINE CPPCO(AP,N,RCOND,Z,INFO)

    CPPCO factors a complex Hermitian positive definite matrix
    stored in packed form and estimates the condition of the matrix.
    If  RCOND  is not needed, CPPFA is slightly faster.
    To solve  A*X = B , follow CPPCO by CPPSL.
    To compute  INVERSE(A)*C , follow CPPCO by CPPSL.
    To compute  DETERMINANT(A) , follow CPPCO by CPPDI.
    To compute  INVERSE(A) , follow CPPCO by CPPDI.
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the packed form of a Hermitian matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    On Return
       AP      an upper triangular matrix  R , stored in packed
               form, so that  A = CTRANS(R)*R .
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is singular to working precision, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    Packed Storage
         The following program segment will pack the upper
         triangle of a Hermitian matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CPPFA
    BLAS CAXPY,CDOTC,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,REAL


CPPDI


      SUBROUTINE CPPDI(AP,N,DET,JOB)

    CPPDI computes the determinant and inverse
    of a complex Hermitian positive definite matrix
    using the factors computed by CPPCO or CPPFA .
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the output from CPPCO or CPPFA.
       N       INTEGER
               the order of the matrix  A .
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       AP      the upper triangular half of the inverse .
               The strict lower triangle is unaltered.
       DET     REAL(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if CPOCO or CPOFA has set INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CSCAL
    Fortran CONJG,MOD,REAL


CPPFA


      SUBROUTINE CPPFA(AP,N,INFO)

    CPPFA factors a complex Hermitian positive definite matrix
    stored in packed form.
    CPPFA is usually called by CPPCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (Time for CPPCO) = (1 + 18/N)*(Time for CPPFA) .
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the packed form of a Hermitian matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    On Return
       AP      an upper triangular matrix  R , stored in packed
               form, so that  A = CTRANS(R)*R .
       INFO    INTEGER
               = 0  for normal return.
               = K  If the leading minor of order  K  is not
                    positive definite.
    Packed Storage
         The following program segment will pack the upper
         triangle of a Hermitian matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CDOTC
    Fortran AIMAG,CMPLX,CONJG,REAL,SQRT


CPPSL


      SUBROUTINE CPPSL(AP,N,B)

    CPPSL solves the complex Hermitian positive definite system
    A * X = B
    using the factors computed by CPPCO or CPPFA.
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the output from CPPCO or CPPFA.
       N       INTEGER
               the order of the matrix  A .
       B       COMPLEX(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically this indicates
       singularity but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CPPCO(AP,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL CPPSL(AP,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTC


CPTSL


      SUBROUTINE CPTSL(N,D,E,B)

    CPTSL given a positive definite tridiagonal matrix and a right
    hand side will find the solution.
    On Entry
       N        INTEGER
                is the order of the tridiagonal matrix.
       D        COMPLEX(N)
                is the diagonal of the tridiagonal matrix.
                On output D is destroyed.
       E        COMPLEX(N)
                is the offdiagonal of the tridiagonal matrix.
                E(1) through E(N-1) should contain the
                offdiagonal.
       B        COMPLEX(N)
                is the right hand side vector.
    On Return
       B        contains the solution.
    LINPACK.  This version dated 08/14/78 .
    Jack Dongarra, Argonne National Laboratory.
    No externals
    Fortran CONJG,MOD


CQRDC


      SUBROUTINE CQRDC(X,LDX,N,P,QRAUX,JPVT,WORK,JOB)

    CQRDC uses Householder transformations to compute the QR
    factorization of an N by P matrix X.  Column pivoting
    based on the 2-norms of the reduced columns may be
    performed at the users option.
    On Entry
       X       COMPLEX(LDX,P), where LDX .GE. N.
               X contains the matrix whose decomposition is to be
               computed.
       LDX     INTEGER.
               LDX is the leading dimension of the array X.
       N       INTEGER.
               N is the number of rows of the matrix X.
       P       INTEGER.
               P is the number of columns of the matrix X.
       JVPT    INTEGER(P).
               JVPT contains integers that control the selection
               of the pivot columns.  The K-th column X(K) of X
               is placed in one of three classes according to the
               value of JVPT(K).
                  If JVPT(K) .GT. 0, then X(K) is an initial
                                     column.
                  If JVPT(K) .EQ. 0, then X(K) is a free column.
                  If JVPT(K) .LT. 0, then X(K) is a final column.
               Before the decomposition is computed, initial columns
               are moved to the beginning of the array X and final
               columns to the end.  Both initial and final columns
               are frozen in place during the computation and only
               free columns are moved.  At the K-th stage of the
               reduction, if X(K) is occupied by a free column
               it is interchanged with the free column of largest
               reduced norm.  JVPT is not referenced if
               JOB .EQ. 0.
       WORK    COMPLEX(P).
               WORK is a work array.  WORK is not referenced if
               JOB .EQ. 0.
       JOB     INTEGER.
               JOB is an integer that initiates column pivoting.
               If JOB .EQ. 0, no pivoting is done.
               If JOB .NE. 0, pivoting is done.
    On Return
       X       X contains in its upper triangle the upper
               triangular matrix R of the QR factorization.
               Below its diagonal X contains information from
               which the unitary part of the decomposition
               can be recovered.  Note that if pivoting has
               been requested, the decomposition is not that
               of the original matrix X but that of X
               with its columns permuted as described by JVPT.
       QRAUX   COMPLEX(P).
               QRAUX contains further information required to recover
               the unitary part of the decomposition.
       JVPT    JVPT(K) contains the index of the column of the
               original matrix that has been interchanged into
               the K-th column, if pivoting was requested.
    LINPACK.  This version dated 08/14/78 .
    Stewart, G. W., University of Maryland, Argonne National Lab.
    CQRDC uses the following functions and subprograms.
    BLAS CAXPY,CDOTC,CSCAL,CSWAP,SCNRM2
    Fortran ABS,AIMAG,AMAX1,CABS,CMPLX,CSQRT,MIN0,REAL


CQRSL


      SUBROUTINE CQRSL(X,LDX,N,K,QRAUX,Y,QY,QTY,B,RSD,XB,JOB,INFO)

    CQRSL applies the output of CQRDC to compute coordinate
    transformations, projections, and least squares solutions.
    For K .LE. MIN(N,P), let XK be the matrix
           XK = (X(JVPT(1)),X(JVPT(2)), ... ,X(JVPT(K)))
    formed from columnns JVPT(1), ... ,JVPT(K) of the original
    N x P matrix X that was input to CQRDC (if no pivoting was
    done, XK consists of the first K columns of X in their
    original order).  CQRDC produces a factored unitary matrix Q
    and an upper triangular matrix R such that
             XK = Q * (R)
                      (0)
    This information is contained in coded form in the arrays
    X and QRAUX.
    On Entry
       X      COMPLEX(LDX,P).
              X contains the output of CQRDC.
       LDX    INTEGER.
              LDX is the leading dimension of the array X.
       N      INTEGER.
              N is the number of rows of the matrix XK.  It must
              have the same value as N in CQRDC.
       K      INTEGER.
              K is the number of columns of the matrix XK.  K
              must not be greater than (N,P), where P is the
              same as in the calling sequence to CQRDC.
       QRAUX  COMPLEX(P).
              QRAUX contains the auxiliary output from CQRDC.
       Y      COMPLEX(N)
              Y contains an N-vector that is to be manipulated
              by CQRSL.
       JOB    INTEGER.
              JOB specifies what is to be computed.  JOB has
              the decimal expansion ABCDE, with the following
              meaning.
                   If A .NE. 0, compute QY.
                   If B,C,D, or E .NE. 0, compute QTY.
                   If C .NE. 0, compute B.
                   If D .NE. 0, compute RSD .
                   If E .NE. 0, compute  XB.
              Note that a request to compute B, RSD, or XB
              automatically triggers the computation of QTY, for
              which an array must be provided in the calling
              sequence.
    On Return
       QY     COMPLEX(N).
              QY contains Q*Y, if its computation has been
              requested.
       QTY    COMPLEX(N).
              QTY contains CTRANS(Q)*Y, if its computation has
              been requested.  Here CTRANS(Q) is the conjugate
              transpose of the matrix Q.
       B      COMPLEX(K)
              B contains the solution of the least squares problem
                   minimize NORM2(Y - XK*B),
              if its computation has been requested.  (Note that
              if pivoting was requested in CQRDC, the J-th
              component of B will be associated with column JVPT(J)
              of the original matrix X that was input into CQRDC.)
       RSD    COMPLEX(N).
              RSD contains the least squares residual Y - XK*B,
              if its computation has been requested.  RSD is
              also the orthogonal projection of Y onto the
              orthogonal complement of the column space of XK.
       XB     COMPLEX(N).
              XB contains the least squares approximation XK*B,
              if its computation has been requested.  XB is also
              the orthogonal projection of Y onto the column space
              of X.
       INFO   INTEGER.
              INFO is zero unless the computation of B has
              been requested and R is exactly singular.  In
              this case, INFO is the index of the first zero
              diagonal element of R and B is left unaltered.
    The parameters QY, QTY, B, RSD, and XB are not referenced
    if their computation is not requested and in this case
    can be replaced by dummy variables in the calling program.
    To save storage, the user may in some cases use the same
    array for different parameters in the calling sequence.  A
    frequently occuring example is when one wishes to compute
    any of B, RSD, or XB and does not need Y or QTY.  In this
    case one may identify Y, QTY, and one of B, RSD, or XB, while
    providing separate arrays for anything else that is to be
    computed.  Thus the calling sequence
         CALL CQRSL(X,LDX,N,K,QRAUX,Y,DUM,Y,B,Y,DUM,110,INFO)
    will result in the computation of B and RSD, with RSD
    overwriting Y.  More generally, each item in the following
    list contains groups of permissible identifications for
    a single callinng sequence.
         1. (Y,QTY,B) (RSD) (XB) (QY)
         2. (Y,QTY,RSD) (B) (XB) (QY)
         3. (Y,QTY,XB) (B) (RSD) (QY)
         4. (Y,QY) (QTY,B) (RSD) (XB)
         5. (Y,QY) (QTY,RSD) (B) (XB)
         6. (Y,QY) (QTY,XB) (B) (RSD)
    In any group the value returned in the array allocated to
    the group corresponds to the last member of the group.
    LINPACK.  This version dated 08/14/78 .
    Stewart, G. W., University of Maryland, Argonne National Lab.
    CQRSL uses the following functions and subprograms.
    BLAS CAXPY,CCOPY,CDOTC
    Fortran ABS,AIMAG,MIN0,MOD,REAL


CROTG


      SUBROUTINE CROTG(CA,CB,C,S)

    Complex Givens transformation
     Construct the Givens transformation
            (C    S)
      G  =  (      ),  C**2 + CABS(S)**2 =1,
            (-S   C)
     which zeros the second entry of the complex 2-vector (CA,CB)**T
   The quantity CA/CABS(CA)*NORM(CA,CB) overwrites CA in storage.
   INPUT:
       CA (Complex)
       CB (Complex)
   Output:
       CA (Complex)      CA/CABS(CA)*NORM(CA,CB)
       C  (Real)
       S  (Complex)


CSCAL


      SUBROUTINE CSCAL(N,CA,CX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      CA  complex scale factor
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
    --Output--
   CSCAL  complex result (unchanged if N .LE. 0)
    replace complex CX by complex CA*CX.
    For I = 0 to N-1, replace CX(1+I*INCX) with  CA * CX(1+I*INCX)


CSCALE


      SUBROUTINE CSCALE(A,NRDA,NROW,NCOL,COLS,COLSAV,ROWS,ROWSAV,ANORM,



CSICO


      SUBROUTINE CSICO(A,LDA,N,KPVT,RCOND,Z)

    CSICO factors a complex symmetric matrix by elimination with
    symmetric pivoting and estimates the condition of the matrix.
    If  RCOND  is not needed, CSIFA is slightly faster.
    To solve  A*X = B , follow CSICO by CSISL.
    To compute  INVERSE(A)*C , follow CSICO by CSISL.
    To compute  INVERSE(A) , follow CSICO by CSIDI.
    To compute  DETERMINANT(A) , follow CSICO by CSIDI.
    On Entry
       A       COMPLEX(LDA, N)
               the symmetric matrix to be factored.
               Only the diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       a block diagonal matrix and the multipliers which
               were used to obtain it.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KVPT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CSIFA
    BLAS CAXPY,CDOTU,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,IABS,REAL


CSIDI


      SUBROUTINE CSIDI(A,LDA,N,KPVT,DET,WORK,JOB)

    CSIDI computes the determinant and inverse
    of a complex symmetric matrix using the factors from CSIFA.
    On Entry
       A       COMPLEX(LDA,N)
               the output from CSIFA.
       LDA     INTEGER
               the leading dimension of the array A .
       N       INTEGER
               the order of the matrix A .
       KVPT    INTEGER(N)
               the pivot vector from CSIFA.
       WORK    COMPLEX(N)
               work vector.  Contents destroyed.
       JOB     INTEGER
               JOB has the decimal expansion  AB  where
                  If  B .NE. 0, the inverse is computed,
                  If  A .NE. 0, the determinant is computed,
               For example, JOB = 11  gives both.
    On Return
       Variables not requested by JOB are not used.
       A      contains the upper triangle of the inverse of
              the original matrix.  The strict lower triangle
              is never referenced.
       DET    COMPLEX(2)
              determinant of original matrix.
              Determinant = DET(1) * 10.0**DET(2)
              with 1.0 .LE. ABS(DET(1)) .LT. 10.0
              or DET(1) = 0.0.
    Error Condition
       A division by zero may occur if the inverse is requested
       and  CSICO  has set RCOND .EQ. 0.0
       or  CSIFA  has set  INFO .NE. 0 .
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CCOPY,CDOTU,CSWAP
    Fortran ABS,CMPLX,IABS,MOD,REAL


CSIFA


      SUBROUTINE CSIFA(A,LDA,N,KPVT,INFO)

    CSIFA factors a complex symmetric matrix by elimination
    with symmetric pivoting.
    To solve  A*X = B , follow CSIFA by CSISL.
    To compute  INVERSE(A)*C , follow CSIFA by CSISL.
    To compute  DETERMINANT(A) , follow CSIFA by CSIDI.
    To compute  INVERSE(A) , follow CSIFA by CSIDI.
    On Entry
       A       COMPLEX(LDA,N)
               the symmetric matrix to be factored.
               Only the diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       a block diagonal matrix and the multipliers which
               were used to obtain it.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KVPT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if the K-th pivot block is singular.  This is
                    not an error condition for this subroutine,
                    but it does indicate that CSISL or CSIDI may
                    divide by zero if called.
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CSWAP,ICAMAX
    Fortran ABS,AIMAG,AMAX1,REAL,SQRT


CSISL


      SUBROUTINE CSISL(A,LDA,N,KPVT,B)

    CSISL solves the complex symmetric system
    A * X = B
    using the factors computed by CSIFA.
    On Entry
       A       COMPLEX(LDA,N)
               the output from CSIFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       KVPT    INTEGER(N)
               the pivot vector from CSIFA.
       B       COMPLEX(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero may occur if  CSICO  has set RCOND .EQ. 0.0
       or  CSIFA  has set INFO .NE. 0  .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CSIFA(A,LDA,N,KVPT,INFO)
          If (INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL CSISL(A,LDA,N,KVPT,C(1,j))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTU
    Fortran IABS


CSPCO


      SUBROUTINE CSPCO(AP,N,KPVT,RCOND,Z)

    CSPCO factors a complex symmetric matrix stored in packed
    form by elimination with symmetric pivoting and estimates
    the condition of the matrix.
    If  RCOND  is not needed, CSPFA is slightly faster.
    To solve  A*X = B , follow CSPCO by CSPSL.
    To compute  INVERSE(A)*C , follow CSPCO by CSPSL.
    To compute  INVERSE(A) , follow CSPCO by CSPDI.
    To compute  DETERMINANT(A) , follow CSPCO by CSPDI.
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    On Return
       AP      a block diagonal matrix and the multipliers which
               were used to obtain it stored in packed form.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KVPT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK CSPFA
    BLAS CAXPY,CDOTU,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,IABS,REAL


CSPDI


      SUBROUTINE CSPDI(AP,N,KPVT,DET,WORK,JOB)

    CSPDI computes the determinant and inverse
    of a complex symmetric matrix using the factors from CSPFA,
    where the matrix is stored in packed form.
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the output from CSPFA.
       N       INTEGER
               the order of the matrix A .
       KVPT    INTEGER(N)
               the pivot vector from CSPFA.
       WORK    COMPLEX(N)
               work vector.  Contents ignored.
       JOB     INTEGER
               JOB has the decimal expansion  AB  where
                  if  B .NE. 0, the inverse is computed,
                  if  A .NE. 0, the determinant is computed.
               For example, JOB = 11  gives both.
    On Return
       Variables not requested by JOB are not used.
       AP     contains the upper triangle of the inverse of
              the original matrix, stored in packed form.
              The columns of the upper triangle are stored
              sequentially in a one-dimensional array.
       DET    COMPLEX(2)
              determinant of original matrix.
              Determinant = DET(1) * 10.0**DET(2)
              with 1.0 .LE. ABS(DET(1)) .LT. 10.0
              or DET(1) = 0.0.
    Error Condition
       A division by zero will occur if the inverse is requested
       and  CSPCO  has set RCOND .EQ. 0.0
       or  CSPFA  has set  INFO .NE. 0 .
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CCOPY,CDOTU,CSWAP
    Fortran ABS,CMPLX,IABS,MOD,REAL


CSPFA


      SUBROUTINE CSPFA(AP,N,KPVT,INFO)

    CSPFA factors a complex symmetric matrix stored in
    packed form by elimination with symmetric pivoting.
    To solve  A*X = B , follow CSPFA by CSPSL.
    To compute  INVERSE(A)*C , follow CSPFA by CSPSL.
    To compute  DETERMINANT(A) , follow CSPFA by CSPDI.
    To compute  INVERSE(A) , follow CSPFA by CSPDI.
    On Entry
       AP      COMPLEX (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    On Return
       AP      a block diagonal matrix and the multipliers which
               were used to obtain it stored in packed form.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KVPT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if the K-th pivot block is singular.  This is
                    not an error condition for this subroutine,
                    but it does indicate that CSPSL or CSPDI may
                    divide by zero if called.
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K)  = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CSWAP,ICAMAX
    Fortran ABS,AIMAG,AMAX1,REAL,SQRT


CSPSL


      SUBROUTINE CSPSL(AP,N,KPVT,B)

    CSISL solves the complex symmetric system
    A * X = B
    using the factors computed by CSPFA.
    On Entry
       AP      COMPLEX(N*(N+1)/2)
               the output from CSPFA.
       N       INTEGER
               the order of the matrix  A .
       KVPT    INTEGER(N)
               the pivot vector from CSPFA.
       B       COMPLEX(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero may occur if  CSPCO  has set RCOND .EQ. 0.0
       or  CSPFA  has set INFO .NE. 0  .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL CSPFA(AP,N,KVPT,INFO)
          IF (INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL CSPSL(AP,N,KVPT,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTU
    Fortran IABS


CSROOT


      SUBROUTINE CSROOT(XR,XI,YR,YI)



CSROT


      SUBROUTINE CSROT(N,CX,INCX,CY,INCY,C,S)

    CSROT applies the complex Givens rotation
         (X)   ( C S)(X)
         (Y) = (-S C)(Y)
    N times where for I = 0,...,N-1
         X = CX(1+I*INCX)
         Y = CY(1+I*INCY)
    Argument Description
       N      (integer)  number of elements in each vector
       CX     (complex array)  beginning of one vector
       INCX   (integer)  memory spacing of successive elements
              of vector CX
       CY     (complex array)  beginning of the other vector
       INCY   (integer)  memory spacing of successive elements
              of vector CY
       C      (real)  cosine term of the rotation
       S      (real)  sine term of the rotation.


CSSCAL


      SUBROUTINE CSSCAL(N,SA,CX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SA  single precision scale factor
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
    --Output--
      CX  scaled result (unchanged if N .LE. 0)
    Replace complex CX by (single precision SA) * (complex CX)
    For I = 0 to N-1, replace CX(1+I*INCX) with  SA * CX(1+I*INCX)


CSVDC


      SUBROUTINE CSVDC(X,LDX,N,P,S,E,U,LDU,V,LDV,WORK,JOB,INFO)

    CSVDC is a subroutine to reduce a complex NxP matrix X by
    unitary transformations U and V to diagonal form.  The
    diagonal elements S(I) are the singular values of X.  The
    columns of U are the corresponding left singular vectors,
    and the columns of V the right singular vectors.
    On Entry
        X         COMPLEX(LDX,P), where LDX .GE. N.
                  X contains the matrix whose singular value
                  decomposition is to be computed.  X is
                  destroyed by CSVDC.
        LDX       INTEGER.
                  LDX is the leading dimension of the array X.
        N         INTEGER.
                  N is the number of columns of the matrix X.
        P         INTEGER.
                  P is the number of rows of the matrix X.
        LDU       INTEGER.
                  LDU is the leading dimension of the array U
                  (see below).
        LDV       INTEGER.
                  LDV is the leading dimension of the array V
                  (see below).
        WORK      COMPLEX(N).
                  WORK is a scratch array.
        JOB       INTEGER.
                  JOB controls the computation of the singular
                  vectors.  It has the decimal expansion AB
                  with the following meaning
                       A .EQ. 0    Do not compute the left singular
                                   vectors.
                       A .EQ. 1    Return the N left singular vectors
                                   in U.
                       A .GE. 2    Return the first MIN(N,P)
                                   left singular vectors in U.
                       B .EQ. 0    Do not compute the right singular
                                   vectors.
                       B .EQ. 1    Return the right singular vectors
                                   in V.
    On Return
        S         COMPLEX(MM), where MM = MIN(N+1,P).
                  The first MIN(N,P) entries of S contain the
                  singular values of X arranged in descending
                  order of magnitude.
        E         COMPLEX(P).
                  E ordinarily contains zeros.  However see the
                  discussion of INFO for exceptions.
        U         COMPLEX(LDU,K), where LDU .GE. N.  If JOBA .EQ. 1
                                  then K .EQ. N.  If JOBA .GE. 2 then
                                  K .EQ. MIN(N,P).
                  U contains the matrix of right singular vectors.
                  U is not referenced if JOBA .EQ. 0.  If N .LE. P
                  or if JOBA .GT. 2, then U may be identified with X
                  in the subroutine call.
        V         COMPLEX(LDV,P), where LDV .GE. P.
                  V contains the matrix of right singular vectors.
                  V is not referenced if JOB .EQ. 0.  If P .LE. N,
                  then V may be identified with X in the
                  subroutine call.
        INFO      INTEGER.
                  The singular values (and their corresponding
                  singular vectors) S(INFO+1),S(INFO+2),...,S(M)
                  are correct (here M=MIN(N,P)).  Thus if
                  INFO.EQ. 0, all the singular values and their
                  vectors are correct.  In any event, the matrix
                  B = CTRANS(U)*X*V is the bidiagonal matrix
                  with the elements of S on its diagonal and the
                  elements of E on its super-diagonal (CTRANS(U)
                  is the conjugate-transpose of U).  Thus the
                  singular values of X and B are the same.
    LINPACK.  This version dated 03/19/79 .
    Stewart, G. W., University of Maryland, Argonne National Lab.
    CSVDC uses the following functions and subprograms.
    External CSROT
    BLAS CAXPY,CDOTC,CSCAL,CSWAP,SCNRM2,SROTG
    Fortran ABS,AIMAG,AMAX1,CABS,CMPLX
    Fortran CONJG,MAX0,MIN0,MOD,REAL,SQRT


CSWAP


      SUBROUTINE CSWAP(N,CX,INCX,CY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
      CY  complex vector with N elements
    INCY  storage spacing between elements of CY
    --Output--
      CX  input vector CY (unchanged if N .LE. 0)
      CY  input vector CX (unchanged if N .LE. 0)
    Interchange complex CX and complex CY
    For I = 0 to N-1, interchange  CX(LX+I*INCX) and CY(LY+I*INCY),
    where LX = 1 if INCX .GT. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


CTRCO


      SUBROUTINE CTRCO(T,LDT,N,RCOND,Z,JOB)

    CTRCO estimates the condition of a complex triangular matrix.
    On Entry
       T       COMPLEX(LDT,N)
               T contains the triangular matrix.  The zero
               elements of the matrix are not referenced, and
               the corresponding elements of the array can be
               used to store other information.
       LDT     INTEGER
               LDT is the leading dimension of the array T.
       N       INTEGER
               N is the order of the system.
       JOB     INTEGER
               = 0         T  is lower triangular.
               = nonzero   T  is upper triangular.
    On Return
       RCOND   REAL
               an estimate of the reciprocal condition of  T .
               For the system  T*X = B , relative perturbations
               in  T  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  T  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       COMPLEX(N)
               a work vector whose contents are usually unimportant.
               If  T  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CSSCAL,SCASUM
    Fortran ABS,AIMAG,AMAX1,CMPLX,CONJG,REAL


CTRDI


      SUBROUTINE CTRDI(T,LDT,N,DET,JOB,INFO)

    CTRDI computes the determinant and inverse of a complex
    triangular matrix.
    On Entry
       T       COMPLEX(LDT,N)
               T contains the triangular matrix.  The zero
               elements of the matrix are not referenced, and
               the corresponding elements of the array can be
               used to store other information.
       LDT     INTEGER
               LDT is the leading dimension of the array T.
       N       INTEGER
               N is the order of the system.
       JOB     INTEGER
               = 010       no det, inverse of lower triangular.
               = 011       no det, inverse of upper triangular.
               = 100       det, no inverse.
               = 110       det, inverse of lower triangular.
               = 111       det, inverse of upper triangular.
    On Return
       T       inverse of original matrix if requested.
               Otherwise unchanged.
       DET     COMPLEX(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. CABS1(DET(1)) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
       INFO    INTEGER
               INFO contains zero if the system is nonsingular
               and the inverse is requested.
               Otherwise INFO contains the index of
               a zero diagonal element of T.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CSCAL
    Fortran ABS,AIMAG,CMPLX,MOD,REAL


CTRSL


      SUBROUTINE CTRSL(T,LDT,N,B,JOB,INFO)

    CTRSL solves systems of the form
                  T * X = B
    or
                  CTRANS(T) * X = B
    where T is a triangular matrix of order N.  Here CTRANS(T)
    denotes the conjugate transpose of the matrix T.
    On Entry
        T         COMPLEX(LDT,N)
                  T contains the matrix of the system.  The zero
                  elements of the matrix are not referenced, and
                  the corresponding elements of the array can be
                  used to store other information.
        LDT       INTEGER
                  LDT is the leading dimension of the array T.
        N         INTEGER
                  N is the order of the system.
        B         COMPLEX(N).
                  B contains the right hand side of the system.
        JOB       INTEGER
                  JOB specifies what kind of system is to be solved.
                  If JOB is
                       00   solve T*X = B, T lower triangular,
                       01   solve T*X = B, T upper triangular,
                       10   solve CTRANS(T)*X = B, T lower triangular,
                       11   solve CTRANS(T)*X = B, T upper triangular.
    On Return
        B         B contains the solution, if INFO .EQ. 0.
                  Otherwise B is unaltered.
        INFO      INTEGER
                  INFO contains zero if the system is nonsingular.
                  Otherwise INFO contains the index of
                  the first zero diagonal element of T.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    Subroutines and Functions
    BLAS CAXPY,CDOTC
    Fortran ABS,AIMAG,CONJG,MOD,REAL


DASUM


      DOUBLE PRECISION FUNCTION DASUM(N,DX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
    --Output--
   DASUM  double precision result (zero if N .LE. 0)
    Returns sum of magnitudes of double precision DX.
    DASUM = sum from 0 to N-1 of DABS(DX(1+I*INCX))


DASYIK


      SUBROUTINE DASYIK(X,FNU,KODE,FLGIK,RA,ARG,IN,Y)



DASYJY


      SUBROUTINE DASYJY(FUNJY,X,FNU,FLGJY,IN,Y,WK,IFLW)

                DASYJY computes Bessel functions J and Y
              for arguments X.GT.0.0 and orders FNU .GE. 35.0
              on FLGJY = 1 and FLGJY = -1 respectively
                                 INPUT
     FUNJY - External function JAIRY or YAIRY
         X - Argument, X.GT.0.0D0
       FNU - Order of the first Bessel function
     FLGJY - Selection flag
             FLGJY =  1.0D0 gives the J function
             FLGJY = -1.0D0 gives the Y function
        IN - Number of functions desired, IN = 1 or 2
                                 OUTPUT
        Y  - A vector whose first IN components contain the sequence
      IFLW - A flag indicating underflow or overflow
                   return variables for BESJ only
     WK(1) = 1 - (X/FNU)**2 = W**2
     WK(2) = DSQRT(DABS(WK(1)))
     WK(3) = DABS(WK(2) - DATAN(WK(2)))  or
             DABS(LN((1 + WK(2))/(X/FNU)) - WK(2))
           = DABS((2/3)*ZETA**(3/2))
     WK(4) = FNU*WK(3)
     WK(5) = (1.5*WK(3)*FNU)**(1/3) = DSQRT(ZETA)*FNU**(1/3)
     WK(6) = DSIGN(1.,W**2)*WK(5)**2 = DSIGN(1.,W**2)*ZETA*FNU**(2/3)
     WK(7) = FNU**(1/3)
                                 Written by
                                 D. E. Amos
    Abstract   **** A Double Precision Routine ****
        DASYJY implements the uniform asymptotic expansion of
        the J and Y Bessel functions for FNU.GE.35 and real
        X.GT.0.0D0. The forms are identical except for a change
        in sign of some of the terms. This change in sign is
        accomplished by means of the flag FLGJY = 1 or -1. On
        FLGJY = 1 the Airy functions AI(X) and DAI(X) are
        supplied by the external function JAIRY, and on
        FLGJY = -1 the Airy functions BI(X) and DBI(X) are
        supplied by the external funtion YAIRY.


DAXPY


      SUBROUTINE DAXPY(N,DA,DX,INCX,DY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DA  double precision scalar multiplier
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
      DY  double precision vector with N elements
    INCY  storage spacing between elements of DY
    --Output--
      DY  double precision result (unchanged if N .LE. 0)
    Overwrite double precision DY with double precision DA*DX + DY.
    For I = 0 to N-1, replace  DY(LY+I*INCY) with DA*DX(LX+I*INCX) +
      DY(LY+I*INCY), where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N
      and LY is defined in a similar way using INCY.


DBDIFF


      SUBROUTINE DBDIFF(L,V)

    DBDIFF computes the sum of B(L,K)*V(K)*(-1)**K where B(L,K)
    are the binomial coefficients.  Truncated sums are computed by
    setting last part of the V vector to zero. On return, the binomial
    sum is in V(L).


DBNDAC


      SUBROUTINE DBNDAC(G,MDG,NB,IP,IR,MT,JT)

      **** Double Precision version of BNDACC ****
    These subroutines solve the least squares problem Ax = b for
    banded matrices A using sequential accumulation of rows of the
    data matrix.  Exactly one right-hand side vector is permitted.
    These subroutines are intended for the type of least squares
    systems that arise in applications such as curve or surface
    fitting of data.  The least squares equations are accumulated and
    processed using only part of the data.  This requires a certain
    user interaction during the solution of Ax = b.
    Specifically, suppose the data matrix (A B) is row partitioned
    into Q submatrices.  Let (E F) be the T-th one of these
    submatrices where E = (0 C 0).  Here the dimension of E is MT by N
    and the dimension of C is MT by NB.  The value of NB is the
    bandwidth of A.  The dimensions of the leading block of zeros in E
    are MT by JT-1.
    The user of the subroutine DBNDAC provides MT,JT,C and F for
    T=1,...,Q.  Not all of this data must be supplied at once.
    Following the processing of the various blocks (E F), the matrix
    (A B) has been transformed to the form (R D) where R is upper
    triangular and banded with bandwidth NB.  The least squares
    system Rx = d is then easily solved using back substitution by
    executing the statement CALL DBNDSL(1,...). The sequence of
    values for JT must be nondecreasing.  This may require some
    preliminary interchanges of rows and columns of the matrix A.
    The primary reason for these subroutines is that the total
    processing can take place in a working array of dimension MU by
    NB+1.  An acceptable value for MU is
                      MU = MAX(MT + N + 1),
    where N is the number of unknowns.
    Here the maximum is taken over all values of MT for T=1,...,Q.
    Notice that MT can be taken to be a small as one, showing that
    MU can be as small as N+2.  The subprogram DBNDAC processes the
    rows more efficiently if MU is large enough so that each new
    block (C F) has a distinct value of JT.
    The four principle parts of these algorithms are obtained by the
    following call statements
    CALL DBNDAC(...)  Introduce new blocks of data.
    CALL DBNDSL(1,...)Compute solution vector and length of
                      residual vector.
    CALL DBNDSL(2,...)Given any row vector H solve YR = H for the
                      row vector Y.
    CALL DBNDSL(3,...)Given any column vector W solve RZ = W for
                      the column vector Z.
    The dots in the above call statements indicate additional
    arguments that will be specified in the following paragraphs.
    The user must dimension the array appearing in the call list..
    G(MDG,NB+1)
    Description of calling sequence for DBNDAC..
    The entire set of parameters for DBNDAC are
    Input.. All Type REAL variables are DOUBLE PRECISION
    G(*,*)            The working array into which the user will
                      place the MT by NB+1 block (C F) in rows IR
                      through IR+MT-1, columns 1 through NB+1.
                      See descriptions of IR and MT below.
    MDG               The number of rows in the working array
                      G(*,*).  The value of MDG should be .GE. MU.
                      The value of MU is defined in the abstract
                      of these subprograms.
    NB                The bandwidth of the data matrix A.
    IP                Set by the user to the value 1 before the
                      first call to DBNDAC.  Its subsequent value
                      is controlled by DBNDAC to set up for the
                      next call to DBNDAC.
    IR                Index of the row of G(*,*) where the user is
                      to place the new block of data (C F).  Set by
                      the user to the value 1 before the first call
                      to DBNDAC.  Its subsequent value is controlled
                      by DBNDAC. A value of IR .GT. MDG is considered
                      an error.
    MT,JT             Set by the user to indicate respectively the
                      number of new rows of data in the block and
                      the index of the first nonzero column in that
                      set of rows (E F) = (0 C 0 F) being processed.
    Output.. All Type REAL variables are DOUBLE PRECISION
    G(*,*)            The working array which will contain the
                      processed rows of that part of the data
                      matrix which has been passed to DBNDAC.
    IP,IR             The values of these arguments are advanced by
                      DBNDAC to be ready for storing and processing
                      a new block of data in G(*,*).
    Description of calling sequence for DBNDSL..
    The user must dimension the arrays appearing in the call list..
    G(MDG,NB+1), X(N)
    The entire set of parameters for DBNDSL are
    Input.. All Type REAL variables are DOUBLE PRECISION
    MODE              Set by the user to one of the values 1, 2, or
                      3.  These values respectively indicate that
                      the solution of AX = B, YR = H or RZ = W is
                      required.
    G(*,*),MDG,       These arguments all have the same meaning and
     NB,IP,IR         contents as following the last call to DBNDAC.
    X(*)              With mode=2 or 3 this array contains,
                      respectively, the right-side vectors H or W of
                      the systems YR = H or RZ = W.
    N                 The number of variables in the solution
                      vector.  If any of the N diagonal terms are
                      zero the subroutine DBNDSL prints an
                      appropriate message.  This condition is
                      considered an error.
    Output.. All Type REAL variables are DOUBLE PRECISION
    X(*)              This array contains the solution vectors X,
                      Y or Z of the systems AX = B, YR = H or
                      RZ = W depending on the value of MODE=1,
                      2 or 3.
    RNORM             If MODE=1 RNORM is the Euclidean length of the
                      residual vector AX-B.  When MODE=2 or 3 RNORM
                      is set to zero.
    Remarks..
    To obtain the upper triangular matrix and transformed right-hand
    side vector D so that the super diagonals of R form the columns
    of G(*,*), execute the following Fortran statements.
    NBP1=NB+1
    DO 10 J=1, NBP1
 10 G(IR,J) = 0.E0
    MT=1
    JT=N+1
    CALL DBNDAC(G,MDG,NB,IP,IR,MT,JT)


DBNDSL


      SUBROUTINE DBNDSL(MODE,G,MDG,NB,IP,IR,X,N,RNORM)

      **** Double Precision version of BNDSOL ****
    These subroutines solve the least squares problem Ax = b for
    banded matrices A using sequential accumulation of rows of the
    data matrix.  Exactly one right-hand side vector is permitted.
    These subroutines are intended for the type of least squares
    systems that arise in applications such as curve or surface
    fitting of data.  The least squares equations are accumulated and
    processed using only part of the data.  This requires a certain
    user interaction during the solution of Ax = b.
    Specifically, suppose the data matrix (A B) is row partitioned
    into Q submatrices.  Let (E F) be the T-th one of these
    submatrices where E = (0 C 0).  Here the dimension of E is MT by N
    and the dimension of C is MT by NB.  The value of NB is the
    bandwidth of A.  The dimensions of the leading block of zeros in E
    are MT by JT-1.
    The user of the subroutine DBNDAC provides MT,JT,C and F for
    T=1,...,Q.  Not all of this data must be supplied at once.
    Following the processing of the various blocks (E F), the matrix
    (A B) has been transformed to the form (R D) where R is upper
    triangular and banded with bandwidth NB.  The least squares
    system Rx = d is then easily solved using back substitution by
    executing the statement CALL DBNDSL(1,...). The sequence of
    values for JT must be nondecreasing.  This may require some
    preliminary interchanges of rows and columns of the matrix A.
    The primary reason for these subroutines is that the total
    processing can take place in a working array of dimension MU by
    NB+1.  An acceptable value for MU is
                      MU = MAX(MT + N + 1),
    where N is the number of unknowns.
    Here the maximum is taken over all values of MT for T=1,...,Q.
    Notice that MT can be taken to be a small as one, showing that
    MU can be as small as N+2.  The subprogram DBNDAC processes the
    rows more efficiently if MU is large enough so that each new
    block (C F) has a distinct value of JT.
    The four principle parts of these algorithms are obtained by the
    following call statements
    CALL DBNDAC(...)  Introduce new blocks of data.
    CALL DBNDSL(1,...)Compute solution vector and length of
                      residual vector.
    CALL DBNDSL(2,...)Given any row vector H solve YR = H for the
                      row vector Y.
    CALL DBNDSL(3,...)Given any column vector W solve RZ = W for
                      the column vector Z.
    The dots in the above call statements indicate additional
    arguments that will be specified in the following paragraphs.
    The user must dimension the array appearing in the call list..
    G(MDG,NB+1)
    Description of calling sequence for DBNDAC..
    The entire set of parameters for DBNDAC are
    Input.. All Type REAL variables are DOUBLE PRECISION
    G(*,*)            The working array into which the user will
                      place the MT by NB+1 block (C F) in rows IR
                      through IR+MT-1, columns 1 through NB+1.
                      See descriptions of IR and MT below.
    MDG               The number of rows in the working array
                      G(*,*).  The value of MDG should be .GE. MU.
                      The value of MU is defined in the abstract
                      of these subprograms.
    NB                The bandwidth of the data matrix A.
    IP                Set by the user to the value 1 before the
                      first call to DBNDAC.  Its subsequent value
                      is controlled by DBNDAC to set up for the
                      next call to DBNDAC.
    IR                Index of the row of G(*,*) where the user is
                      the user to the value 1 before the first call
                      to DBNDAC.  Its subsequent value is controlled
                      by DBNDAC. A value of IR .GT. MDG is considered
                      an error.
    MT,JT             Set by the user to indicate respectively the
                      number of new rows of data in the block and
                      the index of the first nonzero column in that
                      set of rows (E F) = (0 C 0 F) being processed.
    Output.. All Type REAL variables are DOUBLE PRECISION
    G(*,*)            The working array which will contain the
                      processed rows of that part of the data
                      matrix which has been passed to DBNDAC.
    IP,IR             The values of these arguments are advanced by
                      DBNDAC to be ready for storing and processing
                      a new block of data in G(*,*).
    Description of calling sequence for DBNDSL..
    The user must dimension the arrays appearing in the call list..
    G(MDG,NB+1), X(N)
    The entire set of parameters for DBNDSL are
    Input..
    MODE              Set by the user to one of the values 1, 2, or
                      3.  These values respectively indicate that
                      the solution of AX = B, YR = H or RZ = W is
                      required.
    G(*,*),MDG,       These arguments all have the same meaning and
     NB,IP,IR         contents as following the last call to DBNDAC.
    X(*)              With mode=2 or 3 this array contains,
                      respectively, the right-side vectors H or W of
                      the systems YR = H or RZ = W.
    N                 The number of variables in the solution
                      vector.  If any of the N diagonal terms are
                      zero the subroutine DBNDSL prints an
                      appropriate message.  This condition is
                      considered an error.
    Output..
    X(*)              This array contains the solution vectors X,
                      Y or Z of the systems AX = B, YR = H or
                      RZ = W depending on the value of MODE=1,
                      2 or 3.
    RNORM             If MODE=1 RNORM is the Euclidean length of the
                      residual vector AX-B.  When MODE=2 or 3 RNORM
                      is set to zero.
    Remarks..
    To obtain the upper triangular matrix and transformed right-hand
    side vector D so that the super diagonals of R form the columns
    of G(*,*), execute the following Fortran statements.
    NBP1=NB+1
    DO 10 J=1, NBP1
 10 G(IR,J) = 0.E0
    MT=1
    JT=N+1
    CALL DBNDAC(G,MDG,NB,IP,IR,MT,JT)


DBNFAC


      SUBROUTINE DBNFAC(W,NROWW,NROW,NBANDL,NBANDU,IFLAG)



DBNSLV


      SUBROUTINE DBNSLV(W,NROWW,NROW,NBANDL,NBANDU,B)



DCDOT


      SUBROUTINE DCDOT(N,FM,CX,INCX,CY,INCY,DCR,DCI)

 Computes the dot product of 2 complex vectors, CX and CY, e.g.
   CX DOT CY, or, CXconjugate DOT CY.  The real and imaginary
   parts of CX and CY are converted to double precision, the dot
   product accumulation is done in double precision and the output
   is given as 2 double precision numbers, corresponding to the real
   and imaginary part of the result.
    Input
     N:  Number of complex components of CX and CY.
     FM: =+1.0   compute CX DOT CY.
         =-1.0   compute CXconjugate DOT CY.
     CX(N):
     CY(N):  Complex arrays of length N.
     INCX:(Integer)   Spacing of elements of CX to use
     INCY:(Integer)   Spacing of elements of CY to use.
    Output
     DCR:(Double Precision) Real part of dot product.
     DCI:(Double Precision) Imaginary part of dot product.


DCFOD


      SUBROUTINE DCFOD(METH,ELCO,TESCO)



DCHDC


      SUBROUTINE DCHDC(A,LDA,P,WORK,JPVT,JOB,INFO)

    DCHDC computes the Cholesky decomposition of a positive definite
    matrix.  A pivoting option allows the user to estimate the
    condition of a positive definite matrix or determine the rank
    of a positive semidefinite matrix.
    On Entry
        A      DOUBLE PRECISION(LDA,P).
               A contains the matrix whose decomposition is to
               be computed.  Only the upper half of A need be stored.
               The lower part of the array A is not referenced.
        LDA    INTEGER.
               LDA is the leading dimension of the array A.
        P      INTEGER.
               P is the order of the matrix.
        WORK   DOUBLE PRECISION.
               WORK is a work array.
        JPVT   INTEGER(P).
               JPVT contains integers that control the selection
               of the pivot elements, if pivoting has been requested.
               Each diagonal element A(K,K)
               is placed in one of three classes according to the
               value of JPVT(K).
                  If JPVT(K) .GT. 0, then X(K) is an initial
                                     element.
                  If JPVT(K) .EQ. 0, then X(K) is a free element.
                  If JPVT(K) .LT. 0, then X(K) is a final element.
               Before the decomposition is computed, initial elements
               are moved by symmetric row and column interchanges to
               the beginning of the array A and final
               elements to the end.  Both initial and final elements
               are frozen in place during the computation and only
               free elements are moved.  At the K-th stage of the
               reduction, if A(K,K) is occupied by a free element
               it is interchanged with the largest free element
               A(L,L) with L .GE. K.  JPVT is not referenced if
               JOB .EQ. 0.
       JOB     INTEGER.
               JOB is an integer that initiates column pivoting.
               If JOB .EQ. 0, no pivoting is done.
               If JOB .NE. 0, pivoting is done.
    On Return
        A      A contains in its upper half the Cholesky factor
               of the matrix A as it has been permuted by pivoting.
        JPVT   JPVT(J) contains the index of the diagonal element
               of a that was moved into the J-th position,
               provided pivoting was requested.
        INFO   contains the index of the last positive diagonal
               element of the Cholesky factor.
    For positive definite matrices INFO = P is the normal return.
    For pivoting with positive semidefinite matrices INFO will
    in general be less than P.  However, INFO may be greater than
    the rank of A, since rounding error can cause an otherwise zero
    element to be positive.  Indefinite systems will always cause
    INFO to be less than P.
    LINPACK.  This version dated 03/19/79 .
    J.Dongarra J. and Stewart G. W., Argonne National Laboratory and
    University of Maryland.
    BLAS DAXPY,DSWAP
    Fortran DSQRT


DCHDD


      SUBROUTINE DCHDD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S,INFO)

    DCHDD downdates an augmented Cholesky decomposition or the
    triangular factor of an augmented QR decomposition.
    Specifically, given an upper triangular matrix R of order P,  a
    row vector X, a column vector Z, and a scalar Y, DCHDD
    determines an orthogonal matrix U and a scalar ZETA such that
                       (R   Z )     (RR  ZZ)
                   U * (      )  =  (      ) ,
                       (0 ZETA)     ( X   Y)
    where RR is upper triangular.  If R and Z have been obtained
    from the factorization of a least squares problem, then
    RR and ZZ are the factors corresponding to the problem
    with the observation (X,Y) removed.  In this case, if RHO
    is the norm of the residual vector, then the norm of
    the residual vector of the downdated problem is
    DSQRT(RHO**2 - ZETA**2).  DCHDD will simultaneously downdate
    several triplets (Z,Y,RHO) along with R.
    For a less terse description of what DCHDD does and how
    it may be applied, see the LINPACK guide.
    The matrix U is determined as the product U(1)*...*U(P)
    where U(I) is a rotation in the (P+1,I)-plane of the
    form
                      ( C(I)     -S(I)     )
                      (                    ) .
                      ( S(I)       C(I)    )
    The rotations are chosen so that C(I) is double precision.
    The user is warned that a given downdating problem may
    be impossible to accomplish or may produce
    inaccurate results.  For example, this can happen
    if X is near a vector whose removal will reduce the
    rank of R.  Beware.
    On Entry
        R      DOUBLE PRECISION(LDR,P), where LDR .GE. P.
               R contains the upper triangular matrix
               that is to be downdated.  The part of  R
               below the diagonal is not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        P      INTEGER.
               P is the order of the matrix R.
        X      DOUBLE PRECISION(P).
               X contains the row vector that is to
               be removed from R.  X is not altered by DCHDD.
        Z      DOUBLE PRECISION(LDZ,N)Z), where LDZ .GE. P.
               Z is an array of NZ P-vectors which
               are to be downdated along with R.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of vectors to be downdated
               NZ may be zero, in which case Z, Y, and RHO
               are not referenced.
        Y      DOUBLE PRECISION(NZ).
               Y contains the scalars for the downdating
               of the vectors Z.  Y is not altered by DCHDD.
        RHO    DOUBLE PRECISION(NZ).
               RHO contains the norms of the residual
               vectors that are to be downdated.
    On Return
        R
        Z      contain the downdated quantities.
        RHO
        C      DOUBLE PRECISION(P).
               C contains the cosines of the transforming
               rotations.
        S      DOUBLE PRECISION(P).
               S contains the sines of the transforming
               rotations.
        INFO   INTEGER.
               INFO is set as follows.
                  INFO = 0  if the entire downdating
                            was successful.
                  INFO =-1  if R could not be downdated.
                            in this case, all quantities
                            are left unaltered.
                  INFO = 1  if some RHO could not be
                            downdated.  The offending RHO's are
                            set to -1.
    LINPACK.  This version dated 08/14/78 .
    Stewart, G. W., University of Maryland, Argonne National Lab.
    DCHDD uses the following functions and subprograms.
    Fortran DABS
    BLAS DDOT, DNRM2


DCHEX


      SUBROUTINE DCHEX(R,LDR,P,K,L,Z,LDZ,NZ,C,S,JOB)

    DCHEX updates the Cholesky factorization
                  A = TRANS(R)*R
    of a positive definite matrix A of order P under diagonal
    permutations of the form
                  TRANS(E)*A*E
    where E is a permutation matrix.  Specifically, given
    an upper triangular matrix R and a permutation matrix
    E (which is specified by K, L, and JOB), DCHEX determines
    an orthogonal matrix U such that
                          U*R*E = RR,
    where RR is upper triangular.  At the users option, the
    transformation U will be multiplied into the array Z.
    If A = TRANS(X)*X, so that R is the triangular part of the
    QR factorization of X, then RR is the triangular part of the
    QR factorization of X*E, i.e. X with its columns permuted.
    For a less terse description of what DCHEX does and how
    it may be applied, see the LINPACK guide.
    The matrix Q is determined as the product U(L-K)*...*U(1)
    of plane rotations of the form
                          (    C(I)       S(I) )
                          (                    ) ,
                          (    -S(I)      C(I) )
    where C(I) is double precision.  The rows these rotations operate
    on are described below.
    There are two types of permutations, which are determined
    by the value of JOB.
    1. Right circular shift (JOB = 1).
        The columns are rearranged in the following order.
               1,...,K-1,L,K,K+1,...,L-1,L+1,...,P.
        U is the product of L-K rotations U(I), where U(I)
        acts in the (L-I,L-I+1)-plane.
    2. Left circular shift (JOB = 2).
        The columns are rearranged in the following order
               1,...,K-1,K+1,K+2,...,L,K,L+1,...,P.
        U is the product of L-K rotations U(I), where U(I)
        acts in the (K+I-1,K+I)-plane.
    On Entry
        R      DOUBLE PRECISION(LDR,P), where LDR .GE. P.
               R contains the upper triangular factor
               that is to be updated.  Elements of R
               below the diagonal are not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        P      INTEGER.
               P is the order of the matrix R.
        K      INTEGER.
               K is the first column to be permuted.
        L      INTEGER.
               L is the last column to be permuted.
               L must be strictly greater than K.
        Z      DOUBLE PRECISION(LDZ,N)Z), where LDZ .GE. P.
               Z is an array of NZ P-vectors into which the
               transformation U is multiplied.  Z is
               not referenced if NZ = 0.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of columns of the matrix Z.
        JOB    INTEGER.
               JOB determines the type of permutation.
                      JOB = 1  right circular shift.
                      JOB = 2  left circular shift.
    On Return
        R      contains the updated factor.
        Z      contains the updated matrix Z.
        C      DOUBLE PRECISION(P).
               C contains the cosines of the transforming rotations.
        S      DOUBLE PRECISION(P).
               S contains the sines of the transforming rotations.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    DCHEX uses the following functions and subroutines.
    BLAS DROTG
    Fortran MIN0


DCHUD


      SUBROUTINE DCHUD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S)

    DCHUD updates an augmented Cholesky decomposition of the
    triangular part of an augmented QR decomposition.  Specifically,
    given an upper triangular matrix R of order P, a row vector
    X, a column vector Z, and a scalar Y, DCHUD determines a
    untiary matrix U and a scalar ZETA such that
                             (R  Z)     (RR   ZZ )
                        U  * (    )  =  (        ) ,
                             (X  Y)     ( 0  ZETA)
    where RR is upper triangular.  If R and Z have been
    obtained from the factorization of a least squares
    problem, then RR and ZZ are the factors corresponding to
    the problem with the observation (X,Y) appended.  In this
    case, if RHO is the norm of the residual vector, then the
    norm of the residual vector of the updated problem is
    DSQRT(RHO**2 + ZETA**2).  DCHUD will simultaneously update
    several triplets (Z,Y,RHO).
    For a less terse description of what DCHUD does and how
    it may be applied, see the LINPACK guide.
    The matrix U is determined as the product U(P)*...*U(1),
    where U(I) is a rotation in the (I,P+1) plane of the
    form
                      (     C(I)      S(I) )
                      (                    ) .
                      (    -S(I)      C(I) )
    The rotations are chosen so that C(I) is double precision.
    On Entry
        R      DOUBLE PRECISION(LDR,P), where LDR .GE. P.
               R contains the upper triangular matrix
               that is to be updated.  The part of R
               below the diagonal is not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        P      INTEGER.
               P is the order of the matrix R.
        X      DOUBLE PRECISION(P).
               X contains the row to be added to R.  X is
               not altered by DCHUD.
        Z      DOUBLE PRECISION(LDZ,N)Z), where LDZ .GE. P.
               Z is an array containing NZ P-vectors to
               be updated with R.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of vectors to be updated
               NZ may be zero, in which case Z, Y, and RHO
               are not referenced.
        Y      DOUBLE PRECISION(NZ).
               Y contains the scalars for updating the vectors
               Z.  Y is not altered by DCHUD.
        RHO    DOUBLE PRECISION(NZ).
               RHO contains the norms of the residual
               vectors that are to be updated.  If RHO(J)
               is negative, it is left unaltered.
    On Return
        RC
        RHO    contain the updated quantities.
        Z
        C      DOUBLE PRECISION(P).
               C contains the cosines of the transforming
               rotations.
        S      DOUBLE PRECISION(P).
               S contains the sines of the transforming
               rotations.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    DCHUD uses the following functions and subroutines.
    Extended BLAS DROTG
    Fortran DSQRT


DCOPY


      SUBROUTINE DCOPY(N,DX,INCX,DY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
      DY  double precision vector with N elements
    INCY  storage spacing between elements of DY
    --Output--
      DY  copy of vector DX (unchanged if N .LE. 0)
    Copy double precision DX to double precision DY.
    For I = 0 to N-1, copy DX(LX+I*INCX) to DY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


DCOPYM


      SUBROUTINE DCOPYM(N,DX,INCX,DY,INCY)

      Description of Parameters
          The * Flags Output Variables
      N   Number of elements in vector(s)
     DX   Double precision vector with N elements
   INCX   Storage spacing between elements of DX
     DY*  Double precision negative copy of DX
   INCY   Storage spacing between elements of DY
     ***  Note that DY = -DX  ***
 Copy negative of d.p. DX to d.p. DY.  For I=0 to N-1,
  copy  -DX(LX+I*INCX) to DY(LY+I*INCY), where LX=1 if
  INCX .GE. 0, else LX = (-INCX)*N, and LY is defined
  in a similar way using INCY.


DDOT


      DOUBLE PRECISION FUNCTION DDOT(N,DX,INCX,DY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
      DY  double precision vector with N elements
    INCY  storage spacing between elements of DY
    --Output--
    DDOT  double precision dot product (zero if N .LE. 0)
    Returns the dot product of double precision DX and DY.
    DDOT = sum for I = 0 to N-1 of  DX(LX+I*INCX) * DY(LY+I*INCY)
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


DGBCO


      SUBROUTINE DGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)

    DGBCO factors a double precision band matrix by Gaussian
    elimination and estimates the condition of the matrix.
    If  RCOND  is not needed, DGBFA is slightly faster.
    To solve  A*X = B , follow DGBCO by DGBSL.
    To compute  INVERSE(A)*C , follow DGBCO by DGBSL.
    To compute  DETERMINANT(A) , follow DGBCO by DGBDI.
    On Entry
       ABD     DOUBLE PRECISION(LDA, N)
               contains the matrix in band storage.  The columns
               of the matrix are stored in the columns of  ABD  and
               the diagonals of the matrix are stored in rows
               ML+1 through 2*ML+MU+1 of  ABD .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. 2*ML + MU + 1 .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT.  N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT.  N .
               More efficient if  ML .LE. MU .
    On Return
       ABD     an upper triangular matrix in band storage and
               the multipliers which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  M = ML + MU + 1
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-MU)
                     I2 = MIN0(N, J+ML)
                     DO 10 I = I1, I2
                        K = I - J + M
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses rows  ML+1  through  2*ML+MU+1  of  ABD .
          In addition, the first  ML  rows in  ABD  are used for
          elements generated during the triangularization.
          The total number of rows needed in  ABD  is  2*ML+MU+1 .
          The  ML+MU by ML+MU  upper left triangle and the
          ML by ML  lower right triangle are not referenced.
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABD should contain
           *  *  *  +  +  +  , * = not used
           *  * 13 24 35 46  , + = used for pivoting
           * 12 23 34 45 56
          11 22 33 44 55 66
          21 32 43 54 65  *
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and functions used:
    LINPACK DGBFA
    BLAS DAXPY,DDOT,DSCAL,DASUM
    Fortran DABS,DMAX1,MAX0,MIN0,DSIGN


DGBDI


      SUBROUTINE DGBDI(ABD,LDA,N,ML,MU,IPVT,DET)

    DGBDI computes the determinant of a band matrix
    using the factors computed by DGBCO or DGBFA.
    If the inverse is needed, use DGBSL  N  times.
    On Entry
       ABD     DOUBLE PRECISION(LDA, N)
               the output from DGBCO or DGBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from DGBCO or DGBFA.
    On Return
       DET     DOUBLE PRECISION(2)
               determinant of original matrix.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DABS(DET(1)) .LT. 10.0
               or  DET(1) = 0.0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    Fortran DABS


DGBFA


      SUBROUTINE DGBFA(ABD,LDA,N,ML,MU,IPVT,INFO)

    DGBFA factors a double precision band matrix by elimination.
    DGBFA is usually called by DGBCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    On Entry
       ABD     DOUBLE PRECISION(LDA, N)
               contains the matrix in band storage.  The columns
               of the matrix are stored in the columns of  ABD  and
               the diagonals of the matrix are stored in rows
               ML+1 through 2*ML+MU+1 of  ABD .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. 2*ML + MU + 1 .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT.  N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT.  N .
               More efficient if  ML .LE. MU .
    On Return
       ABD     an upper triangular matrix in band storage and
               the multipliers which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if  U(K,K) .EQ. 0.0 .  This is not an error
                    condition for this subroutine, but it does
                    indicate that DGBSL will divide by zero if
                    called.  Use  RCOND  in DGBCO for a reliable
                    indication of singularity.
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  M = ML + MU + 1
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-MU)
                     I2 = MIN0(N, J+ML)
                     DO 10 I = I1, I2
                        K = I - J + M
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses rows  ML+1  through  2*ML+MU+1  of  ABD .
          In addition, the first  ML  rows in  ABD  are used for
          elements generated during the triangularization.
          The total number of rows needed in  ABD  is  2*ML+MU+1 .
          The  ML+MU by ML+MU  upper left triangle and the
          ML by ML  lower right triangle are not referenced.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DSCAL,IDAMAX
    Fortran MAX0,MIN0


DGBSL


      SUBROUTINE DGBSL(ABD,LDA,N,ML,MU,IPVT,B,JOB)

    DGBSL solves the double precision band system
    A * X = B  or  TRANS(A) * X = B
    using the factors computed by DGBCO or DGBFA.
    On Entry
       ABD     DOUBLE PRECISION(LDA, N)
               the output from DGBCO or DGBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from DGBCO or DGBFA.
       B       DOUBLE PRECISION(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B ,
               = nonzero   to solve  TRANS(A)*X = B , where
                           TRANS(A)  is the transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically this indicates singularity
       but it is often caused by improper arguments or improper
       setting of LDA .  It will not occur if the subroutines are
       called correctly and if DGBCO has set RCOND .GT. 0.0
       or DGBFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL DGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)
          IF (RCOND is too small) GO TO ...
          DO 10 J = 1, P
             CALL DGBSL(ABD,LDA,N,ML,MU,IPVT,C(1,J),0)
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DDOT
    Fortran MIN0


DGECO


      SUBROUTINE DGECO(A,LDA,N,IPVT,RCOND,Z)

    DGECO factors a double precision matrix by Gaussian elimination
    and estimates the condition of the matrix.
    If  RCOND  is not needed, DGEFA is slightly faster.
    To solve  A*X = B , follow DGECO by DGESL.
    To compute  INVERSE(A)*C , follow DGECO by DGESL.
    To compute  DETERMINANT(A) , follow DGECO by DGEDI.
    To compute  INVERSE(A) , follow DGECO by DGEDI.
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the matrix to be factored.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix and the multipliers
               which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an INTEGER vector of pivot indices.
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK DGEFA
    BLAS DAXPY,DDOT,DSCAL,DASUM
    Fortran DABS,DMAX1,DSIGN


DGEDI


      SUBROUTINE DGEDI(A,LDA,N,IPVT,DET,WORK,JOB)

    DGEDI computes the determinant and inverse of a matrix
    using the factors computed by DGECO or DGEFA.
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the output from DGECO or DGEFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       IPVT    INTEGER(N)
               the pivot vector from DGECO or DGEFA.
       WORK    DOUBLE PRECISION(N)
               work vector.  Contents destroyed.
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       A       inverse of original matrix if requested.
               Otherwise unchanged.
       DET     DOUBLE PRECISION(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DABS(DET(1)) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if DGECO has set RCOND .GT. 0.0 or DGEFA has set
       INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DSCAL,DSWAP
    Fortran DABS,MOD


DGEFA


      SUBROUTINE DGEFA(A,LDA,N,IPVT,INFO)

    DGEFA factors a double precision matrix by Gaussian elimination.
    DGEFA is usually called by DGECO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (Time for DGECO) = (1 + 9/N)*(Time for DGEFA) .
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the matrix to be factored.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix and the multipliers
               which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if  U(K,K) .EQ. 0.0 .  This is not an error
                    condition for this subroutine, but it does
                    indicate that DGESL or DGEDI will divide by zero
                    if called.  Use  RCOND  in DGECO for a reliable
                    indication of singularity.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DSCAL,IDAMAX


DGEFS


      SUBROUTINE DGEFS(A,LDA,N,V,ITASK,IND,WORK,IWORK)

   Subroutine DGEFS solves a general NxN system of double
   precision linear equations using LINPACK subroutines DGECO
   and DGESL.  That is, if A is an NxN double precision matrix
   and if X and B are double precision N-vectors, then DGEFS
   solves the equation
                         A*X=B.
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to find the
   solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK.GT.1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, N and IWORK must not have been altered by the user follow-
   ing factorization (ITASK=1).  IND will not be changed by DGEFS
   in this case.
 Argument Description ***
   A      DOUBLE PRECISION(LDA,N)
            on entry, the doubly subscripted array with dimension
              (LDA,N) which contains the coefficient matrix.
            on return, an upper triangular matrix U and the
              multipliers necessary to construct a matrix L
              so that A=L*U.
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  The first N elements of
            the array A are the elements of the first column of
            the matrix A.  N must be greater than or equal to 1.
            (terminal error message IND=-2)
   V      DOUBLE PRECISION(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A and IWORK.
            If ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT. 0  see error message corresponding to IND below.
   WORK   DOUBLE PRECISION(N)
            a singly subscripted array of dimension at least N.
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   The matrix A is computationally singular.
                        A solution has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              Note-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


DGESL


      SUBROUTINE DGESL(A,LDA,N,IPVT,B,JOB)

    DGESL solves the double precision system
    A * X = B  or  TRANS(A) * X = B
    using the factors computed by DGECO or DGEFA.
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the output from DGECO or DGEFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       IPVT    INTEGER(N)
               the pivot vector from DGECO or DGEFA.
       B       DOUBLE PRECISION(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B ,
               = nonzero   to solve  TRANS(A)*X = B  where
                           TRANS(A)  is the transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically this indicates singularity
       but it is often caused by improper arguments or improper
       setting of LDA .  It will not occur if the subroutines are
       called correctly and if DGECO has set RCOND .GT. 0.0
       or DGEFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL DGECO(A,LDA,N,IPVT,RCOND,Z)
          IF (RCOND is too small) GO TO ...
          DO 10 J = 1, P
             CALL DGESL(A,LDA,N,IPVT,C(1,J),0)
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DDOT


DGLSS


      SUBROUTINE DGLSS(A,MDA,M,N,B,MDB,NB,RNORM,WORK,LW,IWORK,LIW,INFO)

  **** Double Precision version of SGLSS ****
    DGLSS solves both underdetermined and overdetermined
    LINEAR systems AX = B, where A is an M by N matrix
    and B is an M by NB matrix of right hand sides. If
    M.GE.N, the least squares solution is computed by
    decomposing the matrix A into the product of an
    orthogonal matrix Q and an upper triangular matrix
    R (QR factorization). If M.LT.N, the minimal
    length solution is computed by factoring the
    matrix A into the product of a lower triangular
    matrix L and an orthogonal matrix Q (LQ factor-
    ization). If the matrix A is determined to be rank
    deficient, that is the rank of A is less than
    MIN(M,N), then the minimal length least squares
    solution is computed.
    DGLSS assumes full machine precision in the data.
    If more control over the uncertainty in the data
    is desired, the codes DLLSIA and DULSIA are
    recommended.
    DGLSS requires MDA*N + (MDB + 1)*NB + 5*MIN(M,N) dimensioned
    real space and M+N dimensioned integer space.
  ******************************************************************
  *                                                                *
  *         WARNING - All input arrays are changed on exit.        *
  *                                                                *
  ******************************************************************
    SUBROUTINE DGLSS(A,MDA,M,N,B,MDB,NB,RNORM,WORK,LW,IWORK,LIW,INFO)
    Input..All TYPE REAL variables are DOUBLE PRECISION
    A(,)          Linear coefficient matrix of AX=B, with MDA the
     MDA,M,N      actual first dimension of A in the calling program.
                  M is the row dimension (no. of EQUATIONS of the
                  problem) and N the col dimension (no. of UNKNOWNS).
    B(,)          Right hand side(s), with MDB the actual first
     MDB,NB       dimension of B in the calling program. NB is the
                  number of M by 1 right hand sides. Must have
                  MDB.GE.MAX(M,N). If NB = 0, B is never accessed.
    RNORM()       Vector of length at least NB.  On input the contents
                  of RNORM are unused.
    WORK()        A real work array dimensioned 5*MIN(M,N).
    LW            Actual dimension of WORK.
    IWORK()       Integer work array dimensioned at least N+M.
    LIW           Actual dimension of IWORK.
    INFO          A flag which provides for the efficient
                  solution of subsequent problems involving the
                  same A but different B.
                  If INFO = 0 original call
                     INFO = 1 subsequent calls
                  On subsequent calls, the user must supply A, INFO,
                  LW, IWORK, LIW, and the first 2*MIN(M,N) locations
                  of WORK as output by the original call to DGLSS.
    Output..All TYPE REAL variables are DOUBLE PRECISION
    A(,)          Contains the triangular part of the reduced matrix
                  and the transformation information. It together with
                  the first 2*MIN(M,N) elements of WORK (see below)
                  completely specify the factorization of A.
    B(,)          Contains the N by NB solution matrix X.
    RNORM()       Contains the Euclidean length of the NB residual
                  vectors  B(I)-AX(I), I=1,NB.
    WORK()        The first 2*MIN(M,N) locations of WORK contain value
                  necessary to reproduce the factorization of A.
    IWORK()       The first M+N locations contain the order in
                  which the rows and columns of A were used.
                  If M.GE.N columns then rows. If M.LT.N rows
                  then columns.
    INFO          Flag to indicate status of computation on completion
                 -1   Parameter error(s)
                  0 - Full rank
                  N.GT.0 - Reduced rank  rank=MIN(M,N)-INFO


DGTSL


      SUBROUTINE DGTSL(N,C,D,E,B,INFO)

    DGTSL given a general tridiagonal matrix and a right hand
    side will find the solution.
    On Entry
       N       INTEGER
               is the order of the tridiagonal matrix.
       C       DOUBLE PRECISION(N)
               is the subdiagonal of the tridiagonal matrix.
               C(2) through C(N) should contain the subdiagonal.
               On output C is destroyed.
       D       DOUBLE PRECISION(N)
               is the diagonal of the tridiagonal matrix.
               On output D is destroyed.
       E       DOUBLE PRECISION(N)
               is the superdiagonal of the tridiagonal matrix.
               E(1) through E(N-1) should contain the superdiagonal.
               On output E is destroyed.
       B       DOUBLE PRECISION(N)
               is the right hand side vector.
    On Return
       B       is the solution vector.
       INFO    INTEGER
               = 0 normal value.
               = K if the K-th element of the diagonal becomes
                   exactly zero.  The subroutine returns when
                   this is detected.
    LINPACK.  This version dated 08/14/78 .
    Jack Dongarra, Argonne National Laboratory.
    No externals
    Fortran DABS


DH12


      SUBROUTINE DH12(MODE,LPIVOT,L1,M,U,IUE,UP,C,ICE,ICV,NCV)

     *** DOUBLE PRECISION VERSION OF H12 ******
    SUBROUTINE DH12 (MODE,LPIVOT,L1,M,U,IUE,UP,C,ICE,ICV,NCV)
    C.L.Lawson and R.J.Hanson, Jet Propulsion Laboratory, 1973 Jun 12
    to appear in 'Solving Least Squares Problems', Prentice-Hall, 1974
    Modified at SANDIA LABS, May 1977, to --
    1)  Remove double precision accumulation, and
    2)  Include usage of the Basic Linear Algebra Package for
        vectors longer than a particular threshold.
    Construction and/or application of a single
    Householder transformation..     Q = I + U*(U**T)/B
    MODE    = 1 or 2   to select algorithm  H1  or  H2 .
    LPIVOT is the index of the pivot element.
    L1,M   If L1 .LE. M   the transformation will be constructed to
           zero elements indexed from L1 through M.   If L1 GT. M
           THE SUBROUTINE DOES AN IDENTITY TRANSFORMATION.
    U(),IUE,UP    On entry to H1 U() contains the pivot vector.
                  IUE is the storage increment between elements.
                                      On exit from H1 U() and UP
                  contain quantities defining the vector U of the
                  Householder transformation.   On entry to H2 U()
                  and UP should contain quantities previously computed
                  by H1.  These will not be modified by H2.
    C()    On entry to H1 or H2 C() contains a matrix which will be
           regarded as a set of vectors to which the Householder
           transformation is to be applied.  On exit C() contains the
           set of transformed vectors.
    ICE    Storage increment between elements of vectors in C().
    ICV    Storage increment between vectors in C().
    NCV    Number of vectors in C() to be transformed. If NCV .LE. 0
           no operations will be done on C().


DHFTI


      SUBROUTINE DHFTI(A,MDA,M,N,B,MDB,NB,TAU,KRANK,RNORM,H,G,IP)

      ***** Double Precision version of HFTI *****
    DIMENSION A(MDA,N),(B(MDB,NB) or B(M)),RNORM(NB),H(N),G(N),IP(N)
    This subroutine solves a linear least squares problem or a set of
    linear least squares problems having the same matrix but different
    right-side vectors.  The problem data consists of an M by N matrix
    A, an M by NB matrix B, and an absolute tolerance parameter TAU
    whose usage is described below.  The NB column vectors of B
    represent right-side vectors for NB distinct linear least squares
    problems.
    This set of problems can also be written as the matrix least
    squares problem
                      AX = B,
    where X is the N by NB solution matrix.
    Note that if B is the M by M identity matrix, then X will be the
    pseudo-inverse of A.
    This subroutine first transforms the augmented matrix (A B) to a
    matrix (R C) using premultiplying Householder transformations with
    column interchanges.  All subdiagonal elements in the matrix R are
    zero and its diagonal elements satisfy
                      DABS(R(I,I)).GE.DABS(R(I+1,I+1)),
                      I = 1,...,L-1, where
                      L = MIN(M,N).
    The subroutine will compute an integer, KRANK, equal to the number
    of diagonal terms of R that exceed TAU in magnitude. Then a
    solution of minimum Euclidean length is computed using the first
    KRANK rows of (R C).
    To be specific we suggest that the user consider an easily
    computable matrix norm, such as, the maximum of all column sums of
    magnitudes.
    Now if the relative uncertainty of B is EPS, (norm of uncertainty/
    norm of B), it is suggested that TAU be set approximately equal to
    EPS*(norm of A).
    The user must dimension all arrays appearing in the call list..
    A(MDA,N),(B(MDB,NB) or B(M)),RNORM(NB),H(N),G(N),IP(N).  This
    permits the solution of a range of problems in the same array
    space.
    The entire set of parameters for DHFTI are
    INPUT.. All TYPE REAL variables are DOUBLE PRECISION
    A(*,*),MDA,M,N    The array A(*,*) initially contains the M by N
                      matrix A of the least squares problem AX = B.
                      The first dimensioning parameter of the array
                      A(*,*) is MDA, which must satisfy MDA.GE.M
                      Either M.GE.N or M.LT.N is permitted.  There
                      is no restriction on the rank of A.  The
                      condition MDA.LT.M is considered an error.
    B(*),MDB,NB       If NB = 0 the subroutine will perform the
                      orthogonal decomposition but will make no
                      references to the array B(*).  If NB.GT.0
                      the array B(*) must initially contain the M by
                      NB matrix B of the least squares problem AX =
                      B.  If NB.GE.2 the array B(*) must be doubly
                      subscripted with first dimensioning parameter
                      MDB.GE.MAX(M,N).  If NB = 1 the array B(*) may
                      be either doubly or singly subscripted.  In
                      the latter case the value of MDB is arbitrary
                      but it should be set to some valid integer
                      value such as MDB = M.
                      The condition of NB.GT.1.AND.MDB.LT. MAX(M,N)
                      is considered an error.
    TAU               Absolute tolerance parameter provided by user
                      for pseudorank determination.
    H(*),G(*),IP(*)   Arrays of working space used by DHFTI.
    OUTPUT.. All TYPE REAL variables are DOUBLE PRECISION
    A(*,*)            The contents of the array A(*,*) will be
                      modified by the subroutine. These contents
                      are not generally required by the user.
    B(*)              On return the array B(*) will contain the N by
                      NB solution matrix X.
    KRANK             Set by the subroutine to indicate the
                      pseudorank of A.
    RNORM(*)          On return, RNORM(J) will contain the Euclidean
                      norm of the residual vector for the problem
                      defined by the J-th column vector of the array
                      B(*,*) for J = 1,...,NB.
    H(*),G(*)         On return these arrays respectively contain
                      elements of the pre- and post-multiplying
                      Householder transformations used to compute
                      the minimum Euclidean length solution.
    IP(*)             Array in which the subroutine records indices
                      describing the permutation of column vectors.
                      The contents of arrays H(*),G(*) and IP(*)
                      are not generally required by the user.


DHKSEQ


      SUBROUTINE DHKSEQ(X,M,H,IERR)

     DHKSEQ is an adaptation of subroutine DPSIFN, ACM Trans. Math.
     Software, 1983 to generate H(K,X)=(-X)**(K+1)*(PSI(K,X)-
     -PSI(K,X+0.5))/GAMMA(K+1) for K=0,...,M.


DHSTRT


      SUBROUTINE DHSTRT(DF,NEQ,A,B,Y,YPRIME,ETOL,MORDER,SMALL,BIG,SPY,



DIFF


      FUNCTION DIFF(X,Y)



DLLSIA


      SUBROUTINE DLLSIA(A,MDA,M,N,B,MDB,NB,RE,AE,KEY,MODE,NP,KRANK,

  **** Double Precision version of LLSIA ****
    DLLSIA computes the least squares solution(s) to the problem AX=B
    where A is an M by N matrix with M.GE.N and B is the M by NB
    matrix of right hand sides.  User input bounds on the uncertainty
    in the elements of A are used to detect numerical rank deficiency.
    The algorithm employs a row and column pivot strategy to
    minimize the growth of uncertainty and round-off errors.
    DLLSIA requires (MDA+6)*N + (MDB+1)*NB + M dimensioned space
  ******************************************************************
  *                                                                *
  *         WARNING - All input arrays are changed on exit.        *
  *                                                                *
  ******************************************************************
    SUBROUTINE DLLSIA(A,MDA,M,N,B,MDB,NB,RE,AE,KEY,MODE,NP,
   1   KRANK,KSURE,RNORM,W,LW,IWORK,LIW,INFO)
    Input..All TYPE REAL variables are DOUBLE PRECISION
    A(,)          Linear coefficient matrix of AX=B, with MDA the
     MDA,M,N      actual first dimension of A in the calling program.
                  M is the row dimension (no. of EQUATIONS of the
                  problem) and N the col dimension (no. of UNKNOWNS).
                  Must have MDA.GE.M and M.GE.N.
    B(,)          Right hand side(s), with MDB the actual first
     MDB,NB       dimension of B in the calling program. NB is the
                  number of M by 1 right hand sides. Must have
                  MDB.GE.M. If NB = 0, B is never accessed.
  ******************************************************************
  *                                                                *
  *         Note - Use of RE and AE are what make this             *
  *                code significantly different from               *
  *                other linear least squares solvers.             *
  *                However, the inexperienced user is              *
  *                advised to set RE=0.,AE=0.,KEY=0.               *
  *                                                                *
  ******************************************************************
    RE(),AE(),KEY
    RE()          RE() is a vector of length N such that RE(I) is
                  the maximum relative uncertainty in column I of
                  the matrix A. The values of RE() must be between
                  0 and 1. A minimum of 10*machine precision will
                  be enforced.
    AE()          AE() is a vector of length N such that AE(I) is
                  the maximum absolute uncertainty in column I of
                  the matrix A. The values of AE() must be greater
                  than or equal to 0.
    KEY           For ease of use, RE and AE may be input as either
                  vectors or scalars. If a scalar is input, the algo-
                  rithm will use that value for each column of A.
                  The parameter key indicates whether scalars or
                  vectors are being input.
                       KEY=0     RE scalar  AE scalar
                       KEY=1     RE vector  AE scalar
                       KEY=2     RE scalar  AE vector
                       KEY=3     RE vector  AE vector
    MODE          The integer mode indicates how the routine
                  is to react if rank deficiency is detected.
                  If MODE = 0 return immediately, no solution
                            1 compute truncated solution
                            2 compute minimal length solution
                  The inexperienced user is advised to set MODE=0
    NP            The first NP columns of A will not be interchanged
                  with other columns even though the pivot strategy
                  would suggest otherwise.
                  The inexperienced user is advised to set NP=0.
    WORK()        A real work array dimensioned 5*N. However,if
                  RE or AE have been specified as vectors, dimension
                  WORK 4*N. If both RE and AE have been specified
                  as vectors, dimension WORK 3*N.
    LW            Actual dimension of WORK
    IWORK()       Integer work array dimensioned at least N+M.
    LIW           Actual dimension of IWORK.
    INFO          Is a flag which provides for the efficient
                  solution of subsequent problems involving the
                  same A but different B.
                  If INFO = 0 original call
                     INFO = 1 subsequent calls
                  On subsequent calls, the user must supply A, KRANK,
                  LW, IWORK, LIW, and the first 2*N locations of WORK
                  as output by the original call to DLLSIA. MODE must
                  be equal to the value of MODE in the original call.
                  If MODE.LT.2, only the first N locations of WORK
                  are accessed. AE, RE, KEY, and NP are not accessed.
    Output..All TYPE REAL variable are DOUBLE PRECISION
    A(,)          Contains the upper triangular part of the reduced
                  matrix and the transformation information. It togeth
                  with the first N elements of WORK (see below)
                  completely specify the QR factorization of A.
    B(,)          Contains the N by NB solution matrix for X.
    KRANK,KSURE   The numerical rank of A,  based upon the relative
                  and absolute bounds on uncertainty, is bounded
                  above by KRANK and below by KSURE. The algorithm
                  returns a solution based on KRANK. KSURE provides
                  an indication of the precision of the rank.
    RNORM()       Contains the Euclidean length of the NB residual
                  vectors  B(I)-AX(I), I=1,NB.
    WORK()        The first N locations of WORK contain values
                  necessary to reproduce the Householder transformatio
    IWORK()       The first N locations contain the order in
                  which the columns of A were used. The next
                  M locations contain the order in which the
                  rows of A were used.
    INFO          Flag to indicate status of computation on completion
                 -1   Parameter error(s)
                  0 - Rank deficient, no solution
                  1 - Rank deficient, truncated solution
                  2 - Rank deficient, minimal length solution
                  3 - Numerical rank 0, zero solution
                  4 - Rank .LT. NP
                  5 - Full rank


DNBCO


      SUBROUTINE DNBCO(ABE,LDA,N,ML,MU,IPVT,RCOND,Z)

    DNBCO factors a double precision band matrix by Gaussian
    elimination and estimates the condition of the matrix.
    If RCOND is not needed, DNBFA is slightly faster.
    To solve  A*X = B , follow DNBCO by DNBSL.
    To compute  INVERSE(A)*C , follow DNBCO by DNBSL.
    To compute  DETERMINANT(A) , follow DNBCO by DNBDI.
    On Entry
       ABE     DOUBLE PRECISION(LDA, NC)
               contains the matrix in band storage.  The rows
               of the original matrix are stored in the rows
               of ABE and the diagonals of the original matrix
               are stored in columns 1 through ML+MU+1 of ABE.
               NC must be .GE. 2*ML+MU+1 .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array ABE.
               LDA must be .GE.  N .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT.  N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT.  N .
               More efficient if ML .LE. MU .
    On Return
       ABE     an upper triangular matrix in band storage
               and the multipliers which were used to obtain it.
               The factorization can be written  A = L*U  where
               L is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                        1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  DO 20 I = 1, N
                     J1 = MAX0(1, I-ML)
                     J2 = MIN0(N, I+MU)
                     DO 10 J = J1, J2
                        K = J - I + ML + 1
                        ABE(I,K) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses columns  1  through  ML+MU+1  of ABE .
          Furthermore,  ML  additional columns are needed in
          ABE  starting with column  ML+MU+2  for elements
          generated during the triangularization.  The total
          number of columns needed in  ABE  is  2*ML+MU+1 .
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
           * 11 12 13  +     , * = not used
          21 22 23 24  +     , + = used for pivoting
          32 33 34 35  +
          43 44 45 46  +
          54 55 56  *  +
          65 66  *  *  +
    SLATEC.  This version dated 07/28/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
    SLATEC DNBFA
     DDOT,DAXPY,DSCAL,DASUM
    Fortran  DABS,DMAX1,MAX0,MIN0,DSIGN


DNBDI


      SUBROUTINE DNBDI(ABE,LDA,N,ML,MU,IPVT,DET)

    DNBDI computes the determinant of a band matrix
    using the factors computed by DNBCO or DNBFA.
    If the inverse is needed, use DNBSL  N  times.
    On Entry
       ABE     DOUBLE PRECISION(LDA, NC)
               the output from DNBCO or DNBFA.
               NC must be .GE. 2*ML+MU+1 .
       LDA     INTEGER
               the leading dimension of the array  ABE .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from DNBCO or DNBFA.
    On Return
       DET     DOUBLE PRECISION(2)
               determinant of original matrix.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DABS(DET(1)) .LT. 10.0
               or  DET(1) = 0.0 .
    SLATEC.  This version dated 07/28/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
    Fortran DABS


DNBFA


      SUBROUTINE DNBFA(ABE,LDA,N,ML,MU,IPVT,INFO)

    DNBFA factors a double precision band matrix by elimination.
    DNBFA is usually called by DNBCO, but it can be called
    directly with a saving in time if RCOND is not needed.
    On Entry
       ABE     DOUBLE PRECISION(LDA, NC)
               contains the matrix in band storage.  The rows
               of the original matrix are stored in the rows
               of ABE and the diagonals of the original matrix
               are stored in columns 1 through ML+MU+1 of ABE.
               NC must be .GE. 2*ML+MU+1 .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array ABE.
               LDA must be .GE.  N .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT.  N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT.  N .
               More efficient if ML .LE. MU .
    On Return
       ABE     an upper triangular matrix in band storage
               and the multipliers which were used to obtain it.
               The factorization can be written  A = L*U  where
               L is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               =0  normal value
               =K  if  U(K,K) .EQ. 0.0 .  This is not an error
               condition for this subroutine, but it does
               indicate that DNBSL will divide by zero if
               called.  Use RCOND in DNBCO for a reliable
               indication of singularity.
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  DO 20 I = 1, N
                     J1 = MAX0(1, I-ML)
                     J2 = MIN0(N, I+MU)
                     DO 10 J = J1, J2
                        K = J - I + ML + 1
                        ABE(I,K) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses columns  1  through  ML+MU+1  of ABE .
          Furthermore,  ML  additional columns are needed in
          ABE  starting with column  ML+MU+2  for elements
          generated during the triangularization.  The total
          number of columns needed in  ABE  is  2*ML+MU+1 .
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
           * 11 12 13  +     , * = not used
          21 22 23 24  +     , + = used for pivoting
          32 33 34 35  +
          43 44 45 46  +
          54 55 56  *  +
          65 66  *  *  +
    SLATEC.  This version dated 07/28/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
     IDAMAX,DAXPY,DSCAL,DSWAP
    Fortran  MIN0


DNBFS


      SUBROUTINE DNBFS(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK)

   Subroutine DNBFS solves a general nonsymmetric banded NxN
   system of double precision real linear equations using
   SLATEC subroutines DNBCO and DNBSL.  These are adaptations
   of the LINPACK subroutines DGBCO and DGBSL which require
   a different format for storing the matrix elements.  If
   A  is an NxN double precision matrix and if  X  and  B  are
   double precision N-vectors, then DNBFS solves the equation
                         A*X=B.
   A band matrix is a matrix whose nonzero elements are all
   fairly near the main diagonal, specifically  A(I,J) = 0
   if  I-J is greater than  ML  or  J-I  is greater than
   MU .  The integers ML and MU are called the lower and upper
   band widths and  M = ML+MU+1  is the total band width.
   DNBFS uses less time and storage than the corresponding
   program for general matrices (DGEFS) if 2*ML+MU .LT.  N .
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to find the
   solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, N and IWORK must not have been altered by the user follow-
   ing factorization (ITASK=1).  IND will not be changed by DNBFS
   in this case.
   Band Storage
         If  A  is a band matrix, the following program segment
         will set up the input.
                 ML = (band width below the diagonal)
                 MU = (band width above the diagonal)
                 DO 20 I = 1, N
                    J1 = MAX0(1, I-ML)
                    J2 = MIN0(N, I+MU)
                    DO 10 J = J1, J2
                       K = J - I + ML + 1
                       ABE(I,K) = A(I,J)
              10    CONTINUE
              20 CONTINUE
         This uses columns  1  through  ML+MU+1  of ABE .
         Furthermore,  ML  additional columns are needed in
         ABE  starting with column  ML+MU+2  for elements
         generated during the triangularization.  The total
         number of columns needed in  ABE  is  2*ML+MU+1 .
   Example:  If the original matrix is
         11 12 13  0  0  0
         21 22 23 24  0  0
          0 32 33 34 35  0
          0  0 43 44 45 46
          0  0  0 54 55 56
          0  0  0  0 65 66
    then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
          * 11 12 13  +     , * = not used
         21 22 23 24  +     , + = used for pivoting
         32 33 34 35  +
         43 44 45 46  +
         54 55 56  *  +
         65 66  *  *  +
 Argument Description ***
   ABE    DOUBLE PRECISION(LDA,NC)
            on entry, contains the matrix in band storage as
              described above.  NC  must not be less than
              2*ML+MU+1 .  The user is cautioned to specify  NC
              with care since it is not an argument and cannot
              be checked by DNBFS.  The rows of the original
              matrix are stored in the rows of  ABE  and the
              diagonals of the original matrix are stored in
              columns  1  through  ML+MU+1  of  ABE .
            on return, contains an upper triangular matrix U and
              the multipliers necessary to construct a matrix L
              so that A=L*U.
   LDA    INTEGER
            the leading dimension of array ABE.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater
            than or equal to 1 .  (terminal error message IND=-2)
   ML     INTEGER
            the number of diagonals below the main diagonal.
            ML  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-5)
   MU     INTEGER
            the number of diagonals above the main diagonal.
            MU  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-6)
   V      DOUBLE PRECISION(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A and IWORK.
            If ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT. 0  See error message corresponding to IND below.
   WORK   DOUBLE PRECISION(N)
            a singly subscripted array of dimension at least N.
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   The matrix A is computationally singular.
                        A solution has not been computed.
   IND=-5  terminal   ML is less than zero or is greater than
                        or equal to N .
   IND=-6  terminal   MU is less than zero or is greater than
                        or equal to N .
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              Note-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


DNBSL


      SUBROUTINE DNBSL(ABE,LDA,N,ML,MU,IPVT,B,JOB)

    DNBSL solves the double precision band system
    A * X = B  or  TRANS(A) * X = B
    using the factors computed by DNBCO or DNBFA.
    On Entry
       ABE     DOUBLE PRECISION(LDA, NC)
               the output from DNBCO or DNBFA.
               NC must be .GE. 2*ML+MU+1 .
       LDA     INTEGER
               the leading dimension of the array  ABE .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from DNBCO or DNBFA.
       B       DOUBLE PRECISION(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B .
               = nonzero   to solve  TRANS(A)*X = B , where
                           TRANS(A)  is the transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically this indicates singularity
       but it is often caused by improper arguments or improper
       setting of LDA.  It will not occur if the subroutines are
       called correctly and if DNBCO has set RCOND .GT. 0.0
       or DNBFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL DNBCO(ABE,LDA,N,ML,MU,IPVT,RCOND,Z)
          IF (RCOND is too small) GO TO ...
          DO 10 J = 1, P
            CALL DNBSL(ABE,LDA,N,ML,MU,IPVT,C(1,J),0)
       10 CONTINUE
    SLATEC.  This version dated 07/28/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
     DAXPY,DDOT
    Fortran  MIN0


DNRM2


      DOUBLE PRECISION FUNCTION DNRM2(N,DX,INCX)

               B L A S  Subprogram
   Description of parameters
    --Input--
       N  number of elements in input vector(s)
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
    --Output--
   DNRM2  double precision result (zero if N .LE. 0)
    Euclidean norm of the N-vector stored in DX() with storage
    increment INCX .
    If    N .LE. 0 return with result = 0.
    If N .GE. 1 then INCX must be .GE. 1
          C.L. Lawson, 1978 Jan 08
    Four phase method     using two built-in constants that are
    hopefully applicable to all machines.
        CUTLO = maximum of  DSQRT(U/EPS)  over all known machines.
        CUTHI = minimum of  DSQRT(V)      over all known machines.
    where
        EPS = smallest no. such that EPS + 1. .GT. 1.
        U   = smallest positive no.   (underflow limit)
        V   = largest  no.            (overflow  limit)
    Brief outline of algorithm..
    Phase 1    scans zero components.
    move to phase 2 when a component is nonzero and .LE. CUTLO
    move to phase 3 when a component is .GT. CUTLO
    move to phase 4 when a component is .GE. CUTHI/M
    where M = N for X() real and M = 2*N for complex.
    Values for CUTLO and CUTHI..
    From the environmental parameters listed in the IMSL converter
    document the limiting values are as followS..
    CUTLO, S.P.   U/EPS = 2**(-102) for  Honeywell.  Close seconds are
                  Univac and DEC at 2**(-103)
                  Thus CUTLO = 2**(-51) = 4.44089E-16
    CUTHI, S.P.   V = 2**127 for Univac, Honeywell, and DEC.
                  Thus CUTHI = 2**(63.5) = 1.30438E19
    CUTLO, D.P.   U/EPS = 2**(-67) for Honeywell and DEC.
                  Thus CUTLO = 2**(-33.5) = 8.23181D-11
    CUTHI, D.P.   same as S.P.  CUTHI = 1.30438D19
    SAVE CUTLO, CUTHI, ZERO, ONE
    DATA CUTLO, CUTHI / 8.232D-11,  1.304D19 /
    DATA CUTLO, CUTHI / 4.441E-16,  1.304E19 /


DPBCO


      SUBROUTINE DPBCO(ABD,LDA,N,M,RCOND,Z,INFO)

    DPBCO factors a double precision symmetric positive definite
    matrix stored in band form and estimates the condition of the
    matrix.
    If  RCOND  is not needed, DPBFA is slightly faster.
    To solve  A*X = B , follow DPBCO by DPBSL.
    To compute  INVERSE(A)*C , follow DPBCO by DPBSL.
    To compute  DETERMINANT(A) , follow DPBCO by DPBDI.
    On Entry
       ABD     DOUBLE PRECISION(LDA, N)
               the matrix to be factored.  The columns of the upper
               triangle are stored in the columns of ABD and the
               diagonals of the upper triangle are stored in the
               rows of ABD .  See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. M + 1 .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
               0 .LE. M .LT.  N .
    On Return
       ABD     an upper triangular matrix  R , stored in band
               form, so that  A = TRANS(R)*R .
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  A  is singular to working precision, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    Band Storage
          If  A  is a symmetric positive definite band matrix,
          the following program segment will set up the input.
                  M = (band width above diagonal)
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-M)
                     DO 10 I = I1, J
                        K = I-J+M+1
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses  M + 1  rows of  A , except for the  M by M
          upper left triangle, which is ignored.
    Example:  If the original matrix is
          11 12 13  0  0  0
          12 22 23 24  0  0
          13 23 33 34 35  0
           0 24 34 44 45 46
           0  0 35 45 55 56
           0  0  0 46 56 66
    then  N = 6 , M = 2  and  ABD  should contain
           *  * 13 24 35 46
           * 12 23 34 45 56
          11 22 33 44 55 66
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK DPBFA
    BLAS DAXPY,DDOT,DSCAL,DASUM
    Fortran DABS,DMAX1,MAX0,MIN0,DREAL,DSIGN


DPBDI


      SUBROUTINE DPBDI(ABD,LDA,N,M,DET)

    DPBDI computes the determinant
    of a double precision symmetric positive definite band matrix
    using the factors computed by DPBCO or DPBFA.
    If the inverse is needed, use DPBSL  N  times.
    On Entry
       ABD     DOUBLE PRECISION(LDA, N)
               the output from DPBCO or DPBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
    On Return
       DET     DOUBLE PRECISION(2)
               determinant of original matrix in the form
               DETERMINANT = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.


DPBFA


      SUBROUTINE DPBFA(ABD,LDA,N,M,INFO)

    DPBFA factors a double precision symmetric positive definite
    matrix stored in band form.
    DPBFA is usually called by DPBCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    On Entry
       ABD     DOUBLE PRECISION(LDA, N)
               the matrix to be factored.  The columns of the upper
               triangle are stored in the columns of ABD and the
               diagonals of the upper triangle are stored in the
               rows of ABD .  See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. M + 1 .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
               0 .LE. M .LT. N .
    On Return
       ABD     an upper triangular matrix  R , stored in band
               form, so that  A = TRANS(R)*R .
       INFO    INTEGER
               = 0  for normal return.
               = K  if the leading minor of order  K  is not
                    positive definite.
    Band Storage
          If  A  is a symmetric positive definite band matrix,
          the following program segment will set up the input.
                  M = (band width above diagonal)
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-M)
                     DO 10 I = I1, J
                        K = I-J+M+1
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DDOT
    Fortran MAX0,DSQRT


DPBSL


      SUBROUTINE DPBSL(ABD,LDA,N,M,B)

    DPBSL solves the double precision symmetric positive definite
    band system  A*X = B
    using the factors computed by DPBCO or DPBFA.
    On Entry
       ABD     DOUBLE PRECISION(LDA, N)
               the output from DPBCO or DPBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
       B       DOUBLE PRECISION(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically this indicates
       singularity, but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly, and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL DPBCO(ABD,LDA,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL DPBSL(ABD,LDA,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DDOT
    Fortran MIN0


DPCHCE


      SUBROUTINE DPCHCE(IC,VC,N,X,H,SLOPE,D,INCFD,IERR)

         DPCHCE:  DPCHIC End Derivative Setter.
   Called by DPCHIC to set end derivatives as requested by the user.
   It must be called after interior derivative values have been set.
                     -----
   To facilitate two-dimensional applications, includes an increment
   between successive values of the D-array.

 Calling sequence:
       PARAMETER  (INCFD = ...)
       INTEGER  IC(2), N, IERR
       DOUBLE PRECISION  VC(2), X(N), H(N), SLOPE(N), D(INCFD,N)
       CALL  DPCHCE (IC, VC, N, X, H, SLOPE, D, INCFD, IERR)
  Parameters:
    IC -- (input) integer array of length 2 specifying desired
          boundary conditions:
          IC(1) = IBEG, desired condition at beginning of data.
          IC(2) = IEND, desired condition at end of data.
          ( see prologue to DPCHIC for details. )
    VC -- (input) real*8 array of length 2 specifying desired boundary
          values.  VC(1) need be set only if IC(1) = 2 or 3 .
                   VC(2) need be set only if IC(2) = 2 or 3 .
    N -- (input) number of data points.  (assumes N.GE.2)
    X -- (input) real*8 array of independent variable values.  (the
          elements of X are assumed to be strictly increasing.)
    H -- (input) real*8 array of interval lengths.
    SLOPE -- (input) real*8 array of data slopes.
          If the data are (X(I),Y(I)), I=1(1)N, then these inputs are:
                 H(I) =  X(I+1)-X(I),
             SLOPE(I) = (Y(I+1)-Y(I))/H(I),  I=1(1)N-1.
    D -- (input) real*8 array of derivative values at the data points.
          The value corresponding to X(I) must be stored in
               D(1+(I-1)*INCFD),  I=1(1)N.
         (output) the value of D at X(1) and/or X(N) is changed, if
          necessary, to produce the requested boundary conditions.
          no other entries in D are changed.
    INCFD -- (input) increment between successive values in D.
          This argument is provided primarily for 2-D applications.
    IERR -- (output) error flag.
          Normal return:
             IERR = 0  (no errors).
          Warning errors:
             IERR = 1  if IBEG.LT.0 and D(1) had to be adjusted for
                       monotonicity.
             IERR = 2  if IEND.LT.0 and D(1+(N-1)*INCFD) had to be
                       adjusted for monotonicity.
             IERR = 3  if both of the above are true.
   -------
   WARNING:  This routine does no validity-checking of arguments.
   -------
 Fortran intrinsics used:  DABS, IABS.


DPCHCI


      SUBROUTINE DPCHCI(N,H,SLOPE,D,INCFD)

         DPCHCI:  DPCHIC Initial Derivative Setter.
   Called by DPCHIC to set derivatives needed to determine a monotone
   piecewise cubic Hermite interpolant to the data.
   Default boundary conditions are provided which are compatible
   with monotonicity.  If the data are only piecewise monotonic, the
   interpolant will have an extremum at each point where monotonicity
   switches direction.
   To facilitate two-dimensional applications, includes an increment
   between successive values of the D-array.
   The resulting piecewise cubic Hermite function should be identical
   (within roundoff error) to that produced by DPCHIM.

 Calling sequence:
       PARAMETER  (INCFD = ...)
       INTEGER  N
       DOUBLE PRECISION  H(N), SLOPE(N), D(INCFD,N)
       CALL  DPCHCI (N, H, SLOPE, D, INCFD)
  Parameters:
    N -- (input) number of data points.
          If N=2, simply does linear interpolation.
    H -- (input) real*8 array of interval lengths.
    SLOPE -- (input) real*8 array of data slopes.
          If the data are (X(I),Y(I)), I=1(1)N, then these inputs are:
                 H(I) =  X(I+1)-X(I),
             SLOPE(I) = (Y(I+1)-Y(I))/H(I),  I=1(1)N-1.
    D -- (output) real*8 array of derivative values at data points.
          If the data are monotonic, these values will determine a
          a monotone cubic Hermite function.
          The value corresponding to X(I) is stored in
               D(1+(I-1)*INCFD),  I=1(1)N.
          No other entries in D are changed.
    INCFD -- (input) increment between successive values in D.
          This argument is provided primarily for 2-D applications.
   -------
   WARNING:  This routine does no validity-checking of arguments.
   -------
 Fortran intrinsics used:  DABS, DMAX1, DMIN1.


DPCHCS


      SUBROUTINE DPCHCS(SWITCH,N,H,SLOPE,D,INCFD,IERR)

        DPCHCS:  DPCHIC Monotonicity Switch Derivative Setter.
    Called by  DPCHIC  to adjust the values of D in the vicinity of a
    switch in direction of monotonicity, to produce a more "visually
    pleasing" curve than that given by  DPCHIM .

 Calling sequence:
       PARAMETER  (INCFD = ...)
       INTEGER  N, IERR
       DOUBLE PRECISION  SWITCH, H(N), SLOPE(N), D(INCFD,N)
       CALL  DPCHCS (SWITCH, N, H, SLOPE, D, INCFD, IERR)
  Parameters:
    SWITCH -- (input) indicates the amount of control desired over
          local excursions from data.
    N -- (input) number of data points.  (assumes N.GT.2 .)
    H -- (input) real*8 array of interval lengths.
    SLOPE -- (input) real*8 array of data slopes.
          If the data are (X(I),Y(I)), I=1(1)N, then these inputs are:
                 H(I) =  X(I+1)-X(I),
             SLOPE(I) = (Y(I+1)-Y(I))/H(I),  I=1(1)N-1.
    D -- (input) real*8 array of derivative values at the data points,
          as determined by DPCHCI.
         (output) derivatives in the vicinity of switches in direction
          of monotonicity may be adjusted to produce a more "visually
          pleasing" curve.
          The value corresponding to X(I) is stored in
               D(1+(I-1)*INCFD),  I=1(1)N.
          No other entries in D are changed.
    INCFD -- (input) increment between successive values in D.
          This argument is provided primarily for 2-D applications.
    IERR -- (output) error flag.  should be zero.
          If negative, trouble in DPCHSW.  (should never happen.)
   -------
   WARNING:  This routine does no validity-checking of arguments.
   -------
 Fortran intrinsics used:  DABS, DMAX1, DMIN1.


DPCHDF


      DOUBLE PRECISION FUNCTION DPCHDF(K,X,S,IERR)

         DPCHDF:   DPCHIP Finite Difference Formula
    Uses a divided difference formulation to compute a K-point approx-
    imation to the derivative at X(K) based on the data in X and S.
    Called by  DPCHCE  and  DPCHSP  to compute 3- and 4-point boundary
    derivative approximations.

    On input:
       K      is the order of the desired derivative approximation.
              K must be at least 3 (error return if not).
       X      contains the K values of the independent variable.
              X need not be ordered, but the values **MUST** be
              distinct.  (Not checked here.)
       S      contains the associated slope values:
                 S(I) = (F(I+1)-F(I))/(X(I+1)-X(I)), I=1(1)K-1.
              (Note that S need only be of length K-1.)
    On return:
       S      will be destroyed.
       IERR   will be set to -1 if K.LT.2 .
       DPCHDF  will be set to the desired derivative approximation if
              IERR=0 or to zero if IERR=-1.

 Reference:  Carl de Boor, A Practical Guide to Splines, Springer-
             Verlag (New York, 1978), pp. 10-16.


DPOCO


      SUBROUTINE DPOCO(A,LDA,N,RCOND,Z,INFO)

    DPOCO factors a double precision symmetric positive definite
    matrix and estimates the condition of the matrix.
    If  RCOND  is not needed, DPOFA is slightly faster.
    To solve  A*X = B , follow DPOCO by DPOSL.
    To compute  INVERSE(A)*C , follow DPOCO by DPOSL.
    To compute  DETERMINANT(A) , follow DPOCO by DPODI.
    To compute  INVERSE(A) , follow DPOCO by DPODI.
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the symmetric matrix to be factored.  Only the
               diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix  R  so that  A = TRANS(R)*R
               where  TRANS(R)  is the transpose.
               The strict lower triangle is unaltered.
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK DPOFA
    BLAS DAXPY,DDOT,DSCAL,DASUM
    Fortran DABS,DMAX1,DREAL,DSIGN


DPODI


      SUBROUTINE DPODI(A,LDA,N,DET,JOB)

    DPODI computes the determinant and inverse of a certain
    double precision symmetric positive definite matrix (see below)
    using the factors computed by DPOCO, DPOFA or DQRDC.
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the output  A  from DPOCO or DPOFA
               or the output  X  from DQRDC.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       A       If DPOCO or DPOFA was used to factor  A , then
               DPODI produces the upper half of INVERSE(A) .
               If DQRDC was used to decompose  X , then
               DPODI produces the upper half of inverse(TRANS(X)*X)
               where TRANS(X) is the transpose.
               Elements of  A  below the diagonal are unchanged.
               If the units digit of JOB is zero,  A  is unchanged.
       DET     DOUBLE PRECISION(2)
               determinant of  A  or of  TRANS(X)*X  if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if DPOCO or DPOFA has set INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DSCAL
    Fortran MOD


DPOFA


      SUBROUTINE DPOFA(A,LDA,N,INFO)

    DPOFA factors a double precision symmetric positive definite
    matrix.
    DPOFA is usually called by DPOCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (time for DPOCO) = (1 + 18/N)*(time for DPOFA) .
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the symmetric matrix to be factored.  Only the
               diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix  R  so that  A = TRANS(R)*R
               where  TRANS(R)  is the transpose.
               The strict lower triangle is unaltered.
               If  INFO .NE. 0 , the factorization is not complete.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DDOT
    Fortran DSQRT


DPOFS


      SUBROUTINE DPOFS(A,LDA,N,V,ITASK,IND,WORK)

   Subroutine DPOFS solves a  positive definite symmetric
   NxN system of double precision linear equations using
   LINPACK subroutines DPOCO and DPOSL.  That is, if A is an
   NxN double precision positive definite symmetric matrix and if
   X and B are double precision N-vectors, then DPOFS solves
   the equation
                         A*X=B.
   The matrix A is first factored into upper and lower tri-
   angular matrices R and R-TRANPOSE.  These factors are used to
   find the solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option only to solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, and N must not have been altered by the user following
   factorization (ITASK=1).  IND will not be changed by DPOFS
   in this case.
 Argument Description ***
   A      DOUBLE PRECISION(LDA,N)
            on entry, the doubly subscripted array with dimension
              (LDA,N) which contains the coefficient matrix.  Only
              the upper triangle, including the diagonal, of the
              coefficient matrix need be entered and will subse-
              quently be referenced and changed by the routine.
            on return, A contains in its upper triangle an upper
              triangular matrix R such that A = (R-TRANPOSE) * R .
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater
            than or equal to 1.  (terminal error message IND=-2)
   V      DOUBLE PRECISION(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations  A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK = 1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A.
            If ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT. 0  See error message corresponding to IND below.
   WORK   DOUBLE PRECISION(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  Terminal   N is greater than LDA.
   IND=-2  Terminal   N is less than 1.
   IND=-3  Terminal   ITASK is less than 1.
   IND=-4  Terminal   The matrix A is computationally singular or
                        is not positive definite.  A solution
                        has not been computed.
   IND=-10 Warning    The solution has no apparent significance.
                        The solution may be inaccurate or the
                        matrix A may be poorly scaled.
              Note-  The above Terminal(*fatal*) Error Messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


DPOLCF


      SUBROUTINE DPOLCF(XX,N,X,C,D,WORK)

  **** Double Precision version of POLCOF ****
    Written by Robert E. Huddleston, Sandia Laboratories, Livermore
    Abstract
       Subroutine DPOLCF computes the coefficients of the polynomial
    fit (including Hermite polynomial fits ) produced by a previous
    call to HRMITE or DPLINT. The coefficients of the polynomial,
    expanded about XX, are stored in the array D. The expansion is of
    the form
    P(Z) = D(1) + D(2)*(Z-XX) +D(3)*((Z-XX)**2) + ... +
                                                 D(N)*((Z-XX)**(N-1)).
    Between the call to DPLINT (or to HRMITE) and the call to DPOLCF
    the variable N and the arrays X and C must not be altered.
    *****  INPUT PARAMETERS
     *** All TYPE REAL variables are DOUBLE PRECISION ***
    XX   - The point about which the Taylor expansion is to be made.
    N    - ****
           *     N, X, and C must remain unchanged between the
    X    - *     call to DPLINT or the call to HRMITE and the
           *     call to DPOLCF.
    C    - ****
    *****  OUTPUT PARAMETER
     *** All TYPE REAL variables are DOUBLE PRECISION ***
    D    - The array of coefficients for the Taylor expansion as
           explained in the abstract
    *****  STORAGE PARAMETER
    WORK - This is an array to provide internal working storage. It
           must be dimensioned by at least 2*N in the calling program.
    **** Note - There are two methods for evaluating the fit produced
    by DPLINT or HRMITE. You may call DPOLVL to perform the task, or
    you may call DPOLCF to obtain the coefficients of the Taylor
    expansion and then write your own evaluation scheme. Due to the
    inherent errors in the computations of the Taylor expansion from
    the Newton coefficients produced by DPLINT or HRMITE, much more
    accuracy may be expected by calling DPOLVL as opposed to writing
    your own scheme.


DPOSL


      SUBROUTINE DPOSL(A,LDA,N,B)

    DPOSL solves the double precision symmetric positive definite
    system A * X = B
    using the factors computed by DPOCO or DPOFA.
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the output from DPOCO or DPOFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       B       DOUBLE PRECISION(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically this indicates
       singularity, but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL DPOCO(A,LDA,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL DPOSL(A,LDA,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DDOT


DPPCO


      SUBROUTINE DPPCO(AP,N,RCOND,Z,INFO)

    DPPCO factors a double precision symmetric positive definite
    matrix stored in packed form
    and estimates the condition of the matrix.
    If  RCOND  is not needed, DPPFA is slightly faster.
    To solve  A*X = B , follow DPPCO by DPPSL.
    To compute  INVERSE(A)*C , follow DPPCO by DPPSL.
    To compute  DETERMINANT(A) , follow DPPCO by DPPDI.
    To compute  INVERSE(A) , follow DPPCO by DPPDI.
    On Entry
       AP      DOUBLE PRECISION (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    On Return
       AP      an upper triangular matrix  R , stored in packed
               form, so that  A = TRANS(R)*R .
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  A  is singular to working precision, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK DPPFA
    BLAS DAXPY,DDOT,DSCAL,DASUM
    Fortran DABS,DMAX1,DREAL,DSIGN


DPPDI


      SUBROUTINE DPPDI(AP,N,DET,JOB)

    DPPDI computes the determinant and inverse
    of a double precision symmetric positive definite matrix
    using the factors computed by DPPCO or DPPFA .
    On Entry
       AP      DOUBLE PRECISION (N*(N+1)/2)
               the output from DPPCO or DPPFA.
       N       INTEGER
               the order of the matrix  A .
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       AP      the upper triangular half of the inverse .
               The strict lower triangle is unaltered.
       DET     DOUBLE PRECISION(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               DETERMINANT = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if DPOCO or DPOFA has set INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DSCAL
    Fortran MOD


DPPFA


      SUBROUTINE DPPFA(AP,N,INFO)

    DPPFA factors a double precision symmetric positive definite
    matrix stored in packed form.
    DPPFA is usually called by DPPCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (time for DPPCO) = (1 + 18/N)*(time for DPPFA) .
    On Entry
       AP      DOUBLE PRECISION (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    On Return
       AP      an upper triangular matrix  R , stored in packed
               form, so that  A = TRANS(R)*R .
       INFO    INTEGER
               = 0  for normal return.
               = K  if the leading minor of order  K  is not
                    positive definite.
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DDOT
    Fortran DSQRT


DPPGQ8


      SUBROUTINE DPPGQ8(FUN,LDC,C,XI,LXI,KK,ID,A,B,INPPV,ERR,ANS,IERR)



DPPSL


      SUBROUTINE DPPSL(AP,N,B)

    DPPSL solves the double precision symmetric positive definite
    system A * X = B
    using the factors computed by DPPCO or DPPFA.
    On Entry
       AP      DOUBLE PRECISION (N*(N+1)/2)
               the output from DPPCO or DPPFA.
       N       INTEGER
               the order of the matrix  A .
       B       DOUBLE PRECISION(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically this indicates
       singularity, but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL DPPCO(AP,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL DPPSL(AP,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DDOT


DPTSL


      SUBROUTINE DPTSL(N,D,E,B)

    DPTSL, given a positive definite symmetric tridiagonal matrix and
    a right hand side, will find the solution.
    On Entry
       N        INTEGER
                is the order of the tridiagonal matrix.
       D        DOUBLE PRECISION(N)
                is the diagonal of the tridiagonal matrix.
                On output D is destroyed.
       E        DOUBLE PRECISION(N)
                is the offdiagonal of the tridiagonal matrix.
                E(1) through E(N-1) should contain the
                offdiagonal.
       B        DOUBLE PRECISION(N)
                is the right hand side vector.
    On Return
       B        contains the solution.
    LINPACK.  This version dated 08/14/78 .
    Jack Dongarra, Argonne National Laboratory.
    No externals
    Fortran MOD


DQDOTA


      DOUBLE PRECISION FUNCTION DQDOTA(N,DB,QC,DX,INCX,DY,INCY)

              B L A S  Subprogram
  Description of Parameters
   --Input--
      N  number of elements in input vector(S)
     DB  double precision scalar to be added to inner product
     QC  extended precision scalar to be added to inner product
     DX  double precision vector with N elements
   INCX  storage spacing between elements of DX
     DY  double precision vector with N elements
   INCY  storage spacing between elements of DY
   --Output--
 DQDOTA  double precision result
     QC  extended precision result
   D.P. dot product with extended precision accumulation (and result)
   QC and DQDOTA are set = DB + QC + sum for I = 0 to N-1 of
     DX(LX+I*INCX) * DY(LY+I*INCY),  where QC is an extended
     precision result previously computed by DQDOTI or DQDOTA
     and LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
     defined in a similar way using INCY.  The MP package by
     Richard P. Brent is used for the extended precision arithmetic.
   Fred T. Krogh,  JPL,  1977,  June 1
   The common block for the MP package is name MPCOM.  If local
   variable I1 is zero, DQDOTA calls MPBLAS to initialize
   the MP package and reset I1 to 1.


DQDOTI


      DOUBLE PRECISION FUNCTION DQDOTI(N,DB,QC,DX,INCX,DY,INCY)

               B L A S  Subprogram
   Description of parameters
    --Input--
       N  number of elements in input vector(s)
      DB  double precision scalar to be added to inner product
      QC  extended precision scalar to be added
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
      DY  double precision vector with N elements
    INCY  storage spacing between elements of DY
    --Output--
  DQDOTI  double precision result
      QC  extended precision result
    D.P. dot product with extended precision accumulation (and result)
    QC and DQDOTI are set = DB + sum for I = 0 to N-1 of
      DX(LX+I*INCX) * DY(LY+I*INCY),  where QC is an extended
      precision result which can be used as input to DQDOTA,
      and LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
      defined in a similar way using INCY.  The MP package by
      Richard P. Brent is used for the extended precision arithmetic.
    Fred T. Krogh,  JPL,  1977,  June 1
    The common block for the MP package is named MPCOM.  If local
    variable I1 is zero, DQDOTI calls MPBLAS to initialize the MP
    package and reset I1 to 1.


DQELG


      SUBROUTINE DQELG(N,EPSTAB,RESULT,ABSERR,RES3LA,NRES)

          Epsilon algorithm
          Standard fortran subroutine
          Double precision version
          PARAMETERS
             N      - Integer
                      EPSTAB(N) contains the new element in the
                      first column of the epsilon table.
             EPSTAB - Double precision
                      Vector of dimension 52 containing the elements
                      of the two lower diagonals of the triangular
                      epsilon table. The elements are numbered
                      starting at the right-hand corner of the
                      triangle.
             RESULT - Double precision
                      Resulting approximation to the integral
             ABSERR - Double precision
                      Estimate of the absolute error computed from
                      RESULT and the 3 previous results
             RES3LA - Double precision
                      Vector of dimension 3 containing the last 3
                      results
             NRES   - Integer
                      Number of calls to the routine
                      (should be zero at first call)


DQFORM


      SUBROUTINE DQFORM(M,N,Q,LDQ,WA)



DQRDC


      SUBROUTINE DQRDC(X,LDX,N,P,QRAUX,JPVT,WORK,JOB)

    DQRDC uses Householder transformations to compute the QR
    factorization of an N by P matrix X.  Column pivoting
    based on the 2-norms of the reduced columns may be
    performed at the user's option.
    On Entry
       X       DOUBLE PRECISION(LDX,P), where LDX .GE. N.
               X contains the matrix whose decomposition is to be
               computed.
       LDX     INTEGER.
               LDX is the leading dimension of the array X.
       N       INTEGER.
               N is the number of rows of the matrix X.
       P       INTEGER.
               P is the number of columns of the matrix X.
       JPVT    INTEGER(P).
               JPVT contains integers that control the selection
               of the pivot columns.  The K-th column X(K) of X
               is placed in one of three classes according to the
               value of JPVT(K).
                  If JPVT(K) .GT. 0, then X(K) is an initial
                                     column.
                  If JPVT(K) .EQ. 0, then X(K) is a free column.
                  If JPVT(K) .LT. 0, then X(K) is a final column.
               Before the decomposition is computed, initial columns
               are moved to the beginning of the array X and final
               columns to the end.  Both initial and final columns
               are frozen in place during the computation and only
               free columns are moved.  At the K-th stage of the
               reduction, if X(K) is occupied by a free column
               it is interchanged with the free column of largest
               reduced norm.  JPVT is not referenced if
               JOB .EQ. 0.
       WORK    DOUBLE PRECISION(P).
               WORK is a work array.  WORK is not referenced if
               JOB .EQ. 0.
       JOB     INTEGER.
               JOB is an integer that initiates column pivoting.
               If JOB .EQ. 0, no pivoting is done.
               If JOB .NE. 0, pivoting is done.
    On Return
       X       X contains in its upper triangle the upper
               triangular matrix R of the QR factorization.
               Below its diagonal X contains information from
               which the orthogonal part of the decomposition
               can be recovered.  Note that if pivoting has
               been requested, the decomposition is not that
               of the original matrix X but that of X
               with its columns permuted as described by JPVT.
       QRAUX   DOUBLE PRECISION(P).
               QRAUX contains further information required to recover
               the orthogonal part of the decomposition.
       JPVT    JPVT(K) contains the index of the column of the
               original matrix that has been interchanged into
               the K-th column, if pivoting was requested.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    DQRDC uses the following functions and subprograms.
    BLAS DAXPY,DDOT,DSCAL,DSWAP,DNRM2
    Fortran DABS,DMAX1,MIN0,DSQRT


DQRFAC


      SUBROUTINE DQRFAC(M,N,A,LDA,PIVOT,IPVT,LIPVT,SIGMA,ACNORM,WA)

  **** Double Precision version of QRFAC ****
    **********
    Subroutine DQRFAC
    This subroutine uses Householder transformations with column
    pivoting (optional) to compute a QR factorization of the
    M by N matrix A. That is, DQRFAC determines an orthogonal
    matrix Q, a permutation matrix P, and an upper trapezoidal
    matrix R with diagonal elements of nonincreasing magnitude,
    such that A*P = Q*R. The Householder transformation for
    column K, K = 1,2,...,MIN(M,N), is of the form
                          T
          I - (1/U(K))*U*U
    where U has zeros in the first K-1 positions. The form of
    this transformation and the method of pivoting first
    appeared in the corresponding LINPACK subroutine.
    The subroutine statement is
      SUBROUTINE DQRFAC(M,N,A,LDA,PIVOT,IPVT,LIPVT,SIGMA,ACNORM,WA)
    where
      M is a positive integer input variable set to the number
        of rows of A.
      N is a positive integer input variable set to the number
        of columns of A.
      A is an M by N array. On input A contains the matrix for
        which the QR factorization is to be computed. On output
        the strict upper trapezoidal part of A contains the strict
        upper trapezoidal part of R, and the lower trapezoidal
        part of A contains a factored form of Q (the non-trivial
        elements of the U vectors described above).
      LDA is a positive integer input variable not less than M
        which specifies the leading dimension of the array A.
      PIVOT is a logical input variable. If pivot is set .TRUE.,
        then column pivoting is enforced. If pivot is set .FALSE.,
        then no column pivoting is done.
      IPVT is an integer output array of length LIPVT. IPVT
        defines the permutation matrix P such that A*P = Q*R.
        Column J of P is column IPVT(J) of the identity matrix.
        If pivot is .FALSE., IPVT is not referenced.
      LIPVT is a positive integer input variable. If PIVOT is
            .FALSE., then LIPVT may be as small as 1. If PIVOT is
            .TRUE., then LIPVT must be at least N.
      SIGMA is an output array of length N which contains the
        diagonal elements of R.
      ACNORM is an output array of length N which contains the
        norms of the corresponding columns of the input matrix A.
        If this information is not needed, then ACNORM can coincide
        with SIGMA.
      WA is a work array of length N. If pivot is .FALSE., then WA
        can coincide with SIGMA.
    Subprograms called
      SLATEC-Supplied ... D1MACH,DENORM
      FORTRAN-Supplied ... DMAX1,DSQRT,MIN0
    MINPACK. Version of December 1978.
    Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More
    **********


DQRSL


      SUBROUTINE DQRSL(X,LDX,N,K,QRAUX,Y,QY,QTY,B,RSD,XB,JOB,INFO)

    DQRSL applies the output of DQRDC to compute coordinate
    transformations, projections, and least squares solutions.
    For K .LE. MIN(N,P), let XK be the matrix
           XK = (X(JPVT(1)),X(JPVT(2)), ... ,X(JPVT(K)))
    formed from columnns JPVT(1), ... ,JPVT(K) of the original
    N X P matrix X that was input to DQRDC (if no pivoting was
    done, XK consists of the first K columns of X in their
    original order).  DQRDC produces a factored orthogonal matrix Q
    and an upper triangular matrix R such that
             XK = Q * (R)
                      (0)
    This information is contained in coded form in the arrays
    X and QRAUX.
    On Entry
       X      DOUBLE PRECISION(LDX,P).
              X contains the output of DQRDC.
       LDX    INTEGER.
              LDX is the leading dimension of the array X.
       N      INTEGER.
              N is the number of rows of the matrix XK.  It must
              have the same value as N in DQRDC.
       K      INTEGER.
              K is the number of columns of the matrix XK.  K
              must not be greater than MIN(N,P), where P is the
              same as in the calling sequence to DQRDC.
       QRAUX  DOUBLE PRECISION(P).
              QRAUX contains the auxiliary output from DQRDC.
       Y      DOUBLE PRECISION(N)
              Y contains an N-vector that is to be manipulated
              by DQRSL.
       JOB    INTEGER.
              JOB specifies what is to be computed.  JOB has
              the decimal expansion ABCDE, with the following
              meaning.
                   If A .NE. 0, compute QY.
                   If B,C,D, or E .NE. 0, compute QTY.
                   If C .NE. 0, compute B.
                   If D .NE. 0, compute RSD.
                   If E .NE. 0, compute XB.
              Note that a request to compute B, RSD, or XB
              automatically triggers the computation of QTY, for
              which an array must be provided in the calling
              sequence.
    On Return
       QY     DOUBLE PRECISION(N).
              QY contains Q*Y, if its computation has been
              requested.
       QTY    DOUBLE PRECISION(N).
              QTY contains TRANS(Q)*Y, if its computation has
              been requested.  Here TRANS(Q) is the
              transpose of the matrix Q.
       B      DOUBLE PRECISION(K)
              B contains the solution of the least squares problem
                   minimize norm2(Y - XK*B),
              if its computation has been requested.  (Note that
              if pivoting was requested in DQRDC, the J-th
              component of B will be associated with column JPVT(J)
              of the original matrix X that was input into DQRDC.)
       RSD    DOUBLE PRECISION(N).
              RSD contains the least squares residual Y - XK*B,
              if its computation has been requested.  RSD is
              also the orthogonal projection of Y onto the
              orthogonal complement of the column space of XK.
       XB     DOUBLE PRECISION(N).
              XB contains the least squares approximation XK*B,
              if its computation has been requested.  XB is also
              the orthogonal projection of Y onto the column space
              of X.
       INFO   INTEGER.
              INFO is zero unless the computation of B has
              been requested and R is exactly singular.  In
              this case, INFO is the index of the first zero
              diagonal element of R and B is left unaltered.
    The parameters QY, QTY, B, RSD, and XB are not referenced
    if their computation is not requested and in this case
    can be replaced by dummy variables in the calling program.
    To save storage, the user may in some cases use the same
    array for different parameters in the calling sequence.  A
    frequently occuring example is when one wishes to compute
    any of B, RSD, or XB and does not need Y or QTY.  In this
    case one may identify Y, QTY, and one of B, RSD, or XB, while
    providing separate arrays for anything else that is to be
    computed.  Thus the calling sequence
         CALL DQRSL(X,LDX,N,K,QRAUX,Y,DUM,Y,B,Y,DUM,110,INFO)
    will result in the computation of B and RSD, with RSD
    overwriting Y.  More generally, each item in the following
    list contains groups of permissible identifications for
    a single calling sequence.
         1. (Y,QTY,B) (RSD) (XB) (QY)
         2. (Y,QTY,RSD) (B) (XB) (QY)
         3. (Y,QTY,XB) (B) (RSD) (QY)
         4. (Y,QY) (QTY,B) (RSD) (XB)
         5. (Y,QY) (QTY,RSD) (B) (XB)
         6. (Y,QY) (QTY,XB) (B) (RSD)
    In any group the value returned in the array allocated to
    the group corresponds to the last member of the group.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    DQRSL uses the following functions and subprograms.
    BLAS DAXPY,DCOPY,DDOT
    Fortran DABS,MIN0,MOD


DQRSLV


      SUBROUTINE DQRSLV(N,R,LDR,IPVT,DIAG,QTB,X,SIGMA,WA)

 **** Double Precision version of QRSOLV ****
    **********
    Subroutine DQRSLV
    Given an M by N matrix A, an N by N diagonal matrix D,
    and an M-vector B, the problem is to determine an X which
    solves the system
          A*X = B ,     D*X = 0 ,
    in the least squares sense.
    This subroutine completes the solution of the problem
    if it is provided with the necessary information from the
    QR factorization, with column pivoting, of A. That is, if
    A*P = Q*R, where P is a permutation matrix, Q has orthogonal
    columns, and R is an upper triangular matrix with diagonal
    elements of nonincreasing magnitude, then DQRSLV expects
    the full upper triangle of R, the permutation matrix P,
    and the first N components of (Q TRANSPOSE)*B. The system
    A*X = B, D*X = 0, is then equivalent to
                 T       T
          R*Z = Q *B ,  P *D*P*Z = 0 ,
    where X = P*Z. If this system does not have full rank,
    then a least squares solution is obtained. On output DQRSLV
    also provides an upper triangular matrix S such that
           T   T               T
          P *(A *A + D*D)*P = S *S .
    S is computed within DQRSLV and may be of separate interest.
    The subroutine statement is
      SUBROUTINE DQRSLV(N,R,LDR,IPVT,DIAG,QTB,X,SIGMA,WA)
    where
      N is a positive integer input variable set to the order of R.
      R is an N by N array. On input the full upper triangle
        must contain the full upper triangle of the matrix R.
        On output the full upper triangle is unaltered, and the
        strict lower triangle contains the strict upper triangle
        (transposed) of the upper triangular matrix S.
      LDR is a positive integer input variable not less than N
        which specifies the leading dimension of the array R.
      IPVT is an integer input array of length N which defines the
        permutation matrix P such that A*P = Q*R. Column J of P
        is column IPVT(J) of the identity matrix.
      DIAG is an input array of length N which must contain the
        diagonal elements of the matrix D.
      QTB is an input array of length N which must contain the first
        N elements of the vector (Q TRANSPOSE)*B.
      X is an output array of length N which contains the least
        squares solution of the system A*X = B, D*X = 0.
      SIGMA is an output array of length N which contains the
        diagonal elements of the upper triangular matrix S.
      WA is a work array of length N.
    Subprograms called
      FORTRAN-Supplied ... DABS,DSQRT
    MINPACK. Version of December 1978.
    Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More
    **********


DQWGTC


      DOUBLE PRECISION FUNCTION DQWGTC(X,C,P2,P3,P4,KP)



DQWGTF


      DOUBLE PRECISION FUNCTION DQWGTF(X,OMEGA,P2,P3,P4,INTEGR)



DQWGTS


      DOUBLE PRECISION FUNCTION DQWGTS(X,A,B,ALFA,BETA,INTEGR)



DROT


      SUBROUTINE DROT(N,DX,INCX,DY,INCY,DC,DS)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
      DY  double precision vector with N elements
    INCY  storage spacing between elements of DY
      DC  D.P. element of rotation matrix
      DS  D.P. element of rotation matrix
    --Output--
      DX  rotated vector (unchanged if N .LE. 0)
      DY  rotated vector (unchanged if N .LE. 0)
    Multiply the 2 X 2 matrix  ( DC DS) times the 2 X N matrix (DX**T)
                               (-DS DC)                        (DY**T)
    where **T indicates transpose.  The elements of DX are in
    DX(LX+I*INCX), I = 0 to N-1, where LX = 1 if INCX .GE. 0, else
    LX = (-INCX)*N, and similarly for DY using LY and INCY.


DROTG


      SUBROUTINE DROTG(DA,DB,DC,DS)

               B L A S  Subprogram
   Description of Parameters
    --Input--
      DA  double precision scalar
      DB  double precision scalar
    --Output--
      DA  double precision result R
      DB  double precision result Z
      DC  double precision result
      DS  double precision result
    Designed by C. L. Lawson, JPL, 1977 Sept 08
    Construct the Givens transformation
        ( DC  DS )
    G = (        ) ,    DC**2 + DS**2 = 1 ,
        (-DS  DC )
    which zeros the second entry of the 2-vector  (DA,DB)**T .
    The quantity R = (+/-)DSQRT(DA**2 + DB**2) overwrites DA in
    storage.  The value of DB is overwritten by a value Z which
    allows DC and DS to be recovered by the following algorithm.
          If Z=1  set  DC=0.D0  and  DS=1.D0
          If DABS(Z) .LT. 1  set  DC=DSQRT(1-Z**2)  and  DS=Z
          If DABS(Z) .GT. 1  set  DC=1/Z  and  DS=DSQRT(1-DC**2)
    Normally, the subprogram DROT(N,DX,INCX,DY,INCY,DC,DS) will
    next be called to apply the transformation to a 2 by N matrix.


DROTM


      SUBROUTINE DROTM(N,DX,INCX,DY,INCY,DPARAM)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
      DY  double precision vector with N elements
    INCY  storage spacing between elements of DY
  DPARAM  5-element D.P. vector.  DPARAM(1) is DFLAG described below.
           Elements 2-5 form the transformation matrix H.
    --Output--
      DX  rotated vector (unchanged if N .LE. 0)
      DY  rotated vector (unchanged if N .LE. 0)
    Apply the modified Givens transformation, H, to the 2 by N matrix
    (DX**T), where **T indicates transpose.  The elements of DX are in
    (DY**T)
    DX(LX+I*INCX), I = 0 to N-1, where LX = 1 if INCX .GE. 0, else
    LX = (-INCX)*N, and similarly for SY using LY and INCY.
    With DPARAM(1)=DFLAG, H has one of the following forms.
    DFLAG=-1.D0     DFLAG=0.D0        DFLAG=1.D0     DFLAG=-2.D0
      (DH11  DH12)    (1.D0  DH12)    (DH11  1.D0)    (1.D0  0.D0)
    H=(          )    (          )    (          )    (          )
      (DH21  DH22),   (DH21  1.D0),   (-1.D0 DH22),   (0.D0  1.D0).
    See DROTMG for a description of data storage in DPARAM.


DROTMG


      SUBROUTINE DROTMG(DD1,DD2,DX1,DY1,DPARAM)

               B L A S  Subprogram
   Description of Parameters
    --Input--
     DD1  double precision scalar
     DD2  double precision scalar
     DX1  double precision scalar
     DX2  double precision scalar
  DPARAM  D.P. 5-vector. DPARAM(1)=DFLAG defined below.
            Elements 2-5  define the transformation matrix H.
    --Output--
     DD1  changed to represent the effect of the transformation
     DD2  changed to reflect the transformation
     DX1  changed to reflect the transformation
     DX2  unchanged
    Construct the modified Givens transformation matrix H which zeros
    the second component of the 2-vector  (DSQRT(DD1)*DX1,DSQRT(DD2)*
    DY2)**T.
    With DPARAM(1)=DFLAG, H has one of the following forms..
    DFLAG=-1.D0     DFLAG=0.D0        DFLAG=1.D0     DFLAG=-2.D0
      (DH11  DH12)    (1.D0  DH12)    (DH11  1.D0)    (1.D0  0.D0)
    H=(          )    (          )    (          )    (          )
      (DH21  DH22),   (DH21  1.D0),   (-1.D0 DH22),   (0.D0  1.D0).
    Locations 2-5 of DPARAM contain DH11, DH21, DH12, and DH22
    respectively.  (Values of 1.D0, -1.D0, or 0.D0 implied by the
    value of DPARAM(1) are not stored in DPARAM.)


DRSCO


      SUBROUTINE DRSCO(RSAV,ISAV)

  DRSCO transfers data from arrays to a common block within the
  integrator package DDEBDF.


DSCAL


      SUBROUTINE DSCAL(N,DA,DX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DA  double precision scale factor
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
    --Output--
      DX  double precision result (unchanged if N.LE.0)
    Replace double precision DX by double precision DA*DX.
    For I = 0 to N-1, replace DX(1+I*INCX) with  DA * DX(1+I*INCX)


DSDOT


      DOUBLE PRECISION FUNCTION DSDOT(N,SX,INCX,SY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
      SY  single precision vector with N elements
    INCY  storage spacing between elements of SY
    --Output--
   DSDOT  double precision dot product (zero if N.LE.0)
    Returns D.P. dot product accumulated in D.P., for S.P. SX and SY
    DSDOT = sum for I = 0 to N-1 of  SX(LX+I*INCX) * SY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


DSICO


      SUBROUTINE DSICO(A,LDA,N,KPVT,RCOND,Z)

    DSICO factors a double precision symmetric matrix by elimination
    with symmetric pivoting and estimates the condition of the
    matrix.
    If  RCOND  is not needed, DSIFA is slightly faster.
    To solve  A*X = B , follow DSICO by DSISL.
    To compute  INVERSE(A)*C , follow DSICO by DSISL.
    To compute  INVERSE(A) , follow DSICO by DSIDI.
    To compute  DETERMINANT(A) , follow DSICO by DSIDI.
    To compute  INERTIA(A), follow DSICO by DSIDI.
    On Entry
       A       DOUBLE PRECISION(LDA, N)
               the symmetric matrix to be factored.
               Only the diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    Output
       A       a block diagonal matrix and the multipliers which
               were used to obtain it.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices, TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK DSIFA
    BLAS DAXPY,DDOT,DSCAL,DASUM
    Fortran DABS,DMAX1,IABS,DSIGN


DSIDI


      SUBROUTINE DSIDI(A,LDA,N,KPVT,DET,INERT,WORK,JOB)

    DSIDI computes the determinant, inertia and inverse
    of a double precision symmetric matrix using the factors from
    DSIFA.
    On Entry
       A       DOUBLE PRECISION(LDA,N)
               the output from DSIFA.
       LDA     INTEGER
               the leading dimension of the array A.
       N       INTEGER
               the order of the matrix A.
       KPVT    INTEGER(N)
               the pivot vector from DSIFA.
       WORK    DOUBLE PRECISION(N)
               work vector.  Contents destroyed.
       JOB     INTEGER
               JOB has the decimal expansion  ABC  where
                  if  C .NE. 0, the inverse is computed,
                  if  B .NE. 0, the determinant is computed,
                  if  A .NE. 0, the inertia is computed.
               For example, JOB = 111  gives all three.
    On Return
       Variables not requested by JOB are not used.
       A      contains the upper triangle of the inverse of
              the original matrix.  The strict lower triangle
              is never referenced.
       DET    DOUBLE PRECISION(2)
              determinant of original matrix.
              DETERMINANT = DET(1) * 10.0**DET(2)
              with 1.0 .LE. DABS(DET(1)) .LT. 10.0
              or DET(1) = 0.0.
       INERT  INTEGER(3)
              the inertia of the original matrix.
              INERT(1)  =  number of positive eigenvalues.
              INERT(2)  =  number of negative eigenvalues.
              INERT(3)  =  number of zero eigenvalues.
    Error Condition
       A division by zero may occur if the inverse is requested
       and  DSICO  has set RCOND .EQ. 0.0
       or  DSIFA  has set  INFO .NE. 0 .
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab
    Subroutines and Functions
    BLAS DAXPY,DCOPY,DDOT,DSWAP
    Fortran DABS,IABS,MOD


DSIFA


      SUBROUTINE DSIFA(A,LDA,N,KPVT,INFO)

    DSIFA factors a double precision symmetric matrix by elimination
    with symmetric pivoting.
    To solve  A*X = B , follow DSIFA by DSISL.
    To compute  INVERSE(A)*C , follow DSIFA by DSISL.
    To compute  DETERMINANT(A) , follow DSIFA by DSIDI.
    To compute  INERTIA(A) , follow DSIFA by DSIDI.
    To compute  INVERSE(A) , follow DSIFA by DSIDI.
    On Entry
       A       DOUBLE PRECISION(LDA,N)
               the symmetric matrix to be factored.
               Only the diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       a block diagonal matrix and the multipliers which
               were used to obtain it.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices, TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if the K-th pivot block is singular.  This is
                    not an error condition for this subroutine,
                    but it does indicate that DSISL or DSIDI may
                    divide by zero if called.
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS DAXPY,DSWAP,IDAMAX
    Fortran DABS,DMAX1,DSQRT


DSISL


      SUBROUTINE DSISL(A,LDA,N,KPVT,B)

    DSISL solves the double precision symmetric system
    A * X = B
    using the factors computed by DSIFA.
    On Entry
       A       DOUBLE PRECISION(LDA,N)
               the output from DSIFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       KPVT    INTEGER(N)
               the pivot vector from DSIFA.
       B       DOUBLE PRECISION(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero may occur if  DSICO  has set RCOND .EQ. 0.0
       or  DSIFA  has set INFO .NE. 0  .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL DSIFA(A,LDA,N,KPVT,INFO)
          IF (INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL DSISL(A,LDA,N,KPVT,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS DAXPY,DDOT
    Fortran IABS


DSLVS


      SUBROUTINE DSLVS(WM,IWM,X,TEM)

  DSLVS solves the linear system in the iteration scheme for the
  integrator package DDEBDF.


DSPCO


      SUBROUTINE DSPCO(AP,N,KPVT,RCOND,Z)

    DSPCO factors a double precision symmetric matrix stored in
    packed form by elimination with symmetric pivoting and estimates
    the condition of the matrix.
    IF  RCOND  is not needed, DSPFA is slightly faster.
    To solve  A*X = B , follow DSPCO by DSPSL.
    To compute  INVERSE(A)*C , follow DSPCO by DSPSL.
    To compute  INVERSE(A) , follow DSPCO by DSPDI.
    To compute  DETERMINANT(A) , follow DSPCO by DSPDI.
    To compute  INERTIA(A), follow DSPCO by DSPDI.
    On Entry
       AP      DOUBLE PRECISION (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    Output
       AP      a block diagonal matrix and the multipliers which
               were used to obtain it stored in packed form.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK DSPFA
    BLAS DAXPY,DDOT,DSCAL,DASUM
    Fortran DABS,DMAX1,IABS,DSIGN


DSPDI


      SUBROUTINE DSPDI(AP,N,KPVT,DET,INERT,WORK,JOB)

    DSPDI computes the determinant, inertia and inverse
    of a double precision symmetric matrix using the factors from
    DSPFA, where the matrix is stored in packed form.
    On Entry
       AP      DOUBLE PRECISION (N*(N+1)/2)
               the output from DSPFA.
       N       INTEGER
               the order of the matrix A.
       KPVT    INTEGER(N)
               the pivot vector from DSPFA.
       WORK    DOUBLE PRECISION(N)
               work vector.  Contents ignored.
       JOB     INTEGER
               JOB has the decimal expansion  ABC  where
                  if  C .NE. 0, the inverse is computed,
                  if  B .NE. 0, the determinant is computed,
                  if  A .NE. 0, the inertia is computed.
               For example, JOB = 111  gives all three.
    On Return
       Variables not requested by JOB are not used.
       AP     contains the upper triangle of the inverse of
              the original matrix, stored in packed form.
              The columns of the upper triangle are stored
              sequentially in a one-dimensional array.
       DET    DOUBLE PRECISION(2)
              determinant of original matrix.
              DETERMINANT = DET(1) * 10.0**DET(2)
              with 1.0 .LE. DABS(DET(1)) .LT. 10.0
              or DET(1) = 0.0.
       INERT  INTEGER(3)
              the inertia of the original matrix.
              INERT(1)  =  number of positive eigenvalues.
              INERT(2)  =  number of negative eigenvalues.
              INERT(3)  =  number of zero eigenvalues.
    Error Condition
       A division by zero will occur if the inverse is requested
       and  DSPCO  has set RCOND .EQ. 0.0
       or  DSPFA  has set  INFO .NE. 0 .
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS DAXPY,DCOPY,DDOT,DSWAP
    Fortran DABS,IABS,MOD


DSPFA


      SUBROUTINE DSPFA(AP,N,KPVT,INFO)

    DSPFA factors a double precision symmetric matrix stored in
    packed form by elimination with symmetric pivoting.
    To solve  A*X = B , follow DSPFA by DSPSL.
    To compute  INVERSE(A)*C , follow DSPFA by DSPSL.
    To compute  DETERMINANT(A) , follow DSPFA by DSPDI.
    To compute  INERTIA(A) , follow DSPFA by DSPDI.
    To compute  INVERSE(A) , follow DSPFA by DSPDI.
    On Entry
       AP      DOUBLE PRECISION (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    Output
       AP      a block diagonal matrix and the multipliers which
               were used to obtain it stored in packed form.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices, TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if the K-th pivot block is singular.  This is
                    not an error condition for this subroutine,
                    but it does indicate that DSPSL or DSPDI may
                    divide by zero if called.
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K)  = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS DAXPY,DSWAP,IDAMAX
    Fortran DABS,DMAX1,DSQRT


DSPSL


      SUBROUTINE DSPSL(AP,N,KPVT,B)

    DSISL solves the double precision symmetric system
    A * X = B
    using the factors computed by DSPFA.
    On Entry
       AP      DOUBLE PRECISION(N*(N+1)/2)
               the output from DSPFA.
       N       INTEGER
               the order of the matrix  A .
       KPVT    INTEGER(N)
               the pivot vector from DSPFA.
       B       DOUBLE PRECISION(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero may occur if  DSPCO  has set RCOND .EQ. 0.0
       or  DSPFA  has set INFO .NE. 0  .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL DSPFA(AP,N,KPVT,INFO)
          IF (INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL DSPSL(AP,N,KPVT,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS DAXPY,DDOT
    Fortran IABS


DSVDC


      SUBROUTINE DSVDC(X,LDX,N,P,S,E,U,LDU,V,LDV,WORK,JOB,INFO)

    DSVDC is a subroutine to reduce a double precision NxP matrix X
    by orthogonal transformations U and V to diagonal form.  The
    diagonal elements S(I) are the singular values of X.  The
    columns of U are the corresponding left singular vectors,
    and the columns of V the right singular vectors.
    On Entry
        X         DOUBLE PRECISION(LDX,P), where LDX .GE. N.
                  X contains the matrix whose singular value
                  decomposition is to be computed.  X is
                  destroyed by DSVDC.
        LDX       INTEGER.
                  LDX is the leading dimension of the array X.
        N         INTEGER.
                  N is the number of columns of the matrix X.
        P         INTEGER.
                  P is the number of rows of the matrix X.
        LDU       INTEGER.
                  LDU is the leading dimension of the array U.
                  (See below).
        LDV       INTEGER.
                  LDV is the leading dimension of the array V.
                  (See below).
        WORK      DOUBLE PRECISION(N).
                  WORK is a scratch array.
        JOB       INTEGER.
                  JOB controls the computation of the singular
                  vectors.  It has the decimal expansion AB
                  with the following meaning
                       A .EQ. 0    do not compute the left singular
                                 vectors.
                       A .EQ. 1    return the N left singular vectors
                                 in U.
                       A .GE. 2    return the first MIN(N,P) singular
                                 vectors in U.
                       B .EQ. 0    do not compute the right singular
                                 vectors.
                       B .EQ. 1    return the right singular vectors
                                 in V.
    On Return
        S         DOUBLE PRECISION(MM), where MM=MIN(N+1,P).
                  The first MIN(N,P) entries of S contain the
                  singular values of X arranged in descending
                  order of magnitude.
        E         DOUBLE PRECISION(P).
                  E ordinarily contains zeros.  However see the
                  discussion of INFO for exceptions.
        U         DOUBLE PRECISION(LDU,K), where LDU .GE. N.
                  If JOBA .EQ. 1, then K .EQ. N.
                  If JOBA .GE. 2, then K .EQ. MIN(N,P).
                  U contains the matrix of right singular vectors.
                  U is not referenced if JOBA .EQ. 0.  If N .LE. P
                  or if JOBA .EQ. 2, then U may be identified with X
                  in the subroutine call.
        V         DOUBLE PRECISION(LDV,P), where LDV .GE. P.
                  V contains the matrix of right singular vectors.
                  V is not referenced if JOB .EQ. 0.  If P .LE. N,
                  then V may be identified with X in the
                  subroutine call.
        INFO      INTEGER.
                  The singular values (and their corresponding
                  singular vectors) S(INFO+1),S(INFO+2),...,S(M)
                  are correct (here M=MIN(N,P)).  Thus if
                  INFO .EQ. 0, all the singular values and their
                  vectors are correct.  In any event, the matrix
                  B = TRANS(U)*X*V is the bidiagonal matrix
                  with the elements of S on its diagonal and the
                  elements of E on its super-diagonal (TRANS(U)
                  is the transpose of U).  Thus the singular
                  values of X and B are the same.
    LINPACK.  This version dated 03/19/79 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    DSVDC uses the following functions and subprograms.
    External DROT
    BLAS DAXPY,DDOT,DSCAL,DSWAP,DNRM2,DROTG
    Fortran DABS,DMAX1,MAX0,MIN0,MOD,DSQRT


DSWAP


      SUBROUTINE DSWAP(N,DX,INCX,DY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
      DY  double precision vector with N elements
    INCY  storage spacing between elements of DY
    --Output--
      DX  input vector DY (unchanged if N .LE. 0)
      DY  input vector DX (unchanged if N .LE. 0)
    Interchange double precision DX and double precision DY.
    For I = 0 to N-1, interchange  DX(LX+I*INCX) and DY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


DTRCO


      SUBROUTINE DTRCO(T,LDT,N,RCOND,Z,JOB)

    DTRCO estimates the condition of a double precision triangular
    matrix.
    On Entry
       T       DOUBLE PRECISION(LDT,N)
               T contains the triangular matrix.  The zero
               elements of the matrix are not referenced, and
               the corresponding elements of the array can be
               used to store other information.
       LDT     INTEGER
               LDT is the leading dimension of the array T.
       N       INTEGER
               N is the order of the system.
       JOB     INTEGER
               = 0         T  is lower triangular.
               = nonzero   T  is upper triangular.
    On Return
       RCOND   DOUBLE PRECISION
               an estimate of the reciprocal condition of  T .
               For the system  T*X = B , relative perturbations
               in  T  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  T  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       DOUBLE PRECISION(N)
               a work vector whose contents are usually unimportant.
               If  T  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DSCAL,DASUM
    Fortran DABS,DMAX1,DSIGN


DTRDI


      SUBROUTINE DTRDI(T,LDT,N,DET,JOB,INFO)

    DTRDI computes the determinant and inverse of a double precision
    triangular matrix.
    On Entry
       T       DOUBLE PRECISION(LDT,N)
               T contains the triangular matrix.  The zero
               elements of the matrix are not referenced, and
               the corresponding elements of the array can be
               used to store other information.
       LDT     INTEGER
               LDT is the leading dimension of the array T.
       N       INTEGER
               N is the order of the system.
       JOB     INTEGER
               = 010       no det, inverse of lower triangular.
               = 011       no det, inverse of upper triangular.
               = 100       det, no inverse.
               = 110       det, inverse of lower triangular.
               = 111       det, inverse of upper triangular.
    On Return
       T       inverse of original matrix if requested.
               Otherwise unchanged.
       DET     DOUBLE PRECISION(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               DETERMINANT = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DABS(DET(1)) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
       INFO    INTEGER
               INFO contains zero if the system is nonsingular
               and the inverse is requested.
               Otherwise INFO contains the index of
               a zero diagonal element of T.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DSCAL
    Fortran DABS,MOD


DTRSL


      SUBROUTINE DTRSL(T,LDT,N,B,JOB,INFO)

    DTRSL solves systems of the form
                  T * X = B
    or
                  TRANS(T) * X = B
    where T is a triangular matrix of order N.  Here TRANS(T)
    denotes the transpose of the matrix T.
    On Entry
        T         DOUBLE PRECISION(LDT,N)
                  T contains the matrix of the system.  The zero
                  elements of the matrix are not referenced, and
                  the corresponding elements of the array can be
                  used to store other information.
        LDT       INTEGER
                  LDT is the leading dimension of the array T.
        N         INTEGER
                  N is the order of the system.
        B         DOUBLE PRECISION(N).
                  B contains the right hand side of the system.
        JOB       INTEGER
                  JOB specifies what kind of system is to be solved.
                  If JOB is
                       00   solve T*X=B, T lower triangular,
                       01   solve T*X=B, T upper triangular,
                       10   solve TRANS(T)*X=B, T lower triangular,
                       11   solve TRANS(T)*X=B, T upper triangular.
    On Return
        B         B contains the solution, if INFO .EQ. 0.
                  Otherwise B is unaltered.
        INFO      INTEGER
                  INFO contains zero if the system is nonsingular.
                  Otherwise INFO contains the index of
                  the first zero diagonal element of T.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    Subroutines and Functions
    BLAS DAXPY,DDOT
    Fortran MOD


DU11LS


      SUBROUTINE DU11LS(A,MDA,M,N,UB,DB,MODE,NP,KRANK,KSURE,H,W,EB,IC,

  **** Double Precision version of U11LS ****
      This routine performs a QR factorization of A
      using Householder transformations. Row and
      column pivots are chosen to reduce the growth
      of round-off and to help detect possible rank
      deficiency.


DU11US


      SUBROUTINE DU11US(A,MDA,M,N,UB,DB,MODE,NP,KRANK,KSURE,H,W,EB,IR,

      This routine performs an LQ factorization of the
      matrix A using Householder transformations. Row
      and column pivots are chosen to reduce the growth
      of round-off and to help detect possible rank
      deficiency.


DU12LS


      SUBROUTINE DU12LS(A,MDA,M,N,B,MDB,NB,MODE,KRANK,RNORM,H,W,IC,IR)

       Given the Householder QR factorization of A, this
       subroutine solves the system AX=B. If the system
       is of reduced rank, this routine returns a solution
       according to the selected mode.
      Note - If MODE.NE.2, W is never accessed.


DU12US


      SUBROUTINE DU12US(A,MDA,M,N,B,MDB,NB,MODE,KRANK,RNORM,H,W,IR,IC)

       Given the Householder LQ factorization of A, this
       subroutine solves the system AX=B. If the system
       is of reduced rank, this routine returns a solution
       according to the selected mode.
      Note - If MODE.NE.2, W is never accessed.


DULSIA


      SUBROUTINE DULSIA(A,MDA,M,N,B,MDB,NB,RE,AE,KEY,MODE,NP,KRANK,

      **** Double Precision version of ULSIA ****
    DULSIA computes the minimal length solution(s) to the problem AX=B
    where A is an M by N matrix with M.LE.N and B is the M by NB
    matrix of right hand sides.  User input bounds on the uncertainty
    in the elements of A are used to detect numerical rank deficiency.
    The algorithm employs a row and column pivot strategy to
    minimize the growth of uncertainty and round-off errors.
    DULSIA requires (MDA+1)*N + (MDB+1)*NB + 6*M dimensioned space
  ******************************************************************
  *                                                                *
  *         WARNING - All input arrays are changed on exit.        *
  *                                                                *
  ******************************************************************
    Input.. All TYPE REAL variables are DOUBLE PRECISION
    A(,)          Linear coefficient matrix of AX=B, with MDA the
     MDA,M,N      actual first dimension of A in the calling program.
                  M is the row dimension (no. of EQUATIONS of the
                  problem) and N the col dimension (no. of UNKNOWNS).
                  Must have MDA.GE.M and M.LE.N.
    B(,)          Right hand side(s), with MDB the actual first
     MDB,NB       dimension of B in the calling program. NB is the
                  number of M by 1 right hand sides. Since the solutio
                  is returned in B, must have MDB.GE.N. If NB = 0, B i
                  never accessed.
  ******************************************************************
  *                                                                *
  *         Note - Use of RE and AE are what make this             *
  *                code significantly different from               *
  *                other linear least squares solvers.             *
  *                However, the inexperienced user is              *
  *                advised to set RE=0.,AE=0.,KEY=0.               *
  *                                                                *
  ******************************************************************
    RE(),AE(),KEY
    RE()          RE() is a vector of length N such that RE(I) is
                  the maximum relative uncertainty in row I of
                  the matrix A. The values of RE() must be between
                  0 and 1. A minimum of 10*machine precision will
                  be enforced.
    AE()          AE() is a vector of length N such that AE(I) is
                  the maximum absolute uncertainty in row I of
                  the matrix A. The values of AE() must be greater
                  than or equal to 0.
    KEY           For ease of use, RE and AE may be input as either
                  vectors or scalars. If a scalar is input, the algo-
                  rithm will use that value for each column of A.
                  The parameter KEY indicates whether scalars or
                  vectors are being input.
                       KEY=0     RE scalar  AE scalar
                       KEY=1     RE vector  AE scalar
                       KEY=2     RE scalar  AE vector
                       KEY=3     RE vector  AE vector
    MODE          The integer MODE indicates how the routine
                  is to react if rank deficiency is detected.
                  If MODE = 0 return immediately, no solution
                            1 compute truncated solution
                            2 compute minimal length least squares sol
                  The inexperienced user is advised to set MODE=0
    NP            The first NP rows of A will not be interchanged
                  with other rows even though the pivot strategy
                  would suggest otherwise.
                  The inexperienced user is advised to set NP=0.
    WORK()        A real work array dimensioned 5*M. However,if
                  RE or AE have been specified as vectors, dimension
                  WORK 4*M. If both RE and AE have been specified
                  as vectors, dimension WORK 3*M.
    LW            Actual dimension of WORK
    IWORK()       Integer work array dimensioned at least N+M.
    LIW           Actual dimension of IWORK.
    INFO          Is a flag which provides for the efficient
                  solution of subsequent problems involving the
                  same A but different B.
                  If INFO = 0 original call
                     INFO = 1 subsequent calls
                  On subsequent calls, the user must supply A, KRANK,
                  LW, IWORK, LIW, and the first 2*M locations of WORK
                  as output by the original call to DULSIA. MODE must
                  be equal to the value of MODE in the original call.
                  If MODE.LT.2, only the first N locations of WORK
                  are accessed. AE, RE, KEY, and NP are not accessed.
    Output..All TYPE REAL variables are DOUBLE PRECISION
    A(,)          Contains the lower triangular part of the reduced
                  matrix and the transformation information. It togeth
                  with the first M elements of WORK (see below)
                  completely specify the LQ factorization of A.
    B(,)          Contains the N by NB solution matrix for X.
    KRANK,KSURE   The numerical rank of A,  based upon the relative
                  and absolute bounds on uncertainty, is bounded
                  above by KRANK and below by KSURE. The algorithm
                  returns a solution based on KRANK. KSURE provides
                  an indication of the precision of the rank.
    RNORM()       Contains the Euclidean length of the NB residual
                  vectors  B(I)-AX(I), I=1,NB. If the matrix A is of
                  full rank, then RNORM=0.0.
    WORK()        The first M locations of WORK contain values
                  necessary to reproduce the Householder transformatio
    IWORK()       The first N locations contain the order in
                  which the columns of A were used. The next
                  M locations contain the order in which the
                  rows of A were used.
    INFO          Flag to indicate status of computation on completion
                 -1   Parameter error(s)
                  0 - Rank deficient, no solution
                  1 - Rank deficient, truncated solution
                  2 - Rank deficient, minimal length least squares sol
                  3 - Numerical rank 0, zero solution
                  4 - Rank .LT. NP
                  5 - Full rank


DUSRMT


      SUBROUTINE DUSRMT(I,J,AIJ,INDCAT,PRGOPT,DATTRV,IFLAG)

  The user may supply this code


DVECS


      SUBROUTINE DVECS(NCOMP,LNFC,YHP,WORK,IWORK,INHOMO,IFLAG)



DVNORM


      DOUBLE PRECISION FUNCTION DVNORM(V,NCOMP)



DVNRMS


      DOUBLE PRECISION FUNCTION DVNRMS(N,V,W)



DVOUT


      SUBROUTINE DVOUT(N,DX,IFMT,IDIGIT)

    REVISED FEB. 27, 1981.
    DOUBLE PRECISION VECTOR OUTPUT ROUTINE.
 INPUT..
 N,DX(*) PRINT THE DOUBLE PRECISION ARRAY DX(I),I=1,...,N, ON
         OUTPUT UNIT LOUT. THE HEADING IN THE FORTRAN FORMAT
         STATEMENT IFMT(*), DESCRIBED BELOW, IS PRINTED AS A FIRST
         STEP. THE COMPONENTS DX(I) ARE INDEXED, ON OUTPUT,
         IN A PLEASANT FORMAT.
 IFMT(*) A FORTRAN FORMAT STATEMENT. THIS IS PRINTED ON OUTPUT
         UNIT LOUT WITH THE VARIABLE FORMAT FORTRAN STATEMENT
               WRITE(LOUT,IFMT)
 IDIGIT  PRINT AT LEAST IABS(IDIGIT) DECIMAL DIGITS PER NUMBER.
         THE SUBPROGRAM WILL CHOOSE THAT INTEGER 4,6,10 OR 14
         WHICH WILL PRINT AT LEAST IABS(IDIGIT) NUMBER OF
         PLACES.  IF IDIGIT.LT.0, 72 PRINTING COLUMNS ARE UTILIZED
         TO WRITE EACH LINE OF OUTPUT OF THE ARRAY DX(*). (THIS
         CAN BE USED ON MOST TIME-SHARING TERMINALS). IF
         IDIGIT.GE.0, 133 PRINTING COLUMNS ARE UTILIZED. (THIS CAN
         BE USED ON MOST LINE PRINTERS).
 EXAMPLE..
 PRINT AN ARRAY CALLED (COSTS OF PURCHASES) OF LENGTH 100 SHOWING
 6 DECIMAL DIGITS PER NUMBER. THE USER IS RUNNING ON A TIME-SHARING
 SYSTEM WITH A 72 COLUMN OUTPUT DEVICE.
    DOUBLE PRECISION COSTS(100)
    N = 100
    IDIGIT = -6
    CALL DVOUT(N,COSTS,'(''1COSTS OF PURCHASES'')',IDIGIT)
    AUTHORS    JOHN A. WISNIEWSKI   SANDIA LABS ALBUQUERQUE.
               RICHARD J. HANSON    SANDIA LABS ALBUQUERQUE.
    DATE       JULY 27,1978.


DWNLIT


      SUBROUTINE DWNLIT(W,MDW,M,N,L,IPIVOT,ITYPE,H,SCALE,RNORM,IDOPE,

    THIS IS A COMPANION SUBPROGRAM TO DWNNLS( ).
    THE DOCUMENTATION FOR DWNNLS( ) HAS MORE COMPLETE
    USAGE INSTRUCTIONS.
    NOTE  THE M BY (N+1) MATRIX W( , ) CONTAINS THE RT. HAND SIDE
          B AS THE (N+1)ST COL.
    TRIANGULARIZE L1 BY L1 SUBSYSTEM, WHERE L1=MIN(M,L), WITH
    COL INTERCHANGES.
     REVISED APRIL 19, 1981.


DWNLSM


      SUBROUTINE DWNLSM(W,MDW,MME,MA,N,L,PRGOPT,X,RNORM,MODE,IPIVOT,

  **** Double Precision version of WNLSM ****
    This is a companion subprogram to DWNNLS( ).
    The documentation for DWNNLS( ) has more complete
    usage instructions.
    Written by Karen H. Haskell, Sandia Laboratories,
    with the help of R.J. Hanson, Sandia Laboratories,
    December 1976 - January 1978.
    Revised March 4, 1982.
    In addition to the parameters discussed in the prologue to
    subroutine DWNNLS, the following work arrays are used in
    subroutine DWNLSM  (they are passed through the calling
    sequence from DWNNLS for purposes of variable dimensioning).
    Their contents will in general be of no interest to the user.
        IPIVOT(*)
           An array of length N.  Upon completion it contains the
        pivoting information for the cols of W(*,*).
        ITYPE(*)
           An array of length M which is used to keep track
        of the classification of the equations.  ITYPE(I)=0
        denotes equation I as an equality constraint.
        ITYPE(I)=1 denotes equation I as a least squares
        equation.
        WD(*)
           An array of length N.  Upon completion it contains the
        dual solution vector.
        H(*)
           An array of length N.  Upon completion it contains the
        pivot scalars of the Householder transformations performed
        in the case KRANK.LT.L.
        SCALE(*)
           An array of length M which is used by the subroutine
        to store the diagonal matrix of weights.
        These are used to apply the modified Givens
        transformations.
        Z(*),TEMP(*)
           Working arrays of length N.
        D(*)
           An array of length N that contains the
        column scaling for the matrix (E).
                                      (A)


EISDOC


      SUBROUTINE EISDOC

                **********EISPACK Routines**********
single double complx
------ ------ ------
RS       -    CH     Computes eigenvalues and, optionally,
                     eigenvectors of real symmetric
                     (complex Hermitian) matrix.
RSP      -      -    Compute eigenvalues and, optionally,
                     eigenvectors of real symmetric matrix
                     packed into a one dimensional array.
RG       -    CG     Computes eigenvalues and, optionally,
                     eigenvectors of a real (complex) general
                     matrix.
BISECT   -      -    Compute eigenvalues of symmetric tridiagonal
                     matrix given interval using Sturm sequencing.
IMTQL1   -      -    Computes eigenvalues of symmetric tridiagonal
                     matrix implicit QL method.
IMTQL2   -      -    Computes eigenvalues and eigenvectors of
                     symmetric tridiagonal matrix using
                     implicit QL method.
IMTQLV   -      -    Computes eigenvalues of symmetric tridiagonal
                     matrix by the implicit QL method.
                     Eigenvectors may be computed later.
RATQR    -      -    Computes largest or smallest eigenvalues
                     of symmetric tridiagonal matrix using
                     rational QR method with Newton correction.
RST      -      -    Compute eigenvalues and, optionally,
                     eigenvectors of real symmetric tridiagonal
                     matrix.
RT       -      -    Compute eigenvalues and eigenvectors of
                     a special real tridiagonal matrix.
TQL1     -      -    Compute eigenvalues of symmetric tridiagonal
                     matrix by QL method.
TQL2     -      -    Compute eigenvalues and eigenvectors
                     of symmetric tridiagonal matrix.
TQLRAT   -      -    Computes eigenvalues of symmetric
                     tridiagonal matrix a rational variant
                     of the QL method.
TRIDIB   -      -    Computes eigenvalues of symmetric
                     tridiagonal matrix given interval using
                     Sturm sequencing.
TSTURM   -      -    Computes eigenvalues of symmetric tridiagonal
                     matrix given interval and eigenvectors
                     by Sturm sequencing.  This subroutine
                     is a translation of the ALGOL procedure
                     TRISTURM by Peters and Wilkinson. HANDBOOK
                     FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA,
                     418-439(1971).
BQR      -      -    Computes some of the eigenvalues of a real
                     symmetric matrix using the QR method with
                     shifts of origin.
RSB      -      -    Computes eigenvalues and, optionally,
                     eigenvectors of symmetric band matrix.
RSG      -      -    Computes eigenvalues and, optionally,
                     eigenvectors of symmetric generalized
                     eigenproblem: A*X=(LAMBDA)*B*X
RSGAB    -      -    Computes eigenvalues and, optionally,
                     eigenvectors of symmetric generalized
                     eigenproblem: A*B*X=(LAMBDA)*X
RSGBA    -      -    Computes eigenvalues and, optionally,
                     eigenvectors of symmetric generalized
                     eigenproblem: B*A*X=(LAMBDA)*X
RGG      -      -    Computes eigenvalues and eigenvectors
                     for real generalized eigenproblem:
                     A*X=(LAMBDA)*B*X.
BALANC   -    CBAL   Balances a general real (complex)
                     matrix and isolates eigenvalues whenever
                     possible.
BANDR    -      -    Reduces real symmetric band matrix
                     to symmetric tridiagonal matrix and,
                     optionally, accumulates orthogonal similarity
                     transformations.
HTRID3   -      -    Reduces complex Hermitian (packed) matrix
                     to real symmetric tridiagonal matrix by unitary
                     similarity transformations.
HTRIDI   -      -    Reduces complex Hermitian matrix to real
                     symmetric tridiagonal matrix using unitary
                     similarity transformations.
TRED1    -      -    Reduce real symmetric matrix to symmetric
                     tridiagonal matrix using orthogonal
                     similarity transformations.
TRED2    -      -    Reduce real symmetric matrix to symmetric
                     tridiagonal matrix using and accumulating
                     orthogonal transformations.
TRED3    -      -    Reduce  symmetric matrix stored in packed
                     form to symmetric tridiagonal matrix using
                     orthogonal transformations.
ELMHES   -    COMHES Reduces real (complex) general matrix to
                     upper Hessenberg form using stabilized
                     elementary similarity transformations.
ORTHES   -    CORTH  Reduces real (complex) general matrix to upper
                     Hessenberg form orthogonal (unitary)
                     similarity transformations.
QZHES    -      -    The first step of the QZ algorithm for solving
                     generalized matrix eigenproblems.  Accepts
                     a pair of real general matrices and reduces
                     one of them to upper Hessenberg and the other
                     to upper triangular form using orthogonal
                     transformations. Usually followed by QZIT,
                     QZVAL, QZ
QZIT     -      -    The second step of the QZ algorithm for
                     generalized eigenproblems.  Accepts an upper
                     Hessenberg and an upper triangular matrix
                     and reduces the former to quasi-triangular
                     form while preserving the form of the latter.
                     Usually preceeded by QZHES and followed by QZVAL
                     and QZVEC.
FIGI     -      -    Transforms certain real non-symmetric
                     tridiagonal matrix to symmetric tridiagonal
                     matrix.
FIGI2    -      -    Transforms certain real non-symmetric
                     tridiagonal matrix to symmetric tridiagonal
                     matrix.
REDUC    -      -    Reduces generalized symmetric eigenproblem
                     A*X=(LAMBDA)*B*X, to standard symmetric
                     eigenproblem using Cholesky factorization.
REDUC2   -      -    Reduces certain generalized symmetric
                     eigenproblems standard symmetric eigenproblem,
                     using Cholesky factorization.
  -      -    COMLR  Computes eigenvalues of a complex upper
                     Hessenberg matrix using the modified LR method.
  -      -    COMLR2 Computes eigenvalues and eigenvectors of
                     complex upper Hessenberg matrix using
                     modified LR method.
HQR      -    COMQR  Computes eigenvalues of a real (complex)
                     upper Hessenberg matrix using the QR method.
HQR2     -    COMQR2 Computes eigenvalues and eigenvectors of
                     real (complex) upper Hessenberg matrix
                     using QR method.
INVIT    -    CINVIT Computes eigenvectors of real (complex)
                     Hessenberg matrix associated with specified
                     eigenvalues by inverse iteration.
QZVAL    -      -    The third step of the QZ algorithm for
                     generalized eigenproblems.  Accepts a pair
                     of real matrices, one quasi-triangular form
                     and the other in upper triangular form and
                     computes the eigenvalues of the associated
                     eigenproblem.  Usually preceded by QZHES,
                     QZIT, and followed by QZVEC.
BANDV    -      -    Forms eigenvectors of real symmetric band
                     matrix associated with a set of ordered
                     approximate eigenvalue by inverse iteration.
QZVEC    -      -    The optional fourth step of the QZ algorithm
                     for generalized eigenproblems.  Accepts
                     a matrix in quasi-triangular form and another
                     in upper triangular and computes the
                     eigenvectors of the triangular problem
                     and transforms them back to the original
                     coordinates Usually preceded by QZHES, QZIT,
                     QZVAL.
TINVIT   -      -    Eigenvectors of symmetric tridiagonal
                     matrix corresponding to some specified
                     eigenvalues, using inverse iteration.
BAKVEC   -      -    Forms eigenvectors of certain real
                     non-symmetric tridiagonal matrix from
                     symmetric tridiagonal matrix output from FIGI.
BALBAK   -    CBABK2 Forms eigenvectors of real (complex) general
                     matrix from eigenvectors of matrix output
                     from BALANC (CBAL).
ELMBAK   -    COMBAK Forms eigenvectors of real (complex) general
                     matrix from eigenvectors of upper Hessenberg
                     matrix output from ELMHES (COMHES).
ELTRAN   -      -    Accumulates the stabilized elementary
                     similarity transformations used in the
                     reduction of a real general matrix to upper
                     Hessenberg form by ELMHES.
HTRIB3   -      -    Computes eigenvectors of complex Hermitian
                     matrix from eigenvectors of real symmetric
                     tridiagonal matrix output from HTRID3.
HTRIBK   -      -    Forms eigenvectors of complex Hermitian
                     matrix from eigenvectors of real symmetric
                     tridiagonal matrix output from HTRIDI.
ORTBAK   -    CORTB  Forms eigenvectors of general real (complex)
                     matrix from eigenvectors of upper Hesenberg
                     matrix output from ORTHES (CORTH).
ORTRAN   -      -    Accumulates orthogonal similarity
                     transformations in reduction of real general
                     matrix by ORTHES.
REBAK    -      -    Forms eigenvectors of generalized symmetric
                     eigensystem from eigenvectors of derived
                     matrix output from REDUC or REDUC2.
REBAKB   -      -    Forms eigenvectors of generalized symmetric
                     eigensystem from eigenvectors of derived
                     matrix output from REDUC2
TRBAK1   -      -    Forms the eigenvectors of real symmetric
                     matrix from eigenvectors of symmetric
                     tridiagonal matrix formed by TRED1.
TRBAK3   -      -    Forms eigenvectors of real symmetric matrix
                     from the eigenvectors of symmetric tridiagonal
                     matrix formed by TRED3.
MINFIT   -      -    Compute Singular Value Decomposition
                     of rectangular matrix and solve related
                     Linear Least Squares problem.


ELMBAK


      SUBROUTINE ELMBAK(NM,LOW,IGH,A,INT,M,Z)

    This subroutine is a translation of the ALGOL procedure ELMBAK,
    NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    This subroutine forms the eigenvectors of a REAL GENERAL
    matrix by back transforming those of the corresponding
    upper Hessenberg matrix determined by  ELMHES.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       LOW and IGH are integers determined by the balancing
         subroutine  BALANC.  If  BALANC  has not been used,
         set LOW=1 and IGH equal to the order of the matrix.
       A contains the multipliers which were used in the
         reduction by  ELMHES  in its lower triangle
         below the subdiagonal.
       INT contains information on the rows and columns
         interchanged in the reduction by  ELMHES.
         Only elements LOW through IGH are used.
       M is the number of columns of Z to be back transformed.
       Z contains the real and imaginary parts of the eigen-
         vectors to be back transformed in its first M columns.
    On OUTPUT
       Z contains the real and imaginary parts of the
         transformed eigenvectors in its first M columns.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



ELMHES


      SUBROUTINE ELMHES(NM,N,LOW,IGH,A,INT)

    This subroutine is a translation of the ALGOL procedure ELMHES,
    NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    Given a REAL GENERAL matrix, this subroutine
    reduces a submatrix situated in rows and columns
    LOW through IGH to upper Hessenberg form by
    stabilized elementary similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  BALANC.  If  BALANC  has not been used,
         set LOW=1, IGH=N.
       A contains the input matrix.
    On OUTPUT
       A contains the Hessenberg matrix.  The multipliers
         which were used in the reduction are stored in the
         remaining triangle under the Hessenberg matrix.
       INT contains information on the rows and columns
         interchanged in the reduction.
         Only elements LOW through IGH are used.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



ELTRAN


      SUBROUTINE ELTRAN(NM,N,LOW,IGH,A,INT,Z)

    This subroutine is a translation of the ALGOL procedure ELMTRANS,
    NUM. MATH. 16, 181-204(1970) by Peters and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
    This subroutine accumulates the stabilized elementary
    similarity transformations used in the reduction of a
    REAL GENERAL matrix to upper Hessenberg form by  ELMHES.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  BALANC.  If  BALANC  has not been used,
         set LOW=1, IGH=N.
       A contains the multipliers which were used in the
         reduction by  ELMHES  in its lower triangle
         below the subdiagonal.
       INT contains information on the rows and columns
         interchanged in the reduction by  ELMHES.
         only elements LOW through IGH are used.
    On OUTPUT
       Z contains the transformation matrix produced in the
         reduction by  ELMHES.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



ENORM


      REAL FUNCTION ENORM(N,X)



EPMACH


      FUNCTION EPMACH(DUM)



FIGI


      SUBROUTINE FIGI(NM,N,T,D,E,E2,IERR)

    Given a NONSYMMETRIC TRIDIAGONAL matrix such that the products
    of corresponding pairs of off-diagonal elements are all
    non-negative, this subroutine reduces it to a symmetric
    tridiagonal matrix with the same eigenvalues.  If, further,
    a zero product only occurs when both factors are zero,
    the reduced matrix is similar to the original matrix.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       T contains the input matrix.  Its subdiagonal is
         stored in the last N-1 positions of the first column,
         its diagonal in the N positions of the second column,
         and its superdiagonal in the first N-1 positions of
         the third column.  T(1,1) and T(N,3) are arbitrary.
    On OUTPUT
       T is unaltered.
       D contains the diagonal elements of the symmetric matrix.
       E contains the subdiagonal elements of the symmetric
         matrix in its last N-1 positions.  E(1) is not set.
       E2 contains the squares of the corresponding elements of E.
         E2 may coincide with E if the squares are not needed.
       IERR is set to
         Zero       for normal return,
         N+I        if T(I,1)*T(I-1,3) is negative,
         -(3*N+I)   if T(I,1)*T(I-1,3) is zero with one factor
                    non-zero.  In this case, the eigenvectors of
                    the symmetric matrix are not simply related
                    to those of  T  and should not be sought.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



FIGI2


      SUBROUTINE FIGI2(NM,N,T,D,E,Z,IERR)

    Given a NONSYMMETRIC TRIDIAGONAL matrix such that the products
    of corresponding pairs of off-diagonal elements are all
    non-negative, and zero only when both factors are zero, this
    subroutine reduces it to a SYMMETRIC TRIDIAGONAL matrix
    using and accumulating diagonal similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       T contains the input matrix.  Its subdiagonal is
         stored in the last N-1 positions of the first column,
         its diagonal in the N positions of the second column,
         and its superdiagonal in the first N-1 positions of
         the third column.  T(1,1) and T(N,3) are arbitrary.
    On OUTPUT
       T is unaltered.
       D contains the diagonal elements of the symmetric matrix.
       E contains the subdiagonal elements of the symmetric
         matrix in its last N-1 positions.  E(1) is not set.
       Z contains the transformation matrix produced in
         the reduction.
       IERR is set to
         Zero       for normal return,
         N+I        if T(I,1)*T(I-1,3) is negative,
         2*N+I      if T(I,1)*T(I-1,3) is zero with
                    one factor non-zero.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



HFTI


      SUBROUTINE HFTI(A,MDA,M,N,B,MDB,NB,TAU,KRANK,RNORM,H,G,IP)

    DIMENSION A(MDA,N),(B(MDB,NB) or B(M)),RNORM(NB),H(N),G(N),IP(N)
    This subroutine solves a linear least squares problem or a set of
    linear least squares problems having the same matrix but different
    right-side vectors.  The problem data consists of an M by N matrix
    A, an M by NB matrix B, and an absolute tolerance parameter TAU
    whose usage is described below.  The NB column vectors of B
    represent right-side vectors for NB distinct linear least squares
    problems.
    This set of problems can also be written as the matrix least
    squares problem
                      AX = B,
    where X is the N by NB solution matrix.
    Note that if B is the M by M identity matrix, then X will be the
    pseudo-inverse of A.
    This subroutine first transforms the augmented matrix (A B) to a
    matrix (R C) using premultiplying Householder transformations with
    column interchanges.  All subdiagonal elements in the matrix R are
    zero and its diagonal elements satisfy
                      ABS(R(I,I)).GE.ABS(R(I+1,I+1)),
                      I = 1,...,L-1, where
                      L = MIN(M,N).
    The subroutine will compute an integer, KRANK, equal to the number
    of diagonal terms of R that exceed TAU in magnitude. Then a
    solution of minimum Euclidean length is computed using the first
    KRANK rows of (R C).
    To be specific we suggest that the user consider an easily
    computable matrix norm, such as, the maximum of all column sums of
    magnitudes.
    Now if the relative uncertainty of B is EPS, (norm of uncertainty/
    norm of B), it is suggested that TAU be set approximately equal to
    EPS*(norm of A).
    The user must dimension all arrays appearing in the call list..
    A(MDA,N),(B(MDB,NB) or B(M)),RNORM(NB),H(N),G(N),IP(N).  This
    permits the solution of a range of problems in the same array
    space.
    The entire set of parameters for HFTI are
    INPUT..
    A(*,*),MDA,M,N    The array A(*,*) initially contains the M by N
                      matrix A of the least squares problem AX = B.
                      The first dimensioning parameter of the array
                      A(*,*) is MDA, which must satisfy MDA.GE.M
                      Either M.GE.N or M.LT.N is permitted.  There
                      is no restriction on the rank of A.  The
                      condition MDA.LT.M is considered an error.
    B(*),MDB,NB       If NB = 0 the subroutine will perform the
                      orthogonal decomposition but will make no
                      references to the array B(*).  If NB.GT.0
                      the array B(*) must initially contain the M by
                      NB matrix B of the least squares problem AX =
                      B.  If NB.GE.2 the array B(*) must be doubly
                      subscripted with first dimensioning parameter
                      MDB.GE.MAX(M,N).  If NB = 1 the array B(*) may
                      be either doubly or singly subscripted.  In
                      the latter case the value of MDB is arbitrary
                      but it should be set to some valid integer
                      value such as MDB = M.
                      The condition of NB.GT.1.AND.MDB.LT. MAX(M,N)
                      is considered an error.
    TAU               Absolute tolerance parameter provided by user
                      for pseudorank determination.
    H(*),G(*),IP(*)   Arrays of working space used by HFTI.
    OUTPUT..
    A(*,*)            The contents of the array A(*,*) will be
                      modified by the subroutine. These contents
                      are not generally required by the user.
    B(*)              On return the array B(*) will contain the N by
                      NB solution matrix X.
    KRANK             Set by the subroutine to indicate the
                      pseudorank of A.
    RNORM(*)          On return, RNORM(J) will contain the Euclidean
                      norm of the residual vector for the problem
                      defined by the J-th column vector of the array
                      B(*,*) for J = 1,...,NB.
    H(*),G(*)         On return these arrays respectively contain
                      elements of the pre- and post-multiplying
                      Householder transformations used to compute
                      the minimum Euclidean length solution.
    IP(*)             Array in which the subroutine records indices
                      describing the permutation of column vectors.
                      The contents of arrays H(*),G(*) and IP(*)
                      are not generally required by the user.


HKSEQ


      SUBROUTINE HKSEQ(X,M,H,IERR)

     HKSEQ is an adaptation of subroutine PSIFN, ACM Trans. Math.
     Software, 1983 to generate H(K,X)=(-X)**(K+1)*(PSI(K,X)-
     -PSI(K,X+0.5))/GAMMA(K+1) FOR K=0,...,M.


HQR


      SUBROUTINE HQR(NM,N,LOW,IGH,H,WR,WI,IERR)

    This subroutine is a translation of the ALGOL procedure HQR,
    NUM. MATH. 14, 219-231(1970) by Martin, Peters, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 359-371(1971).
    This subroutine finds the eigenvalues of a REAL
    UPPER Hessenberg matrix by the QR method.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  BALANC.  If  BALANC  has not been used,
         set LOW=1, IGH=N.
       H contains the upper Hessenberg matrix.  Information about
         the transformations used in the reduction to Hessenberg
         form by  ELMHES  or  ORTHES, if performed, is stored
         in the remaining triangle under the Hessenberg matrix.
    On OUTPUT
       H has been destroyed.  Therefore, it must be saved
         before calling  HQR  if subsequent calculation and
         back transformation of eigenvectors is to be performed.
       WR and WI contain the real and imaginary parts,
         respectively, of the eigenvalues.  The eigenvalues
         are unordered except that complex conjugate pairs
         of values appear consecutively with the eigenvalue
         having the positive imaginary part first.  If an
         error exit is made, the eigenvalues should be correct
         for indices IERR+1,...,N.
       IERR is set to
         Zero       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after a total of 30*N iterations.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



HQR2


      SUBROUTINE HQR2(NM,N,LOW,IGH,H,WR,WI,Z,IERR)

    This subroutine is a translation of the ALGOL procedure HQR2,
    NUM. MATH. 16, 181-204(1970) by Peters and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
    This subroutine finds the eigenvalues and eigenvectors
    of a REAL UPPER Hessenberg matrix by the QR method.  The
    eigenvectors of a REAL GENERAL matrix can also be found
    if  ELMHES  and  ELTRAN  or  ORTHES  and  ORTRAN  have
    been used to reduce this general matrix to Hessenberg form
    and to accumulate the similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  BALANC.  If  BALANC  has not been used,
         set LOW=1, IGH=N.
       H contains the upper Hessenberg matrix.
       Z contains the transformation matrix produced by  ELTRAN
         after the reduction by  ELMHES, or by  ORTRAN  after the
         reduction by  ORTHES, if performed.  If the eigenvectors
         of the Hessenberg matrix are desired, Z must contain the
         identity matrix.
    On OUTPUT
       H has been destroyed.
       WR and WI contain the real and imaginary parts,
         respectively, of the eigenvalues.  The eigenvalues
         are unordered except that complex conjugate pairs
         of values appear consecutively with the eigenvalue
         having the positive imaginary part first.  If an
         error exit is made, the eigenvalues should be correct
         for indices IERR+1,...,N.
       Z contains the real and imaginary parts of the eigenvectors.
         If the I-th eigenvalue is real, the I-th column of Z
         contains its eigenvector.  If the I-th eigenvalue is complex
         with positive imaginary part, the I-th and (I+1)-th
         columns of Z contain the real and imaginary parts of its
         eigenvector.  The eigenvectors are unnormalized.  If an
         error exit is made, none of the eigenvectors has been found.
       IERR is set to
         Zero       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after a total of 30*N iterations.
    Calls CDIV for complex division.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



HSTART


      SUBROUTINE HSTART(F,NEQ,A,B,Y,YPRIME,ETOL,MORDER,SMALL,BIG,SPY,



HTRIB3


      SUBROUTINE HTRIB3(NM,N,A,TAU,M,ZR,ZI)

    This subroutine is a translation of a complex analogue of
    the ALGOL procedure TRBAK3, NUM. MATH. 11, 181-195(1968)
    by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine forms the eigenvectors of a COMPLEX HERMITIAN
    matrix by back transforming those of the corresponding
    real symmetric tridiagonal matrix determined by  HTRID3.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       A contains information about the unitary transformations
         used in the reduction by  HTRID3.
       TAU contains further information about the transformations.
       M IS the number of eigenvectors to be back transformed.
       ZR contains the eigenvectors to be back transformed
         in its first M columns.
    On OUTPUT
       ZR and ZI contain the real and imaginary parts,
         respectively, of the transformed eigenvectors
         in their first M columns.
    NOTE that the last component of each returned vector
    is real and that vector Euclidean norms are preserved.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



HTRIBK


      SUBROUTINE HTRIBK(NM,N,AR,AI,TAU,M,ZR,ZI)

    This subroutine is a translation of a complex analogue of
    the ALGOL procedure TRBAK1, NUM. MATH. 11, 181-195(1968)
    by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine forms the eigenvectors of a COMPLEX HERMITIAN
    matrix by back transforming those of the corresponding
    real symmetric tridiagonal matrix determined by  HTRIDI.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       AR and AI contain information about the unitary trans-
         formations used in the reduction by  HTRIDI  in their
         full lower triangles except for the diagonal of AR.
       TAU contains further information about the transformations.
       M is the number of eigenvectors to be back transformed.
       ZR contains the eigenvectors to be back transformed
         in its first M columns.
    On OUTPUT
       ZR and ZI contain the real and imaginary parts,
         respectively, of the transformed eigenvectors
         in their first M columns.
    Note that the last component of each returned vector
    is real and that vector Euclidean norms are preserved.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



HTRID3


      SUBROUTINE HTRID3(NM,N,A,D,E,E2,TAU)

    This subroutine is a translation of a complex analogue of
    the ALGOL procedure TRED3, NUM. MATH. 11, 181-195(1968)
    by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine reduces a COMPLEX HERMITIAN matrix, stored as
    a single square array, to a real symmetric tridiagonal matrix
    using unitary similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       A contains the lower triangle of the complex hermitian input
         matrix.  The real parts of the matrix elements are stored
         in the full lower triangle of A, and the imaginary parts
         are stored in the transposed positions of the strict upper
         triangle of A.  No storage is required for the zero
         imaginary parts of the diagonal elements.
    On OUTPUT
       A contains information about the unitary transformations
         used in the reduction.
       D contains the diagonal elements of the the tridiagonal matrix.
       E contains the subdiagonal elements of the tridiagonal
         matrix in its last N-1 positions.  E(1) is set to zero.
       E2 contains the squares of the corresponding elements of E.
         E2 may coincide with E if the squares are not needed.
       TAU contains further information about the transformations.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



HTRIDI


      SUBROUTINE HTRIDI(NM,N,AR,AI,D,E,E2,TAU)

    This subroutine is a translation of a complex analogue of
    the ALGOL procedure TRED1, NUM. MATH. 11, 181-195(1968)
    by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine reduces a COMPLEX HERMITIAN matrix
    to a real symmetric tridiagonal matrix using
    unitary similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       AR and AI contain the real and imaginary parts,
         respectively, of the complex hermitian input matrix.
         Only the lower triangle of the matrix need be supplied.
    On OUTPUT
       AR and AI contain information about the unitary trans-
         formations used in the reduction in their full lower
         triangles.  Their strict upper triangles and the
         diagonal of AR are unaltered.
       D contains the diagonal elements of the tridiagonal matrix.
       E contains the subdiagonal elements of the tridiagonal
         matrix in its last N-1 positions.  E(1) is set to zero.
       E2 contains the squares of the corresponding elements of E.
         E2 may coincide with E if the squares are not needed.
       TAU contains further information about the transformations.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



ICAMAX


      INTEGER FUNCTION ICAMAX(N,CX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
    --Output--
  ICAMAX  smallest index (zero if N .LE. 0)
     Returns the index of the component of CX having the
     largest sum of magnitudes of real and imaginary parts.
    ICAMAX = first I, I = 1 to N, to minimize
       ABS(REAL(CX(1-INCX+I*INCX))) + ABS(IMAG(CX(1-INCX+I*INCX)))


IDAMAX


      INTEGER FUNCTION IDAMAX(N,DX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      DX  double precision vector with N elements
    INCX  storage spacing between elements of DX
    --Output--
  IDAMAX  smallest index (zero if N .LE. 0)
    Find smallest index of maximum magnitude of double precision DX.
    IDAMAX =  first I, I = 1 to N, to minimize  ABS(DX(1-INCX+I*INCX)


IMTQL1


      SUBROUTINE IMTQL1(N,D,E,IERR)

    This subroutine is a translation of the ALGOL procedure IMTQL1,
    NUM. MATH. 12, 377-383(1968) by Martin and Wilkinson,
    as modified in NUM. MATH. 15, 450(1970) by Dubrulle.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 241-248(1971).
    This subroutine finds the eigenvalues of a SYMMETRIC
    TRIDIAGONAL matrix by the implicit QL method.
    On INPUT
       N is the order of the matrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
     On OUTPUT
       D contains the eigenvalues in ascending order.  If an
         error exit is made, the eigenvalues are correct and
         ordered for indices 1,2,...IERR-1, but may not be
         the smallest eigenvalues.
       E has been destroyed.
       IERR is set to
         ZERO       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after 30 iterations.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



IMTQL2


      SUBROUTINE IMTQL2(NM,N,D,E,Z,IERR)

    This subroutine is a translation of the ALGOL procedure IMTQL2,
    NUM. MATH. 12, 377-383(1968) by Martin and Wilkinson,
    as modified in NUM. MATH. 15, 450(1970) by Dubrulle.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 241-248(1971).
    This subroutine finds the eigenvalues and eigenvectors
    of a SYMMETRIC TRIDIAGONAL matrix by the implicit QL method.
    The eigenvectors of a FULL SYMMETRIC matrix can also
    be found if  TRED2  has been used to reduce this
    full matrix to tridiagonal form.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
       Z contains the transformation matrix produced in the
         reduction by  TRED2, if performed.  If the eigenvectors
         of the tridiagonal matrix are desired, Z must contain
         the identity matrix.
     On OUTPUT
       D contains the eigenvalues in ASCENDING order.  If an
         error exit is made, the eigenvalues are correct but
         UNORDERED for indices 1,2,...,IERR-1.
       E has been destroyed.
       Z contains orthonormal eigenvectors of the symmetric
         tridiagonal (or full) matrix.  If an error exit is made,
         Z contains the eigenvectors associated with the stored
         eigenvalues.
       IERR is set to
         ZERO       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after 30 iterations.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



IMTQLV


      SUBROUTINE IMTQLV(N,D,E,E2,W,IND,IERR,RV1)

    This subroutine is a variant of  IMTQL1  which is a translation of
    ALGOL procedure IMTQL1, NUM. MATH. 12, 377-383(1968) by Martin and
    Wilkinson, as modified in NUM. MATH. 15, 450(1970) by Dubrulle.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 241-248(1971).
    This subroutine finds the eigenvalues of a SYMMETRIC TRIDIAGONAL
    matrix by the implicit QL method and associates with them
    their corresponding submatrix indices.
    On INPUT
       N is the order of the matrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
       E2 contains the squares of the corresponding elements of E.
         E2(1) is arbitrary.
    On OUTPUT
       D and E are unaltered.
       Elements of E2, corresponding to elements of E regarded
         as negligible, have been replaced by zero causing the
         matrix to split into a direct sum of submatrices.
         E2(1) is also set to zero.
       W contains the eigenvalues in ascending order.  If an
         error exit is made, the eigenvalues are correct and
         ordered for indices 1,2,...IERR-1, but may not be
         the smallest eigenvalues.
       IND contains the submatrix indices associated with the
         corresponding eigenvalues in W -- 1 for eigenvalues
         belonging to the first submatrix from the top,
         2 for those belonging to the second submatrix, etc.
       IERR is set to
         ZERO       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after 30 iterations.
       RV1 is a temporary storage array.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



INDXA


      SUBROUTINE INDXA(I,IR,IDXA,NA)



INDXB


      SUBROUTINE INDXB(I,IR,IDX,IDP)



INDXC


      SUBROUTINE INDXC(I,IR,IDXC,NC)



INVIT


      SUBROUTINE INVIT(NM,N,A,WR,WI,SELECT,MM,M,Z,IERR,RM1,RV1,RV2)

    This subroutine is a translation of the ALGOL procedure INVIT
    by Peters and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 418-439(1971).
    This subroutine finds those eigenvectors of a REAL UPPER
    Hessenberg matrix corresponding to specified eigenvalues,
    using inverse iteration.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       A contains the Hessenberg matrix.
       WR and WI contain the real and imaginary parts, respectively,
         of the eigenvalues of the matrix.  The eigenvalues MUST be
         stored in a manner IDENTICAL to that of subroutine  HQR,
         which recognizes possible splitting of the matrix.
       SELECT specifies the eigenvectors to be found. The
         eigenvector corresponding to the J-th eigenvalue is
         specified by setting SELECT(J) to .TRUE.
       MM should be set to an upper bound for the number of
         columns required to store the eigenvectors to be found.
         NOTE that two columns are required to store the
         eigenvector corresponding to a complex eigenvalue.
    On OUTPUT
       A and WI are unaltered.
       WR may have been altered since close eigenvalues are perturbed
         slightly in searching for independent eigenvectors.
       SELECT may have been altered.  If the elements corresponding
         to a pair of conjugate complex eigenvalues were each
         initially set to .TRUE., the program resets the second of
         the two elements to .FALSE.
       M is the number of columns actually used to store
         the eigenvectors.
       Z contains the real and imaginary parts of the eigenvectors.
         If the next selected eigenvalue is real, the next column
         of Z contains its eigenvector.  If the eigenvalue is
         complex, the next two columns of Z contain the real and
         imaginary parts of its eigenvector.  The eigenvectors are
         normalized so that the component of largest magnitude is 1.
         Any vector which fails the acceptance test is set to zero.
       IERR is set to
         ZERO       for normal return,
         -(2*N+1)   if more than MM columns of Z are necessary
                    to store the eigenvectors corresponding to
                    the specified eigenvalues.
         -K         if the iteration corresponding to the K-th
                    value fails,
         -(N+K)     if both error situations occur.
       RM1, RV1, and RV2 are temporary storage arrays.  Note that RM1
         is square of dimension N by N and, augmented by two columns
         of Z, is the transpose of the corresponding ALGOL B array.
    The ALGOL procedure GUESSVEC appears in INVIT in line.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Calls CDIV for complex division.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



INXCA


      SUBROUTINE INXCA(I,IR,IDXA,NA)



INXCB


      SUBROUTINE INXCB(I,IR,IDX,IDP)



INXCC


      SUBROUTINE INXCC(I,IR,IDXC,NC)



IPLOC


      INTEGER FUNCTION IPLOC(LOC,SX,IX)

    RELATIVE ADDRESS DETERMINATION FUNCTION.
    REVISED 810306-0800
    REVISED YYMMDD-HHMM
    AUTHORS  JOHN A. WISNIEWSKI AND R.J. HANSON
    GIVEN A VIRTUAL LOCATION, FUNCTION IPLOC() RETURNS THE RELATIVE
    WORKING MEMORY ADDRESS OF THE VECTOR COMPONENT STORED IN SX(*),
    IX(*). ANY NECESSARY PAGE SWAPS ARE PERFORMED AUTOMATICALLY
    FOR THE USER IN THIS FUNCTION SUBPROGRAM.
    LOC       IS THE VIRTUAL ADDRESS OF THE DATA TO BE RETRIEVED.
  SX(*),IX(*) REPRESENT THE MATRIX WHERE THE DATA IS STORED.
    EXTERNAL REFERENCES   PRWPGE,XERROR


ISAMAX


      INTEGER FUNCTION ISAMAX(N,SX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
    --Output--
  ISAMAX  smallest index (zero if N .LE. 0)
    Find smallest index of maximum magnitude of single precision SX.
    ISAMAX =  first I, I = 1 to N, to minimize  ABS(SX(1-INCX+I*INCX)


ISWAP


      SUBROUTINE ISWAP(N,IX,INCX,IY,INCY)

               Extended B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      IX  integer vector with N elements
    INCX  storage spacing between elements of IX
      IY  integer vector with N elements
    INCY  storage spacing between elements of IY
    --Output--
      IX  input vector IY (unchanged if N .LE. 0)
      IY  input vector IX (unchanged if N .LE. 0)
    Interchange integer IX and integer IY.
    For I = 0 to N-1, interchange  IX(LX+I*INCX) and IY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


IVOUT


      SUBROUTINE IVOUT(N,IX,IFMT,IDIGIT)

    REVISED FEB. 27, 1981.
    INTEGER VECTOR OUTPUT ROUTINE.
 INPUT..
 N,IX(*) PRINT THE INTEGER ARRAY IX(I),I=1,...,N, ON OUTPUT
         UNIT LOUT. THE HEADING IN THE FORTRAN FORMAT
         STATEMENT IFMT(*), DESCRIBED BELOW, IS PRINTED AS A FIRST
         STEP. THE COMPONENTS IX(I) ARE INDEXED, ON OUTPUT,
         IN A PLEASANT FORMAT.
 IFMT(*) A FORTRAN FORMAT STATEMENT. THIS IS PRINTED ON OUTPUT
         UNIT LOUT WITH THE VARIABLE FORMAT FORTRAN STATEMENT
               WRITE(LOUT,IFMT)
 IDIGIT  PRINT UP TO IABS(IDIGIT) DECIMAL DIGITS PER NUMBER.
         THE SUBPROGRAM WILL CHOOSE THAT INTEGER 4,6,10 OR 14
         WHICH WILL PRINT AT LEAST IABS(IDIGIT) NUMBER OF
         PLACES.  IF IDIGIT.LT.0, 72 PRINTING COLUMNS ARE UTILIZED
         TO WRITE EACH LINE OF OUTPUT OF THE ARRAY IX(*). (THIS
         CAN BE USED ON MOST TIME-SHARING TERMINALS). IF
         IDIGIT.GE.0, 133 PRINTING COLUMNS ARE UTILIZED. (THIS CAN
         BE USED ON MOST LINE PRINTERS).
 EXAMPLE..
 PRINT AN ARRAY CALLED (COSTS OF PURCHASES) OF LENGTH 100 SHOWING
 6 DECIMAL DIGITS PER NUMBER. THE USER IS RUNNING ON A TIME-SHARING
 SYSTEM WITH A 72 COLUMN OUTPUT DEVICE.
    DIMENSION ICOSTS(100)
    N = 100
    IDIGIT = -6
    CALL IVOUT(N,ICOSTS,'(''1COSTS OF PURCHASES'')',IDIGIT)
    AUTHORS    JOHN A. WISNIEWSKI   SANDIA LABS ALBUQUERQUE.
               RICHARD J. HANSON    SANDIA LABS ALBUQUERQUE.
    DATE       JULY 27,1978.


J4SAVE


      FUNCTION J4SAVE(IWHICH,IVALUE,ISET)

    Abstract
       J4SAVE saves and recalls several global variables needed
       by the library error handling routines.
    Description of Parameters
     --Input--
       IWHICH - Index of item desired.
               = 1 Refers to current error number.
               = 2 Refers to current error control flag.
                = 3 Refers to current unit number to which error
                   messages are to be sent.  (0 means use standard.)
                = 4 Refers to the maximum number of times any
                    message is to be printed (as set by XERMAX).
                = 5 Refers to the total number of units to which
                    each error message is to be written.
                = 6 Refers to the 2nd unit for error messages
                = 7 Refers to the 3rd unit for error messages
                = 8 Refers to the 4th unit for error messages
                = 9 Refers to the 5th unit for error messages
       IVALUE - The value to be set for the IWHICH-th parameter,
                if ISET is .TRUE. .
       ISET   - If ISET=.TRUE., the IWHICH-th parameter will BE
                given the value, IVALUE.  If ISET=.FALSE., the
                IWHICH-th parameter will be unchanged, and IVALUE
                is a dummy parameter.
     --Output--
       The (old) value of the IWHICH-th parameter will be returned
       in the function value, J4SAVE.
    Written by Ron Jones, with SLATEC Common Math Library Subcommittee
   Adapted from Bell Laboratories PORT Library Error Handler
    Latest revision ---  1 August 1985


JAIRY


      SUBROUTINE JAIRY(X,RX,C,AI,DAI)



LA05AD


      SUBROUTINE LA05AD(A,IND,NZ,IA,N,IP,IW,W,G,U)

    THIS SUBPROGRAM IS A SLIGHT MODIFICATION OF A SUBPROGRAM
    FROM THE C. 1979 AERE HARWELL LIBRARY.  THE NAME OF THE
    CORRESPONDING HARWELL CODE CAN BE OBTAINED BY DELETING
    THE FINAL LETTER =D= IN THE NAMES USED HERE.
    REVISIONS MADE BY R J HANSON, SNLA, AUGUST, 1979.
    REVISED SEP. 13, 1979.
    ROYALTIES HAVE BEEN PAID TO AERE-UK FOR USE OF THEIR CODES
    IN THE PACKAGE GIVEN HERE.  ANY PRIMARY USAGE OF THE HARWELL
    SUBROUTINES REQUIRES A ROYALTY AGREEMENT AND PAYMENT BETWEEN
    THE USER AND AERE-UK.  ANY USAGE OF THE SANDIA WRITTEN CODES
    DSPLP( ) (WHICH USES THE HARWELL SUBROUTINES) IS PERMITTED.
IP(I,1),IP(I,2) POINT TO THE START OF ROW/COL I.
IW(I,1),IW(I,2) HOLD THE NUMBER OF NON-ZEROS IN ROW/COL I.
DURING THE MAIN BODY OF THIS SUBROUTINE THE VECTORS IW(.,3),IW(.,5),
    IW(.,7) ARE USED TO HOLD DOUBLY LINKED LISTS OF ROWS THAT HAVE
    NOT BEEN PIVOTAL AND HAVE EQUAL NUMBERS OF NON-ZEROS.
IW(.,4),IW(.,6),IW(.,8) HOLD SIMILAR LISTS FOR THE COLUMNS.
IW(I,3),IW(I,4) HOLD FIRST ROW/COLUMN TO HAVE I NON-ZEROS
    OR ZERO IF THERE ARE NONE.
IW(I,5), IW(I,6) HOLD ROW/COL NUMBER OF ROW/COL PRIOR TO ROW/COL I
    IN ITS LIST, OR ZERO IF NONE.
IW(I,7), IW(I,8) HOLD ROW/COL NUMBER OF ROW/COL AFTER ROW/COL I
    IN ITS LIST, OR ZERO IF NONE.
FOR ROWS/COLS THAT HAVE BEEN PIVOTAL IW(I,5),IW(I,6) HOLD NEGATION OF
    POSITION OF ROW/COL I IN THE PIVOTAL ORDERING.


LA05AS


      SUBROUTINE LA05AS(A,IND,NZ,IA,N,IP,IW,W,G,U)

    THIS SUBPROGRAM IS A SLIGHT MODIFICATION OF A SUBPROGRAM
    FROM THE C. 1979 AERE HARWELL LIBRARY.  THE NAME OF THE
    CORRESPONDING HARWELL CODE CAN BE OBTAINED BY DELETING
    THE FINAL LETTER =S= IN THE NAMES USED HERE.
    REVISIONS MADE BY R J HANSON, SNLA, AUGUST, 1979.
    REVISED SEP. 13, 1979.
    ROYALTIES HAVE BEEN PAID TO AERE-UK FOR USE OF THEIR CODES
    IN THE PACKAGE GIVEN HERE.  ANY PRIMARY USAGE OF THE HARWELL
    SUBROUTINES REQUIRES A ROYALTY AGREEMENT AND PAYMENT BETWEEN
    THE USER AND AERE-UK.  ANY USAGE OF THE SANDIA WRITTEN CODES
    SPLP( ) (WHICH USES THE HARWELL SUBROUTINES) IS PERMITTED.
IP(I,1),IP(I,2) POINT TO THE START OF ROW/COL I.
IW(I,1),IW(I,2) HOLD THE NUMBER OF NON-ZEROS IN ROW/COL I.
DURING THE MAIN BODY OF THIS SUBROUTINE THE VECTORS IW(.,3),IW(.,5),
    IW(.,7) ARE USED TO HOLD DOUBLY LINKED LISTS OF ROWS THAT HAVE
    NOT BEEN PIVOTAL AND HAVE EQUAL NUMBERS OF NON-ZEROS.
IW(.,4),IW(.,6),IW(.,8) HOLD SIMILAR LISTS FOR THE COLUMNS.
IW(I,3),IW(I,4) HOLD FIRST ROW/COLUMN TO HAVE I NON-ZEROS
    OR ZERO IF THERE ARE NONE.
IW(I,5), IW(I,6) HOLD ROW/COL NUMBER OF ROW/COL PRIOR TO ROW/COL I
    IN ITS LIST, OR ZERO IF NONE.
IW(I,7), IW(I,8) HOLD ROW/COL NUMBER OF ROW/COL AFTER ROW/COL I
    IN ITS LIST, OR ZERO IF NONE.
FOR ROWS/COLS THAT HAVE BEEN PIVOTAL IW(I,5),IW(I,6) HOLD NEGATION OF
    POSITION OF ROW/COL I IN THE PIVOTAL ORDERING.


LA05BD


      SUBROUTINE LA05BD(A,IND,IA,N,IP,IW,W,G,B,TRANS)

    THIS SUBPROGRAM IS A SLIGHT MODIFICATION OF A SUBPROGRAM
    FROM THE C. 1979 AERE HARWELL LIBRARY.  THE NAME OF THE
    CORRESPONDING HARWELL CODE CAN BE OBTAINED BY DELETING
    THE FINAL LETTER =D= IN THE NAMES USED HERE.
    REVISED SEP. 13, 1979.
    ROYALTIES HAVE BEEN PAID TO AERE-UK FOR USE OF THEIR CODES
    IN THE PACKAGE GIVEN HERE.  ANY PRIMARY USAGE OF THE HARWELL
    SUBROUTINES REQUIRES A ROYALTY AGREEMENT AND PAYMENT BETWEEN
    THE USER AND AERE-UK.  ANY USAGE OF THE SANDIA WRITTEN CODES
    DSPLP( ) (WHICH USES THE HARWELL SUBROUTINES) IS PERMITTED.
IP(I,1),IP(I,2) POINT TO START OF ROW/COLUMN I OF U.
IW(I,1),IW(I,2) ARE LENGTHS OF ROW/COL I OF U.
IW(.,3),IW(.,4) HOLD ROW/COL NUMBERS IN PIVOTAL ORDER.


LA05BS


      SUBROUTINE LA05BS(A,IND,IA,N,IP,IW,W,G,B,TRANS)

    THIS SUBPROGRAM IS A SLIGHT MODIFICATION OF A SUBPROGRAM
    FROM THE C. 1979 AERE HARWELL LIBRARY.  THE NAME OF THE
    CORRESPONDING HARWELL CODE CAN BE OBTAINED BY DELETING
    THE FINAL LETTER =S= IN THE NAMES USED HERE.
    REVISED SEP. 13, 1979.
    ROYALTIES HAVE BEEN PAID TO AERE-UK FOR USE OF THEIR CODES
    IN THE PACKAGE GIVEN HERE.  ANY PRIMARY USAGE OF THE HARWELL
    SUBROUTINES REQUIRES A ROYALTY AGREEMENT AND PAYMENT BETWEEN
    THE USER AND AERE-UK.  ANY USAGE OF THE SANDIA WRITTEN CODES
    SPLP( ) (WHICH USES THE HARWELL SUBROUTINES) IS PERMITTED.
IP(I,1),IP(I,2) POINT TO START OF ROW/COLUMN I OF U.
IW(I,1),IW(I,2) ARE LENGTHS OF ROW/COL I OF U.
IW(.,3),IW(.,4) HOLD ROW/COL NUMBERS IN PIVOTAL ORDER.


LA05CD


      SUBROUTINE LA05CD(A,IND,IA,N,IP,IW,W,G,U,MM)

    THIS SUBPROGRAM IS A SLIGHT MODIFICATION OF A SUBPROGRAM
    FROM THE C. 1979 AERE HARWELL LIBRARY.  THE NAME OF THE
    CORRESPONDING HARWELL CODE CAN BE OBTAINED BY DELETING
    THE FINAL LETTER =D= IN THE NAMES USED HERE.
    REVISED SEP. 13, 1979.
    ROYALTIES HAVE BEEN PAID TO AERE-UK FOR USE OF THEIR CODES
    IN THE PACKAGE GIVEN HERE.  ANY PRIMARY USAGE OF THE HARWELL
    SUBROUTINES REQUIRES A ROYALTY AGREEMENT AND PAYMENT BETWEEN
    THE USER AND AERE-UK.  ANY USAGE OF THE SANDIA WRITTEN CODES
    DSPLP( ) (WHICH USES THE HARWELL SUBROUTINES) IS PERMITTED.


LA05CS


      SUBROUTINE LA05CS(A,IND,IA,N,IP,IW,W,G,U,MM)

    THIS SUBPROGRAM IS A SLIGHT MODIFICATION OF A SUBPROGRAM
    FROM THE C. 1979 AERE HARWELL LIBRARY.  THE NAME OF THE
    CORRESPONDING HARWELL CODE CAN BE OBTAINED BY DELETING
    THE FINAL LETTER =S= IN THE NAMES USED HERE.
    REVISED SEP. 13, 1979.
    ROYALTIES HAVE BEEN PAID TO AERE-UK FOR USE OF THEIR CODES
    IN THE PACKAGE GIVEN HERE.  ANY PRIMARY USAGE OF THE HARWELL
    SUBROUTINES REQUIRES A ROYALTY AGREEMENT AND PAYMENT BETWEEN
    THE USER AND AERE-UK.  ANY USAGE OF THE SANDIA WRITTEN CODES
    SPLP( ) (WHICH USES THE HARWELL SUBROUTINES) IS PERMITTED.


LA05ED


      SUBROUTINE LA05ED(A,IRN,IP,N,IW,IA,REALS)

    THIS SUBPROGRAM IS A SLIGHT MODIFICATION OF A SUBPROGRAM
    FROM THE C. 1979 AERE HARWELL LIBRARY.  THE NAME OF THE
    CORRESPONDING HARWELL CODE CAN BE OBTAINED BY DELETING
    THE FINAL LETTER =D= IN THE NAMES USED HERE.
    REVISED SEP. 13, 1979.
    ROYALTIES HAVE BEEN PAID TO AERE-UK FOR USE OF THEIR CODES
    IN THE PACKAGE GIVEN HERE.  ANY PRIMARY USAGE OF THE HARWELL
    SUBROUTINES REQUIRES A ROYALTY AGREEMENT AND PAYMENT BETWEEN
    THE USER AND AERE-UK.  ANY USAGE OF THE SANDIA WRITTEN CODES
    DSPLP( ) (WHICH USES THE HARWELL SUBROUTINES) IS PERMITTED.


LA05ES


      SUBROUTINE LA05ES(A,IRN,IP,N,IW,IA,REALS)

    THIS SUBPROGRAM IS A SLIGHT MODIFICATION OF A SUBPROGRAM
    FROM THE C. 1979 AERE HARWELL LIBRARY.  THE NAME OF THE
    CORRESPONDING HARWELL CODE CAN BE OBTAINED BY DELETING
    THE FINAL LETTER =S= IN THE NAMES USED HERE.
    REVISED SEP. 13, 1979.
    ROYALTIES HAVE BEEN PAID TO AERE-UK FOR USE OF THEIR CODES
    IN THE PACKAGE GIVEN HERE.  ANY PRIMARY USAGE OF THE HARWELL
    SUBROUTINES REQUIRES A ROYALTY AGREEMENT AND PAYMENT BETWEEN
    THE USER AND AERE-UK.  ANY USAGE OF THE SANDIA WRITTEN CODES
    SPLP( ) (WHICH USES THE HARWELL SUBROUTINES) IS PERMITTED.


LLSIA


      SUBROUTINE LLSIA(A,MDA,M,N,B,MDB,NB,RE,AE,KEY,MODE,NP,KRANK,

    LLSIA computes the least squares solution(s) to the problem AX=B
    where A is an M by N matrix with M.GE.N and B is the M by NB
    matrix of right hand sides.  User input bounds on the uncertainty
    in the elements of A are used to detect numerical rank deficiency.
    The algorithm employs a row and column pivot strategy to
    minimize the growth of uncertainty and round-off errors.
    LLSIA requires (MDA+6)*N + (MDB+1)*NB + M dimensioned space
  ******************************************************************
  *                                                                *
  *         WARNING - All input arrays are changed on exit.        *
  *                                                                *
  ******************************************************************
    SUBROUTINE LLSIA(A,MDA,M,N,B,MDB,NB,RE,AE,KEY,MODE,NP,
   1   KRANK,KSURE,RNORM,W,LW,IWORK,LIW,INFO)
    Input..
    A(,)          Linear coefficient matrix of AX=B, with MDA the
     MDA,M,N      actual first dimension of A in the calling program.
                  M is the row dimension (no. of EQUATIONS of the
                  problem) and N the col dimension (no. of UNKNOWNS).
                  Must have MDA.GE.M and M.GE.N.
    B(,)          Right hand side(s), with MDB the actual first
     MDB,NB       dimension of B in the calling program. NB is the
                  number of M by 1 right hand sides. Must have
                  MDB.GE.M. If NB = 0, B is never accessed.
  ******************************************************************
  *                                                                *
  *         Note - Use of RE and AE are what make this             *
  *                code significantly different from               *
  *                other linear least squares solvers.             *
  *                However, the inexperienced user is              *
  *                advised to set RE=0.,AE=0.,KEY=0.               *
  *                                                                *
  ******************************************************************
    RE(),AE(),KEY
    RE()          RE() is a vector of length N such that RE(I) is
                  the maximum relative uncertainty in column I of
                  the matrix A. The values of RE() must be between
                  0 and 1. A minimum of 10*machine precision will
                  be enforced.
    AE()          AE() is a vector of length N such that AE(I) is
                  the maximum absolute uncertainty in column I of
                  the matrix A. The values of AE() must be greater
                  than or equal to 0.
    KEY           For ease of use, RE and AE may be input as either
                  vectors or scalars. If a scalar is input, the algo-
                  rithm will use that value for each column of A.
                  The parameter key indicates whether scalars or
                  vectors are being input.
                       KEY=0     RE scalar  AE scalar
                       KEY=1     RE vector  AE scalar
                       KEY=2     RE scalar  AE vector
                       KEY=3     RE vector  AE vector
    MODE          The integer mode indicates how the routine
                  is to react if rank deficiency is detected.
                  If MODE = 0 return immediately, no solution
                            1 compute truncated solution
                            2 compute minimal length solution
                  The inexperienced user is advised to set MODE=0
    NP            The first NP columns of A will not be interchanged
                  with other columns even though the pivot strategy
                  would suggest otherwise.
                  The inexperienced user is advised to set NP=0.
    WORK()        A real work array dimensioned 5*N. However,if
                  RE or AE have been specified as vectors, dimension
                  WORK 4*N. If both RE and AE have been specified
                  as vectors, dimension WORK 3*N.
    LW            Actual dimension of WORK
    IWORK()       Integer work array dimensioned at least N+M.
    LIW           Actual dimension of IWORK.
    INFO          Is a flag which provides for the efficient
                  solution of subsequent problems involving the
                  same A but different B.
                  If INFO = 0 original call
                     INFO = 1 subsequent calls
                  On subsequent calls, the user must supply A, KRANK,
                  LW, IWORK, LIW, and the first 2*N locations of WORK
                  as output by the original call to LLSIA. MODE must
                  be equal to the value of MODE in the original call.
                  If MODE.LT.2, only the first N locations of WORK
                  are accessed. AE, RE, KEY, and NP are not accessed.
    Output..
    A(,)          Contains the upper triangular part of the reduced
                  matrix and the transformation information. It togeth
                  with the first N elements of WORK (see below)
                  completely specify the QR factorization of A.
    B(,)          Contains the N by NB solution matrix for X.
    KRANK,KSURE   The numerical rank of A,  based upon the relative
                  and absolute bounds on uncertainty, is bounded
                  above by KRANK and below by KSURE. The algorithm
                  returns a solution based on KRANK. KSURE provides
                  an indication of the precision of the rank.
    RNORM()       Contains the Euclidean length of the NB residual
                  vectors  B(I)-AX(I), I=1,NB.
    WORK()        The first N locations of WORK contain values
                  necessary to reproduce the Householder transformatio
    IWORK()       The first N locations contain the order in
                  which the columns of A were used. The next
                  M locations contain the order in which the
                  rows of A were used.
    INFO          Flag to indicate status of computation on completion
                 -1   Parameter error(s)
                  0 - Rank deficient, no solution
                  1 - Rank deficient, truncated solution
                  2 - Rank deficient, minimal length solution
                  3 - Numerical rank 0, zero solution
                  4 - Rank .LT. NP
                  5 - Full rank


LMPAR


      SUBROUTINE LMPAR(N,R,LDR,IPVT,DIAG,QTB,DELTA,PAR,X,SIGMA,WA1,WA2)



LPDP


      SUBROUTINE LPDP(A,MDA,M,N1,N2,PRGOPT,X,WNORM,MODE,WS,IS)



MINFIT


      SUBROUTINE MINFIT(NM,M,N,A,W,IP,B,IERR,RV1)

    This subroutine is a translation of the ALGOL procedure MINFIT,
    NUM. MATH. 14, 403-420(1970) by Golub and Reinsch.
    HANDBOOK FOR AUTO. COMP., VOL II-LINEAR ALGEBRA, 134-151(1971).
    This subroutine determines, towards the solution of the linear
                                                       T
    system AX=B, the singular value decomposition A=USV  of a real
                                        T
    M by N rectangular matrix, forming U B rather than U.  Householder
    bidiagonalization and a variant of the QR algorithm are used.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.  Note that NM must be at least
         as large as the maximum of M and N.
       M is the number of rows of A and B.
       N is the number of columns of A and the order of V.
       A contains the rectangular coefficient matrix of the system.
       IP is the number of columns of B.  IP can be zero.
       B contains the constant column matrix of the system
         if IP is not zero.  Otherwise B is not referenced.
    On OUTPUT
       A has been overwritten by the matrix V (orthogonal) of the
         decomposition in its first N rows and columns.  If an
         error exit is made, the columns of V corresponding to
         indices of correct singular values should be correct.
       W contains the N (non-negative) singular values of A (the
         diagonal elements of S).  They are UNORDERED.  If an
         error exit is made, the singular values should be correct
         for indices IERR+1,IERR+2,...,N.
                                  T
       B has been overwritten by U B.  If an error exit is made,
                      T
         the rows of U B corresponding to indices of correct
         singular values should be correct.
       IERR is set to
         Zero       for normal return,
         K          if the K-th singular value has not been
                    determined after 30 iterations.
       RV1 is a temporary storage array.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



MINSO4


      SUBROUTINE MINSO4(USOL,IDMN,ZN,ZM,PERTB)



MINSOL


      SUBROUTINE MINSOL(USOL,IDMN,ZN,ZM,PERTB)



MPADD


      SUBROUTINE MPADD(X,Y,Z)



MPADD2


      SUBROUTINE MPADD2(X,Y,Z,Y1,TRUNC)



MPADD3


      SUBROUTINE MPADD3(X,Y,S,MED,RE)



MPBLAS


      SUBROUTINE MPBLAS(I1)



MPCDM


      SUBROUTINE MPCDM(DX,Z)



MPCHK


      SUBROUTINE MPCHK(I,J)



MPCMD


      SUBROUTINE MPCMD(X,DZ)



MPDIVI


      SUBROUTINE MPDIVI(X,IY,Z)



MPERR


      SUBROUTINE MPERR



MPMAXR


      SUBROUTINE MPMAXR(X)



MPMLP


      SUBROUTINE MPMLP(U,V,W,J)



MPMUL


      SUBROUTINE MPMUL(X,Y,Z)



MPMUL2


      SUBROUTINE MPMUL2(X,IY,Z,TRUNC)



MPMULI


      SUBROUTINE MPMULI(X,IY,Z)



MPNZR


      SUBROUTINE MPNZR(RS,RE,Z,TRUNC)



MPOVFL


      SUBROUTINE MPOVFL(X)



MPSTR


      SUBROUTINE MPSTR(X,Y)



MPUNFL


      SUBROUTINE MPUNFL(X)



NUMXER


      FUNCTION NUMXER(NERR)

    Abstract
       NUMXER returns the most recent error number,
       in both NUMXER and the parameter NERR.
    Written by Ron Jones, with SLATEC Common Math Library Subcommittee
    Latest revision ---  7 JUNE 1978


OHTROL


      SUBROUTINE OHTROL(Q,N,NRDA,DIAG,IRANK,DIV,TD)



OHTROR


      SUBROUTINE OHTROR(Q,N,NRDA,DIAG,IRANK,DIV,TD)



ORTBAK


      SUBROUTINE ORTBAK(NM,LOW,IGH,A,ORT,M,Z)

    This subroutine is a translation of the ALGOL procedure ORTBAK,
    NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    This subroutine forms the eigenvectors of a REAL GENERAL
    matrix by back transforming those of the corresponding
    upper Hessenberg matrix determined by  ORTHES.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       LOW and IGH are integers determined by the balancing
         subroutine  BALANC.  If  BALANC  has not been used,
         set LOW=1 and IGH equal to the order of the matrix.
       A contains information about the orthogonal trans-
         formations used in the reduction by  ORTHES
         in its strict lower triangle.
       ORT contains further information about the trans-
         formations used in the reduction by  ORTHES.
         Only elements LOW through IGH are used.
       M is the number of columns of Z to be back transformed.
       Z contains the real and imaginary parts of the eigen-
         vectors to be back transformed in its first M columns.
    On OUTPUT
       Z contains the real and imaginary parts of the
         transformed eigenvectors in its first M columns.
       ORT has been altered.
    NOTE that ORTBAK preserves vector Euclidean norms.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



ORTHES


      SUBROUTINE ORTHES(NM,N,LOW,IGH,A,ORT)

    This subroutine is a translation of the ALGOL procedure ORTHES,
    NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    Given a REAL GENERAL matrix, this subroutine
    reduces a submatrix situated in rows and columns
    LOW through IGH to upper Hessenberg form by
    orthogonal similarity transformations.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  BALANC.  If  BALANC  has not been used,
         set LOW=1, IGH=N.
       A contains the input matrix.
    On OUTPUT
       A contains the Hessenberg matrix.  Information about
         the orthogonal transformations used in the reduction
         is stored in the remaining triangle under the
         Hessenberg matrix.
       ORT contains further information about the transformations.
         only elements LOW through IGH are used.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



ORTHO4


      SUBROUTINE ORTHO4(USOL,IDMN,ZN,ZM,PERTRB)



ORTHOG


      SUBROUTINE ORTHOG(USOL,IDMN,ZN,ZM,PERTRB)



ORTHOL


      SUBROUTINE ORTHOL(A,M,N,NRDA,IFLAG,IRANK,ISCALE,DIAG,KPIVOT,



ORTHOR


      SUBROUTINE ORTHOR(A,N,M,NRDA,IFLAG,IRANK,ISCALE,DIAG,KPIVOT,



ORTRAN


      SUBROUTINE ORTRAN(NM,N,LOW,IGH,A,ORT,Z)

    This subroutine is a translation of the ALGOL procedure ORTRANS,
    NUM. MATH. 16, 181-204(1970) by Peters and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
    This subroutine accumulates the orthogonal similarity
    transformations used in the reduction of a REAL GENERAL
    matrix to upper Hessenberg form by  ORTHES.
    On INPUT
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       LOW and IGH are integers determined by the balancing
         subroutine  BALANC.  If  BALANC  has not been used,
         set LOW=1, IGH=N.
       A contains information about the orthogonal trans-
         formations used in the reduction by  ORTHES
         in its strict lower triangle.
       ORT contains further information about the trans-
         formations used in the reduction by  ORTHES.
         only elements LOW through IGH are USED.
    On OUTPUT
       Z contains the transformation matrix produced in the
         reduction by  ORTHES.
       ORT has been altered.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



PASSB


      SUBROUTINE PASSB(NAC,IDO,IP,L1,IDL1,CC,C1,C2,CH,CH2,WA)



PASSB2


      SUBROUTINE PASSB2(IDO,L1,CC,CH,WA1)



PASSB3


      SUBROUTINE PASSB3(IDO,L1,CC,CH,WA1,WA2)



PASSB4


      SUBROUTINE PASSB4(IDO,L1,CC,CH,WA1,WA2,WA3)



PASSB5


      SUBROUTINE PASSB5(IDO,L1,CC,CH,WA1,WA2,WA3,WA4)



PASSF


      SUBROUTINE PASSF(NAC,IDO,IP,L1,IDL1,CC,C1,C2,CH,CH2,WA)



PASSF2


      SUBROUTINE PASSF2(IDO,L1,CC,CH,WA1)



PASSF3


      SUBROUTINE PASSF3(IDO,L1,CC,CH,WA1,WA2)



PASSF4


      SUBROUTINE PASSF4(IDO,L1,CC,CH,WA1,WA2,WA3)



PASSF5


      SUBROUTINE PASSF5(IDO,L1,CC,CH,WA1,WA2,WA3,WA4)



PCHCE


      SUBROUTINE PCHCE(IC,VC,N,X,H,SLOPE,D,INCFD,IERR)

         PCHCE:  PCHIC End Derivative Setter.
   Called by PCHIC to set end derivatives as requested by the user.
   It must be called after interior derivative values have been set.
                     -----
   To facilitate two-dimensional applications, includes an increment
   between successive values of the D-array.

 Calling sequence:
       PARAMETER  (INCFD = ...)
       INTEGER  IC(2), N, IERR
       REAL  VC(2), X(N), H(N), SLOPE(N), D(INCFD,N)
       CALL  PCHCE (IC, VC, N, X, H, SLOPE, D, INCFD, IERR)
  Parameters:
    IC -- (input) integer array of length 2 specifying desired
          boundary conditions:
          IC(1) = IBEG, desired condition at beginning of data.
          IC(2) = IEND, desired condition at end of data.
          ( see prologue to PCHIC for details. )
    VC -- (input) real array of length 2 specifying desired boundary
          values.  VC(1) need be set only if IC(1) = 2 or 3 .
                   VC(2) need be set only if IC(2) = 2 or 3 .
    N -- (input) number of data points.  (assumes N.GE.2)
    X -- (input) real array of independent variable values.  (the
          elements of X are assumed to be strictly increasing.)
    H -- (input) real array of interval lengths.
    SLOPE -- (input) real array of data slopes.
          If the data are (X(I),Y(I)), I=1(1)N, then these inputs are:
                 H(I) =  X(I+1)-X(I),
             SLOPE(I) = (Y(I+1)-Y(I))/H(I),  I=1(1)N-1.
    D -- (input) real array of derivative values at the data points.
          The value corresponding to X(I) must be stored in
               D(1+(I-1)*INCFD),  I=1(1)N.
         (output) the value of D at X(1) and/or X(N) is changed, if
          necessary, to produce the requested boundary conditions.
          no other entries in D are changed.
    INCFD -- (input) increment between successive values in D.
          This argument is provided primarily for 2-D applications.
    IERR -- (output) error flag.
          Normal return:
             IERR = 0  (no errors).
          Warning errors:
             IERR = 1  if IBEG.LT.0 and D(1) had to be adjusted for
                       monotonicity.
             IERR = 2  if IEND.LT.0 and D(1+(N-1)*INCFD) had to be
                       adjusted for monotonicity.
             IERR = 3  if both of the above are true.
   -------
   WARNING:  This routine does no validity-checking of arguments.
   -------
 Fortran intrinsics used:  ABS, IABS.


PCHCI


      SUBROUTINE PCHCI(N,H,SLOPE,D,INCFD)

         PCHCI:  PCHIC Initial Derivative Setter.
   Called by PCHIC to set derivatives needed to determine a monotone
   piecewise cubic Hermite interpolant to the data.
   Default boundary conditions are provided which are compatible
   with monotonicity.  If the data are only piecewise monotonic, the
   interpolant will have an extremum at each point where monotonicity
   switches direction.
   To facilitate two-dimensional applications, includes an increment
   between successive values of the D-array.
   The resulting piecewise cubic Hermite function should be identical
   (within roundoff error) to that produced by PCHIM.

 Calling sequence:
       PARAMETER  (INCFD = ...)
       INTEGER  N
       REAL  H(N), SLOPE(N), D(INCFD,N)
       CALL  PCHCI (N, H, SLOPE, D, INCFD)
  Parameters:
    N -- (input) number of data points.
          If N=2, simply does linear interpolation.
    H -- (input) real array of interval lengths.
    SLOPE -- (input) real array of data slopes.
          If the data are (X(I),Y(I)), I=1(1)N, then these inputs are:
                 H(I) =  X(I+1)-X(I),
             SLOPE(I) = (Y(I+1)-Y(I))/H(I),  I=1(1)N-1.
    D -- (output) real array of derivative values at the data points.
          If the data are monotonic, these values will determine a
          a monotone cubic Hermite function.
          The value corresponding to X(I) is stored in
               D(1+(I-1)*INCFD),  I=1(1)N.
          No other entries in D are changed.
    INCFD -- (input) increment between successive values in D.
          This argument is provided primarily for 2-D applications.
   -------
   WARNING:  This routine does no validity-checking of arguments.
   -------
 Fortran intrinsics used:  ABS, AMAX1, AMIN1.


PCHCS


      SUBROUTINE PCHCS(SWITCH,N,H,SLOPE,D,INCFD,IERR)

        PCHCS:  PCHIC Monotonicity Switch Derivative Setter.
    Called by  PCHIC  to adjust the values of D in the vicinity of a
    switch in direction of monotonicity, to produce a more "visually
    pleasing" curve than that given by  PCHIM .

 Calling sequence:
       PARAMETER  (INCFD = ...)
       INTEGER  N, IERR
       REAL  SWITCH, H(N), SLOPE(N), D(INCFD,N)
       CALL  PCHCS (SWITCH, N, H, SLOPE, D, INCFD, IERR)
  Parameters:
    SWITCH -- (input) indicates the amount of control desired over
          local excursions from data.
    N -- (input) number of data points.  (assumes N.GT.2 .)
    H -- (input) real array of interval lengths.
    SLOPE -- (input) real array of data slopes.
          If the data are (X(I),Y(I)), I=1(1)N, then these inputs are:
                 H(I) =  X(I+1)-X(I),
             SLOPE(I) = (Y(I+1)-Y(I))/H(I),  I=1(1)N-1.
    D -- (input) real array of derivative values at the data points,
          as determined by PCHCI.
         (output) derivatives in the vicinity of switches in direction
          of monotonicity may be adjusted to produce a more "visually
          pleasing" curve.
          The value corresponding to X(I) is stored in
               D(1+(I-1)*INCFD),  I=1(1)N.
          No other entries in D are changed.
    INCFD -- (input) increment between successive values in D.
          This argument is provided primarily for 2-D applications.
    IERR -- (output) error flag.  should be zero.
          If negative, trouble in PCHSW.  (should never happen.)
   -------
   WARNING:  This routine does no validity-checking of arguments.
   -------
 Fortran intrinsics used:  ABS, AMAX1, AMIN1.


PCHDF


      REAL FUNCTION PCHDF(K,X,S,IERR)

         PCHDF:   PCHIP Finite Difference Formula
    Uses a divided difference formulation to compute a K-point approx-
    imation to the derivative at X(K) based on the data in X and S.
    Called by  PCHCE  and  PCHSP  to compute 3- and 4-point boundary
    derivative approximations.

    On input:
       K      is the order of the desired derivative approximation.
              K must be at least 3 (error return if not).
       X      contains the K values of the independent variable.
              X need not be ordered, but the values **MUST** be
              distinct.  (Not checked here.)
       S      contains the associated slope values:
                 S(I) = (F(I+1)-F(I))/(X(I+1)-X(I)), I=1(1)K-1.
              (Note that S need only be of length K-1.)
    On return:
       S      will be destroyed.
       IERR   will be set to -1 if K.LT.2 .
       PCHDF  will be set to the desired derivative approximation if
              IERR=0 or to zero if IERR=-1.

 Reference:  Carl de Boor, A Practical Guide to Splines, Springer-
             Verlag (New York, 1978), pp. 10-16.

 Programmed by:  Fred N. Fritsch,  FTS 532-4275, (415) 422-4275,
                 Mathematics and Statistics Division,
                 Lawrence Livermore National Laboratory.
 Change record:
    82-08-05   Converted to SLATEC library version.

 Programming notes:
    To produce a double precision version, simply:
       a. Change PCHDF to DPCHDF wherever it occurs,
       b. Change the real declarations to double precision, and
       c. Change the constant ZERO to double precision.


QZHES


      SUBROUTINE QZHES(NM,N,A,B,MATZ,Z)

    This subroutine is the first step of the QZ algorithm
    for solving generalized matrix eigenvalue problems,
    SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART.
    This subroutine accepts a pair of REAL GENERAL matrices and
    reduces one of them to upper Hessenberg form and the other
    to upper triangular form using orthogonal transformations.
    It is usually followed by  QZIT,  QZVAL  and, possibly,  QZVEC.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrices.
       A contains a real general matrix.
       B contains a real general matrix.
       MATZ should be set to .TRUE. if the right hand transformations
         are to be accumulated for later use in computing
         eigenvectors, and to .FALSE. otherwise.
    On Output
       A has been reduced to upper Hessenberg form.  The elements
         below the first subdiagonal have been set to zero.
       B has been reduced to upper triangular form.  The elements
         below the main diagonal have been set to zero.
       Z contains the product of the right hand transformations if
         MATZ has been set to .TRUE.  Otherwise, Z is not referenced.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



QZIT


      SUBROUTINE QZIT(NM,N,A,B,EPS1,MATZ,Z,IERR)

    This subroutine is the second step of the QZ algorithm
    for solving generalized matrix eigenvalue problems,
    SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART,
    as modified in technical note NASA TN D-7305(1973) by WARD.
    This subroutine accepts a pair of REAL matrices, one of them
    in upper Hessenberg form and the other in upper triangular form.
    It reduces the Hessenberg matrix to quasi-triangular form using
    orthogonal transformations while maintaining the triangular form
    of the other matrix.  It is usually preceded by  QZHES  and
    followed by  QZVAL  and, possibly,  QZVEC.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrices.
       A contains a real upper Hessenberg matrix.
       B contains a real upper triangular matrix.
       EPS1 is a tolerance used to determine negligible elements.
         EPS1 = 0.0 (or negative) may be input, in which case an
         element will be neglected only if it is less than roundoff
         error times the norm of its matrix.  If the input EPS1 is
         positive, then an element will be considered negligible
         if it is less than EPS1 times the norm of its matrix.  A
         positive value of EPS1 may result in faster execution,
         but less accurate results.
       MATZ should be set to .TRUE. If the right hand transformations
         are to be accumulated for later use in computing
         eigenvectors, and to .FALSE. otherwise.
       Z contains, if MATZ has been set to .TRUE., the
         transformation matrix produced in the reduction
         by  QZHES, if performed, or else the identity matrix.
         If MATZ has been set to .FALSE., Z is not referenced.
    On Output
       A has been reduced to quasi-triangular form.  The elements
         below the first subdiagonal are still zero and no two
         consecutive subdiagonal elements are nonzero.
       B is still in upper triangular form, although its elements
         have been altered.  The location B(N,1) is used to store
         EPS1 times the norm of B for later use by  QZVAL  and  QZVEC.
       Z contains the product of the right hand transformations
         (for both steps) if MATZ has been set to .TRUE.
       IERR is set to
         ZERO       for normal return,
         J          if neither A(J,J-1) nor A(J-1,J-2) has become
                    zero after a total of 30*N iterations.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



QZVAL


      SUBROUTINE QZVAL(NM,N,A,B,ALFR,ALFI,BETA,MATZ,Z)

    This subroutine is the third step of the QZ algorithm
    for solving generalized matrix eigenvalue problems,
    SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART.
    This subroutine accepts a pair of REAL matrices, one of them
    in quasi-triangular form and the other in upper triangular form.
    It reduces the quasi-triangular matrix further, so that any
    remaining 2-by-2 blocks correspond to pairs of complex
    eigenvalues, and returns quantities whose ratios give the
    generalized eigenvalues.  It is usually preceded by  QZHES
    and  QZIT  and may be followed by  QZVEC.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrices.
       A contains a real upper quasi-triangular matrix.
       B contains a real upper triangular matrix.  In addition,
         location B(N,1) contains the tolerance quantity (EPSB)
         computed and saved in  QZIT.
       MATZ should be set to .TRUE. If the right hand transformations
         are to be accumulated for later use in computing
         eigenvectors, and to .FALSE. otherwise.
       Z contains, if MATZ has been set to .TRUE., the
         transformation matrix produced in the reductions by QZHES
         and QZIT, if performed, or else the identity matrix.
         If MATZ has been set to .FALSE., Z is not referenced.
    On Output
       A has been reduced further to a quasi-triangular matrix
         in which all nonzero subdiagonal elements correspond to
         pairs of complex eigenvalues.
       B is still in upper triangular form, although its elements
         have been altered.  B(N,1) is unaltered.
       ALFR and ALFI contain the real and imaginary parts of the
         diagonal elements of the triangular matrix that would be
         obtained if a were reduced completely to triangular form
         by unitary transformations.  Non-zero values of ALFI occur
         in pairs, the first member positive and the second negative.
       BETA contains the diagonal elements of the corresponding B,
         normalized to be real and non-negative.  The generalized
         eigenvalues are then the ratios ((ALFR+I*ALFI)/BETA).
       Z contains the product of the right hand transformations
         (for all three steps) if MATZ has been set to .TRUE.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



QZVEC


      SUBROUTINE QZVEC(NM,N,A,B,ALFR,ALFI,BETA,Z)

    This subroutine is the optional fourth step of the QZ algorithm
    for solving generalized matrix eigenvalue problems,
    SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART.
    This subroutine accepts a pair of REAL matrices, one of them in
    quasi-triangular form (in which each 2-by-2 block corresponds to
    a pair of complex eigenvalues) and the other in upper triangular
    form.  It computes the eigenvectors of the triangular problem and
    transforms the results back to the original coordinate system.
    It is usually preceded by  QZHES,  QZIT, and  QZVAL.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrices.
       A contains a real upper quasi-triangular matrix.
       B contains a real upper triangular matrix.  In addition,
         location B(N,1) contains the tolerance quantity (EPSB)
         computed and saved in  QZIT.
       ALFR, ALFI, and BETA  are vectors with components whose
         ratios ((ALFR+I*ALFI)/BETA) are the generalized
         eigenvalues.  They are usually obtained from  QZVAL.
       Z contains the transformation matrix produced in the
         reductions by  QZHES,  QZIT, and  QZVAL, if performed.
         If the eigenvectors of the triangular problem are
         desired, Z must contain the identity matrix.
    On Output
       A is unaltered.  Its subdiagonal elements provide information
          about the storage of the complex eigenvectors.
       B has been destroyed.
       ALFR, ALFI, and BETA are unaltered.
       Z contains the real and imaginary parts of the eigenvectors.
         If ALFI(I) .EQ. 0.0, the I-th eigenvalue is real and
           the I-th column of Z contains its eigenvector.
         If ALFI(I) .NE. 0.0, the I-th eigenvalue is complex.
           If ALFI(I) .GT. 0.0, the eigenvalue is the first of
             a complex pair and the I-th and (I+1)-th columns
             of Z contain its eigenvector.
           If ALFI(I) .LT. 0.0, the eigenvalue is the second of
             a complex pair and the (I-1)-th and I-th columns
             of Z contain the conjugate of its eigenvector.
         Each eigenvector is normalized so that the modulus
         of its largest component is 1.0 .
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RATQR


      SUBROUTINE RATQR(N,EPS1,D,E,E2,M,W,IND,BD,TYPE,IDEF,IERR)

    This subroutine is a translation of the ALGOL procedure RATQR,
    NUM. MATH. 11, 264-272(1968) by REINSCH and BAUER.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 257-265(1971).
    This subroutine finds the algebraically smallest or largest
    eigenvalues of a SYMMETRIC TRIDIAGONAL matrix by the
    rational QR method with Newton corrections.
    On Input
       N is the order of the matrix.
       EPS1 is a theoretical absolute error tolerance for the
         computed eigenvalues.  If the input EPS1 is non-positive,
         or indeed smaller than its default value, it is reset
         at each iteration to the respective default value,
         namely, the product of the relative machine precision
         and the magnitude of the current eigenvalue iterate.
         The theoretical absolute error in the K-th eigenvalue
         is usually not greater than K times EPS1.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
       E2 contains the squares of the corresponding elements of E.
         E2(1) is arbitrary.
       M is the number of eigenvalues to be found.
       IDEF should be set to 1 if the input matrix is known to be
         positive definite, to -1 if the input matrix is known to
         be negative definite, and to 0 otherwise.
       TYPE should be set to .TRUE. if the smallest eigenvalues
         are to be found, and to .FALSE. If the largest eigenvalues
         are to be found.
    On Output
       EPS1 is unaltered unless it has been reset to its
         (last) default value.
       D and E are unaltered (unless W overwrites D).
       ELEMENTS of E2, corresponding to elements of E regarded
         as negligible, have been replaced by zero causing the
         matrix to split into a direct sum of submatrices.
         E2(1) is set to 0.0e0 if the smallest eigenvalues have been
         found, and to 2.0e0 if the largest eigenvalues have been
         found.  E2 is otherwise unaltered (unless overwritten by BD).
       W contains the M algebraically smallest eigenvalues in
         ascending order, or the M largest eigenvalues in
         descending order.  If an error exit is made because of
         an incorrect specification of IDEF, no eigenvalues
         are found.  If the Newton iterates for a particular
         eigenvalue are not monotone, the best estimate obtained
         is returned and IERR is set.  W may coincide with D.
       IND contains in its first M positions the submatrix indices
         associated with the corresponding eigenvalues in W --
         1 for eigenvalues belonging to the first submatrix from
         the top, 2 for those belonging to the second submatrix, etc.
       BD contains refined bounds for the theoretical errors of the
         corresponding eigenvalues in W.  These bounds are usually
         within the tolerance specified by EPS1.  BD may coincide
         with E2.
       IERR is set to
         Zero       for normal return,
         6*N+1      if  IDEF  is set to 1 and  type  to .TRUE.
                    when the matrix is NOT positive definite, or
                    if  IDEF  is set to -1 and  type  to .FALSE.
                    when the matrix is NOT negative definite,
         5*N+K      if successive iterates to the K-th eigenvalue
                    are NOT monotone increasing, where K refers
                    to the last such occurrence.
    Note that subroutine TRIDIB is generally faster and more
    accurate than RATQR if the eigenvalues are clustered.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



REBAK


      SUBROUTINE REBAK(NM,N,B,DL,M,Z)

    This subroutine is a translation of the ALGOL procedure REBAKA,
    NUM. MATH. 11, 99-110(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 303-314(1971).
    This subroutine forms the eigenvectors of a generalized
    SYMMETRIC eigensystem by back transforming those of the
    derived symmetric matrix determined by  REDUC.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix system.
       B contains information about the similarity transformation
         (Cholesky decomposition) used in the reduction by  REDUC
         in its strict lower triangle.
       DL contains further information about the transformation.
       M is the number of eigenvectors to be back transformed.
       Z contains the eigenvectors to be back transformed
         in its first M columns.
    On Output
       Z contains the transformed eigenvectors
         in its first M columns.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



REBAKB


      SUBROUTINE REBAKB(NM,N,B,DL,M,Z)

    This subroutine is a translation of the ALGOL procedure REBAKB,
    NUM. MATH. 11, 99-110(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 303-314(1971).
    This subroutine forms the eigenvectors of a generalized
    SYMMETRIC eigensystem by back transforming those of the
    derived symmetric matrix determined by  REDUC2.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix system.
       B contains information about the similarity transformation
         (cholesky decomposition) used in the reduction by  REDUC2
         in its strict lower triangle.
       DL contains further information about the transformation.
       M is the number of eigenvectors to be back transformed.
       Z contains the eigenvectors to be back transformed
         in its first M columns.
    On Output
       Z contains the transformed eigenvectors
         in its first M columns.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



REDUC


      SUBROUTINE REDUC(NM,N,A,B,DL,IERR)

    This subroutine is a translation of the ALGOL procedure REDUC1,
    NUM. MATH. 11, 99-110(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 303-314(1971).
    This subroutine reduces the generalized SYMMETRIC eigenproblem
    Ax=(LAMBDA)Bx, where B is POSITIVE DEFINITE, to the standard
    symmetric eigenproblem using the Cholesky factorization of B.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrices A and B.  If the Cholesky
         factor L of B is already available, N should be prefixed
         with a minus sign.
       A and B contain the real symmetric input matrices.  Only the
         full upper triangles of the matrices need be supplied.  If
         N is negative, the strict lower triangle of B contains,
         instead, the strict lower triangle of its Cholesky factor L.
       DL contains, if N is negative, the diagonal elements of L.
    On Output
       A contains in its full lower triangle the full lower triangle
         of the symmetric matrix derived from the reduction to the
         standard form.  The strict upper triangle of A is unaltered.
       B contains in its strict lower triangle the strict lower
         triangle of its Cholesky factor L.  The full upper
         triangle of B is unaltered.
       DL contains the diagonal elements of L.
       IERR is set to
         Zero       for normal return,
         7*N+1      if B is not positive definite.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



REDUC2


      SUBROUTINE REDUC2(NM,N,A,B,DL,IERR)

    This subroutine is a translation of the ALGOL procedure REDUC2,
    NUM. MATH. 11, 99-110(1968) by Martin and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 303-314(1971).
    This subroutine reduces the generalized SYMMETRIC eigenproblems
    ABx=(LAMBDA)x OR BAy=(LAMBDA)y, where B is POSITIVE DEFINITE,
    to the standard symmetric eigenproblem using the Cholesky
    factorization of B.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrices A and B.  If the Cholesky
         factor L of B is already available, N should be prefixed
         with a minus sign.
       A AND B contain the real symmetric input matrices.  Only the
         full upper triangles of the matrices need be supplied.  If
         N is negative, the strict lower triangle of B contains,
         instead, the strict lower triangle of its Cholesky factor L.
       DL contains, if N is negative, the diagonal elements of L.
    On Output
       A contains in its full lower triangle the full lower triangle
         of the symmetric matrix derived from the reduction to the
         standard form.  The strict upper triangle of A is unaltered.
       B contains in its strict lower triangle the strict lower
         triangle of its Cholesky factor L.  The full upper
         triangle of B is unaltered.
       DL contains the diagonal elements of L.
       IERR is set to
         Zero       for normal return,
         7*N+1      if B is not positive definite.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



REORT


      SUBROUTINE REORT(NCOMP,Y,YP,YHP,NIV,W,S,P,IP,STOWA,IFLAG)



RG


      SUBROUTINE RG(NM,N,A,WR,WI,MATZ,Z,IV1,FV1,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    To find the eigenvalues and eigenvectors (if desired)
    of a REAL GENERAL matrix.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrix  A.
       A  contains the real general matrix.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       WR  and  WI  contain the real and imaginary parts,
       respectively, of the eigenvalues.  Complex conjugate
       pairs of eigenvalues appear consecutively with the
       eigenvalue having the positive imaginary part first.
       Z  contains the real and imaginary parts of the eigenvectors
       if MATZ is not zero.  If the J-th eigenvalue is real, the
       J-th column of  Z  contains its eigenvector.  If the J-TH
       eigenvalue is complex with positive imaginary part, the
       J-th and (J+1)-th columns of  Z  contain the real and
       imaginary parts of its eigenvector.  The conjugate of this
       vector is the eigenvector for the conjugate eigenvalue.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       IV1  and  FV1  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RGG


      SUBROUTINE RGG(NM,N,A,B,ALFR,ALFI,BETA,MATZ,Z,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    for the REAL GENERAL GENERALIZED eigenproblem  Ax = (LAMBDA)Bx.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrices  A  and  B.
       A  contains a real general matrix.
       B  contains a real general matrix.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       ALFR  and  ALFI  contain the real and imaginary parts,
       respectively, of the numerators of the eigenvalues.
       BETA  contains the denominators of the eigenvalues,
       which are thus given by the ratios  (ALFR+I*ALFI)/BETA.
       Complex conjugate pairs of eigenvalues appear consecutively
       with the eigenvalue having the positive imaginary part first.
       Z  contains the real and imaginary parts of the eigenvectors
       if MATZ is not zero.  If the J-th eigenvalue is real, the
       J-th column of  Z  contains its eigenvector.  If the J-th
       eigenvalue is complex with positive imaginary part, the
       J-th and (J+1)-th columns of  Z  contain the real and
       imaginary parts of its eigenvector.  The conjugate of this
       vector is the eigenvector for the conjugate eigenvalue.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RS


      SUBROUTINE RS(NM,N,A,W,MATZ,Z,FV1,FV2,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    of a REAL SYMMETRIC matrix.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrix  A.
       A  contains the real symmetric matrix.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       W  contains the eigenvalues in ascending order.
       Z  contains the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       FV1  and  FV2  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RSB


      SUBROUTINE RSB(NM,N,MB,A,W,MATZ,Z,FV1,FV2,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    of a REAL SYMMETRIC BAND matrix.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrix  A.
       MB  is the half band width of the matrix, defined as the
       number of adjacent diagonals, including the principal
       diagonal, required to specify the non-zero portion of the
       lower triangle of the matrix.
       A  contains the lower triangle of the real symmetric
       band matrix.  Its lowest subdiagonal is stored in the
       last  N+1-MB  positions of the first column, its next
       subdiagonal in the last  N+2-MB  positions of the
       second column, further subdiagonals similarly, and
       finally its principal diagonal in the  N  positions
       of the last column.  Contents of storages not part
       of the matrix are arbitrary.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       W  contains the eigenvalues in ascending order.
       Z  contains the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       FV1  and  FV2  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RSCO


      SUBROUTINE RSCO(RSAV,ISAV)



RSG


      SUBROUTINE RSG(NM,N,A,B,W,MATZ,Z,FV1,FV2,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    To find the eigenvalues and eigenvectors (if desired)
    for the REAL SYMMETRIC generalized eigenproblem  Ax = (LAMBDA)Bx.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrices  A  and  B.
       A  contains a real symmetric matrix.
       B  contains a positive definite real symmetric matrix.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       W  contains the eigenvalues in ascending order.
       Z  contains the eigenvectors if MATZ is not ZERO.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       FV1  and  FV2  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RSGAB


      SUBROUTINE RSGAB(NM,N,A,B,W,MATZ,Z,FV1,FV2,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    for the REAL SYMMETRIC generalized eigenproblem  ABx = (LAMBDA)x.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrices  A  and  B.
       A  contains a real symmetric matrix.
       B  contains a positive definite real symmetric matrix.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       W  contains the eigenvalues in ascending order.
       Z  contains the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       FV1  and  FV2  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RSGBA


      SUBROUTINE RSGBA(NM,N,A,B,W,MATZ,Z,FV1,FV2,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    for the REAL SYMMETRIC generalized eigenproblem  BAx = (LAMBDA)x.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrices  A  and  B.
       A  contains a real symmetric matrix.
       B  contains a positive definite real symmetric matrix.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       W  contains the eigenvalues in ascending order.
       Z  contains the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       FV1  and  FV2  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RSP


      SUBROUTINE RSP(NM,N,NV,A,W,MATZ,Z,FV1,FV2,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    of a REAL SYMMETRIC PACKED matrix.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrix  A.
       NV  is an integer variable set equal to the
       dimension of the array  A  as specified for
       A  in the calling program.  NV  must not be
       less than  N*(N+1)/2.
       A  contains the lower triangle of the real symmetric
       packed matrix stored row-wise.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       W  contains the eigenvalues in ascending order.
       Z  contains the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B OF the
       documentation.  The normal completion code is zero.
       FV1  and  FV2  are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RST


      SUBROUTINE RST(NM,N,W,E,MATZ,Z,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    of a REAL SYMMETRIC TRIDIAGONAL matrix.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrix.
       W  contains the diagonal elements of the real
       symmetric tridiagonal matrix.
       E  contains the subdiagonal elements of the matrix in
       its last N-1 POSITIONS.  E(1) is arbitrary.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       W  contains the eigenvalues in ascending order.
       Z  contains the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



RT


      SUBROUTINE RT(NM,N,A,W,MATZ,Z,FV1,IERR)

    This subroutine calls the recommended sequence of
    subroutines from the eigensystem subroutine package (EISPACK)
    to find the eigenvalues and eigenvectors (if desired)
    of a special REAL TRIDIAGONAL matrix.
    On Input
       NM  must be set to the row dimension of the two-dimensional
       array parameters as declared in the calling program
       dimension statement.
       N  is the order of the matrix  A.
       A  contains the special real tridiagonal matrix in its
       first three columns.  The subdiagonal elements are stored
       in the last  N-1  positions of the first column, the
       diagonal elements in the second column, and the superdiagonal
       elements in the first  N-1  positions of the third column.
       elements  A(1,1)  and  A(N,3)  are arbitrary.
       MATZ  is an integer variable set equal to zero if
       only eigenvalues are desired.  Otherwise it is set to
       any non-zero integer for both eigenvalues and eigenvectors.
    On Output
       W  contains the eigenvalues in ascending order.
       Z  contains the eigenvectors if MATZ is not zero.
       IERR  is an integer output variable set equal to an
       error completion code described in section 2B of the
       documentation.  The normal completion code is zero.
       FV1  is a temporary storage array.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



SASUM


      REAL FUNCTION SASUM(N,SX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(S)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
    --Output--
   SASUM  single precision result (zero if N .LE. 0)
    Returns sum of magnitudes of single precision SX.
    SASUM = sum from 0 to N-1 of  ABS(SX(1+I*INCX))


SAXPY


      SUBROUTINE SAXPY(N,SA,SX,INCX,SY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SA  single precision scalar multiplier
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
      SY  single precision vector with N elements
    INCY  storage spacing between elements of SY
    --Output--
      SY  single precision result (unchanged if N .LE. 0)
    Overwrite single precision SY with single precision SA*SX +SY.
    For I = 0 to N-1, replace  SY(LY+I*INCY) with SA*SX(LX+I*INCX) +
      SY(LY+I*INCY), where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N
      and LY is defined in a similar way using INCY.


SCASUM


      FUNCTION SCASUM(N,CX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
    --Output--
  SCASUM  single precision result (zero if N .LE. 0)
    Returns sums of magnitudes of real and imaginary parts of
    components of CX.  Note that this is not the L1 norm of CX.
    CASUM = sum from 0 to N-1 of ABS(REAL(CX(1+I*INCX))) +
            ABS(IMAG(CX(1+I*INCX)))


SCHDC


      SUBROUTINE SCHDC(A,LDA,P,WORK,JPVT,JOB,INFO)

    SCHDC computes the Cholesky decomposition of a positive definite
    matrix.  A pivoting option allows the user to estimate the
    condition of a positive definite matrix or determine the rank
    of a positive semidefinite matrix.
    On Entry
        A      REAL(LDA,P).
               A contains the matrix whose decomposition is to
               be computed.  Only the upper half of A need be stored.
               The lower part of the array A is not referenced.
        LDA    INTEGER.
               LDA is the leading dimension of the array A.
        P      INTEGER.
               P is the order of the matrix.
        WORK   REAL.
               WORK is a work array.
        JPVT   INTEGER(P).
               JPVT contains integers that control the selection
               of the pivot elements, if pivoting has been requested.
               Each diagonal element A(K,K)
               is placed in one of three classes according to the
               value of JPVT(K).
                  If JPVT(K) .GT. 0, then X(K) is an initial
                                     element.
                  If JPVT(K) .EQ. 0, then X(K) is a free element.
                  If JPVT(K) .LT. 0, then X(K) is a final element.
               Before the decomposition is computed, initial elements
               are moved by symmetric row and column interchanges to
               the beginning of the array A and final
               elements to the end.  Both initial and final elements
               are frozen in place during the computation and only
               free elements are moved.  At the K-th stage of the
               reduction, if A(K,K) is occupied by a free element
               it is interchanged with the largest free element
               A(L,L) with L .GE. K.  JPVT is not referenced if
               JOB .EQ. 0.
       JOB     INTEGER.
               JOB is an integer that initiates column pivoting.
               If JOB .EQ. 0, no pivoting is done.
               If JOB .NE. 0, pivoting is done.
    On Return
        A      A contains in its upper half the Cholesky factor
               of the matrix A as it has been permuted by pivoting.
        JPVT   JPVT(J) contains the index of the diagonal element
               of a that was moved into the J-th position,
               provided pivoting was requested.
        INFO   contains the index of the last positive diagonal
               element of the Cholesky factor.
    For positive definite matrices INFO = P is the normal return.
    For pivoting with positive semidefinite matrices INFO will
    in general be less than P.  However, INFO may be greater than
    the rank of A, since rounding error can cause an otherwise zero
    element to be positive.  Indefinite systems will always cause
    INFO to be less than P.
    LINPACK.  This version dated 03/19/79 .
    J. J. Dongarra and G. W. Stewart, Argonne National Laboratory and
    University of Maryland.
    BLAS SAXPY,SSWAP
    Fortran SQRT


SCHDD


      SUBROUTINE SCHDD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S,INFO)

    SCHDD downdates an augmented Cholesky decomposition or the
    triangular factor of an augmented QR decomposition.
    Specifically, given an upper triangular matrix R of order P, a
    row vector X, a column vector Z, and a scalar Y, SCHDD
    determines an orthogonal matrix U and a scalar ZETA such that
                       (R   Z )     (RR  ZZ)
                   U * (      )  =  (      ) ,
                       (0 ZETA)     ( X   Y)
    where RR is upper triangular.  If R and Z have been obtained
    from the factorization of a least squares problem, then
    RR and ZZ are the factors corresponding to the problem
    with the observation (X,Y) removed.  In this case, if RHO
    is the norm of the residual vector, then the norm of
    the residual vector of the downdated problem is
    SQRT(RHO**2 - ZETA**2). SCHDD will simultaneously downdate
    several triplets (Z,Y,RHO) along with R.
    For a less terse description of what SCHDD does and how
    it may be applied, see the LINPACK guide.
    The matrix U is determined as the product U(1)*...*U(P)
    where U(I) is a rotation in the (P+1,I)-plane of the
    form
                      ( C(I)     -S(I)     )
                      (                    ) .
                      ( S(I)       C(I)    )
    The rotations are chosen so that C(I) is real.
    The user is warned that a given downdating problem may
    be impossible to accomplish or may produce
    inaccurate results.  For example, this can happen
    if X is near a vector whose removal will reduce the
    rank of R.  Beware.
    On Entry
        R      REAL(LDR,P), where LDR .GE. P.
               R contains the upper triangular matrix
               that is to be downdated.  The part of  R
               below the diagonal is not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        P      INTEGER.
               P is the order of the matrix R.
        X      REAL(P).
               X contains the row vector that is to
               be removed from R.  X is not altered by SCHDD.
        Z      REAL(LDZ,NZ), where LDZ .GE. P.
               Z is an array of NZ P-vectors which
               are to be downdated along with R.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of vectors to be downdated
               NZ may be zero, in which case Z, Y, and RHO
               are not referenced.
        Y      REAL(NZ).
               Y contains the scalars for the downdating
               of the vectors Z.  Y is not altered by SCHDD.
        RHO    REAL(NZ).
               RHO contains the norms of the residual
               vectors that are to be downdated.
    On Return
        R
        Z      contain the downdated quantities.
        RHO
        C      REAL(P).
               C contains the cosines of the transforming
               rotations.
        S      REAL(P).
               S contains the sines of the transforming
               rotations.
        INFO   INTEGER.
               INFO is set as follows.
                  INFO = 0  if the entire downdating
                            was successful.
                  INFO =-1  if R could not be downdated.
                            In this case, all quantities
                            are left unaltered.
                  INFO = 1  if some RHO could not be
                            downdated.  The offending RHOs are
                            set to -1.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    SCHDD uses the following functions and subprograms.
    Fortran ABS
    BLAS SDOT, SNRM2


SCHEX


      SUBROUTINE SCHEX(R,LDR,P,K,L,Z,LDZ,NZ,C,S,JOB)

    SCHEX updates the Cholesky factorization
                  A = TRANS(R)*R
    of a positive definite matrix A of order P under diagonal
    permutations of the form
                  TRANS(E)*A*E
    where E is a permutation matrix.  Specifically, given
    an upper triangular matrix R and a permutation matrix
    E (which is specified by K, L, and JOB), SCHEX determines
    an orthogonal matrix U such that
                          U*R*E = RR,
    where RR is upper triangular.  At the users option, the
    transformation U will be multiplied into the array Z.
    If A = TRANS(X)*X, so that R is the triangular part of the
    QR factorization of X, then RR is the triangular part of the
    QR factorization of X*E, i.e., X with its columns permuted.
    For a less terse description of what SCHEX does and how
    it may be applied, see the LINPACK guide.
    The matrix Q is determined as the product U(L-K)*...*U(1)
    of plane rotations of the form
                          (    C(I)       S(I) )
                          (                    ) ,
                          (    -S(I)      C(I) )
    where C(I) is real.  The rows these rotations operate on
    are described below.
    There are two types of permutations, which are determined
    by the value of JOB.
    1. Right circular shift (JOB = 1).
        The columns are rearranged in the following order.
               1,...,K-1,L,K,K+1,...,L-1,L+1,...,P.
        U is the product of L-K rotations U(I), where U(I)
        acts in the (L-I,L-I+1)-plane.
    2. Left circular shift (JOB = 2).
        The columns are rearranged in the following order
               1,...,K-1,K+1,K+2,...,L,K,L+1,...,P.
        U is the product of L-K rotations U(I), where U(I)
        acts in the (K+I-1,K+I)-plane.
    On Entry
        R      REAL(LDR,P), where LDR .GE. P.
               R contains the upper triangular factor
               that is to be updated.  Elements of R
               below the diagonal are not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        P      INTEGER.
               P is the order of the matrix R.
        K      INTEGER.
               K is the first column to be permuted.
        L      INTEGER.
               L is the last column to be permuted.
               L must be strictly greater than K.
        Z      REAL(LDZ,NZ), where LDZ.GE.P.
               Z is an array of NZ P-vectors into which the
               transformation U is multiplied.  Z is
               not referenced if NZ = 0.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of columns of the matrix Z.
        JOB    INTEGER.
               JOB determines the type of permutation.
                      JOB = 1  right circular shift.
                      JOB = 2  left circular shift.
    On Return
        R      contains the updated factor.
        Z      contains the updated matrix Z.
        C      REAL(P).
               C contains the cosines of the transforming rotations.
        S      REAL(P).
               S contains the sines of the transforming rotations.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    SCHEX uses the following functions and subroutines.
    BLAS SROTG
    Fortran MIN0


SCHUD


      SUBROUTINE SCHUD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S)

    SCHUD updates an augmented Cholesky decomposition of the
    triangular part of an augmented QR decomposition.  Specifically,
    given an upper triangular matrix R of order P, a row vector
    X, a column vector Z, and a scalar Y, SCHUD determines a
    unitary matrix U and a scalar ZETA such that
                             (R  Z)     (RR   ZZ )
                        U  * (    )  =  (        ) ,
                             (X  Y)     ( 0  ZETA)
    where RR is upper triangular.  If R and Z have been
    obtained from the factorization of a least squares
    problem, then RR and ZZ are the factors corresponding to
    the problem with the observation (X,Y) appended.  In this
    case, if RHO is the norm of the residual vector, then the
    norm of the residual vector of the updated problem is
    SQRT(RHO**2 + ZETA**2).  SCHUD will simultaneously update
    several triplets (Z,Y,RHO).
    For a less terse description of what SCHUD does and how
    it may be applied, see the LINPACK guide.
    The matrix U is determined as the product U(P)*...*U(1),
    where U(I) is a rotation in the (I,P+1) plane of the
    form
                      (     C(I)      S(I) )
                      (                    ) .
                      (    -S(I)      C(I) )
    The rotations are chosen so that C(I) is real.
    On Entry
        R      REAL(LDR,P), where LDR .GE. P.
               R contains the upper triangular matrix
               that is to be updated.  The part of R
               below the diagonal is not referenced.
        LDR    INTEGER.
               LDR is the leading dimension of the array R.
        P      INTEGER.
               P is the order of the matrix R.
        X      REAL(P).
               X contains the row to be added to R.  X is
               not altered by SCHUD.
        Z      REAL(LDZ,NZ), where LDZ .GE. P.
               Z is an array containing NZ P-vectors to
               be updated with R.
        LDZ    INTEGER.
               LDZ is the leading dimension of the array Z.
        NZ     INTEGER.
               NZ is the number of vectors to be updated.
               NZ may be zero, in which case Z, Y, and RHO
               are not referenced.
        Y      REAL(NZ).
               Y contains the scalars for updating the vectors
               Z.  Y is not altered by SCHUD.
        RHO    REAL(NZ).
               RHO contains the norms of the residual
               vectors that are to be updated.  If RHO(J)
               is negative, it is left unaltered.
    On Return
        RC
        RHO    contain the updated quantities.
        Z
        C      REAL(P).
               C contains the cosines of the transforming
               rotations.
        S      REAL(P).
               S contains the sines of the transforming
               rotations.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    SCHUD uses the following functions and subroutines.
    Extended BLAS SROTG
    Fortran SQRT


SCLOSM


      SUBROUTINE SCLOSM(IPAGE)

    1. UNLOAD, RELEASE, OR CLOSE UNIT NUMBER IPAGEF.
    THIS IS FORTRAN 77 STANDARD FOR THE 'CLOSE' STATEMENT.


SCNRM2


      REAL FUNCTION SCNRM2(N,CX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      CX  complex vector with N elements
    INCX  storage spacing between elements of CX
    --Output--
  SCNRM2  single precision result (zero if N .LE. 0)
    unitary norm of the complex N-vector stored in CX() with storage
    increment INCX .
    If N .LE. 0, return with result = 0.
    If N .GE. 1, then INCX must be .GE. 1
          C. L. Lawson, 1978 Jan 08
    Four phase method     using two built-in constants that are
    hopefully applicable to all machines.
        CUTLO = maximum of  SQRT(U/EPS)  over all known machines.
        CUTHI = minimum of  SQRT(V)      over all known machines.
    where
        EPS = smallest no. such that EPS + 1. .GT. 1.
        U   = smallest positive no.   (underflow limit)
        V   = largest  no.            (overflow  limit)
    Brief outline of algorithm..
    Phase 1    scans zero components.
    Move to phase 2 when a component is nonzero and .LE. CUTLO
    Move to phase 3 when a component is .GT. CUTLO
    Move to phase 4 when a component is .GE. CUTHI/M
    where M = N for X() real and M = 2*N for complex.
    Values for CUTLO and CUTHI..
    From the environmental parameters listed in the IMSL converter
    document the limiting values are as follows..
    CUTLO, S.P.   U/EPS = 2**(-102) for  Honeywell.  Close seconds are
                  Univac and DEC at 2**(-103)
                  Thus CUTLO = 2**(-51) = 4.44089E-16
    CUTHI, S.P.   V = 2**127 for Univac, Honeywell, and DEC.
                  Thus CUTHI = 2**(63.5) = 1.30438E19
    CUTLO, D.P.   U/EPS = 2**(-67) for Honeywell and DEC.
                  Thus CUTLO = 2**(-33.5) = 8.23181D-11
    CUTHI, D.P.   same as S.P.  CUTHI = 1.30438D19
    SAVE CUTLO, CUTHI, ZERO, ONE
    DATA CUTLO, CUTHI / 8.232D-11,  1.304D19 /
    DATA CUTLO, CUTHI / 4.441E-16,  1.304E19 /


SCOEF


      SUBROUTINE SCOEF(YH,YP,NCOMP,NROWB,NFC,NIC,B,BETA,COEF,INHOMO,RE,



SCOPY


      SUBROUTINE SCOPY(N,SX,INCX,SY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
      SY  single precision vector with N elements
    INCY  storage spacing between elements of SY
    --Output--
      SY  copy of vector SX (unchanged if N .LE. 0)
    Copy single precision SX to single precision SY.
    For I = 0 to N-1, copy  SX(LX+I*INCX) to SY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


SCOPYM


      SUBROUTINE SCOPYM(N,SX,INCX,SY,INCY)

      Description of Parameters
          The * Flags Output Variables
      N   Number of elements in vector(s)
     SX   Real vector with N elements
   INCX   Storage spacing between elements of SX
     SY*  Real negative copy of SX
   INCY   Storage spacing between elements of SY
     ***  Note that SY = -SX  ***
 Copy negative of real SX to real SY.  For I=0 to N-1,
  copy  -SX(LX+I*INCX) to SY(LY+I*INCY), where LX=1 if
  INCX .GE. 0, else LX = (-INCX)*N, and LY is defined
  in a similar way using INCY.


SDOT


      REAL FUNCTION SDOT(N,SX,INCX,SY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
      SY  single precision vector with N elements
    INCY  storage spacing between elements of SY
    --Output--
    SDOT  single precision dot product (zero if N .LE. 0)
    Returns the dot product of single precision SX and SY.
    SDOT = sum for I = 0 to N-1 of  SX(LX+I*INCX) * SY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


SDSDOT


      REAL FUNCTION SDSDOT(N,SB,SX,INCX,SY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SB  single precision scalar to be added to inner product
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
      SY  single precision vector with N elements
    INCY  storage spacing between elements of SY
    --Output--
  SDSDOT  single precision dot product (zero if N .LE. 0)
    Returns S.P. result with dot product accumulated in D.P.
    SDSDOT = SB + sum for I = 0 to N-1 of SX(LX+I*INCX)*SY(LY+I*INCY)
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


SGBCO


      SUBROUTINE SGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)

    SBGCO factors a real band matrix by Gaussian
    elimination and estimates the condition of the matrix.
    If  RCOND  is not needed, SGBFA is slightly faster.
    To solve  A*X = B , follow SBGCO by SGBSL.
    To compute  INVERSE(A)*C , follow SBGCO by SGBSL.
    To compute  DETERMINANT(A) , follow SBGCO by SGBDI.
    On Entry
       ABD     REAL(LDA, N)
               contains the matrix in band storage.  The columns
               of the matrix are stored in the columns of  ABD  and
               the diagonals of the matrix are stored in rows
               ML+1 through 2*ML+MU+1 of  ABD .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. 2*ML + MU + 1 .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT. N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT. N .
               More efficient if  ML .LE. MU .
    On Return
       ABD     an upper triangular matrix in band storage and
               the multipliers which were used to obtain it.
               The factorization can be written  A = L*U  where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  M = ML + MU + 1
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-MU)
                     I2 = MIN0(N, J+ML)
                     DO 10 I = I1, I2
                        K = I - J + M
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses rows  ML+1  through  2*ML+MU+1  of  ABD .
          In addition, the first  ML  rows in  ABD  are used for
          elements generated during the triangularization.
          The total number of rows needed in  ABD  is  2*ML+MU+1 .
          The  ML+MU by ML+MU  upper left triangle and the
          ML by ML  lower right triangle are not referenced.
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABD should contain
           *  *  *  +  +  +  , * = not used
           *  * 13 24 35 46  , + = used for pivoting
           * 12 23 34 45 56
          11 22 33 44 55 66
          21 32 43 54 65  *
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK SGBFA
    BLAS SAXPY,SDOT,SSCAL,SASUM
    Fortran ABS,AMAX1,MAX0,MIN0,SIGN


SGBDI


      SUBROUTINE SGBDI(ABD,LDA,N,ML,MU,IPVT,DET)

    SGBDI computes the determinant of a band matrix
    using the factors computed by SBGCO or SGBFA.
    If the inverse is needed, use SGBSL  N  times.
    On Entry
       ABD     REAL(LDA, N)
               the output from SBGCO or SGBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from SBGCO or SGBFA.
    On Return
       DET     REAL(2)
               determinant of original matrix.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. ABS(DET(1)) .LT. 10.0
               or  DET(1) = 0.0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    Fortran ABS


SGBFA


      SUBROUTINE SGBFA(ABD,LDA,N,ML,MU,IPVT,INFO)

    SGBFA factors a real band matrix by elimination.
    SGBFA is usually called by SBGCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    On Entry
       ABD     REAL(LDA, N)
               contains the matrix in band storage.  The columns
               of the matrix are stored in the columns of  ABD  and
               the diagonals of the matrix are stored in rows
               ML+1 through 2*ML+MU+1 of  ABD .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. 2*ML + MU + 1 .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT. N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT. N .
               More efficient if  ML .LE. MU .
    On Return
       ABD     an upper triangular matrix in band storage and
               the multipliers which were used to obtain it.
               The factorization can be written  A = L*U , where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if  U(K,K) .EQ. 0.0 .  This is not an error
                    condition for this subroutine, but it does
                    indicate that SGBSL will divide by zero if
                    called.  Use  RCOND  in SBGCO for a reliable
                    indication of singularity.
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  M = ML + MU + 1
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-MU)
                     I2 = MIN0(N, J+ML)
                     DO 10 I = I1, I2
                        K = I - J + M
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses rows  ML+1  through  2*ML+MU+1  of  ABD .
          In addition, the first  ML  rows in  ABD  are used for
          elements generated during the triangularization.
          The total number of rows needed in  ABD  is  2*ML+MU+1 .
          The  ML+MU by ML+MU  upper left triangle and the
          ML by ML  lower right triangle are not referenced.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SSCAL,ISAMAX
    Fortran MAX0,MIN0


SGBSL


      SUBROUTINE SGBSL(ABD,LDA,N,ML,MU,IPVT,B,JOB)

    SGBSL solves the real band system
    A * X = B  or  TRANS(A) * X = B
    using the factors computed by SBGCO or SGBFA.
    On Entry
       ABD     REAL(LDA, N)
               the output from SBGCO or SGBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from SBGCO or SGBFA.
       B       REAL(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B ,
               = nonzero   to solve  TRANS(A)*X = B , where
                           TRANS(A)  is the transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically, this indicates singularity,
       but it is often caused by improper arguments or improper
       setting of LDA .  It will not occur if the subroutines are
       called correctly and if SBGCO has set RCOND .GT. 0.0
       or SGBFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL SBGCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)
          If (RCOND is too small) GO TO ...
          DO 10 J = 1, P
             CALL SGBSL(ABD,LDA,N,ML,MU,IPVT,C(1,J),0)
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SDOT
    Fortran MIN0


SGECO


      SUBROUTINE SGECO(A,LDA,N,IPVT,RCOND,Z)

    SGECO factors a real matrix by Gaussian elimination
    and estimates the condition of the matrix.
    If  RCOND  is not needed, SGEFA is slightly faster.
    To solve  A*X = B , follow SGECO by SGESL.
    To compute  INVERSE(A)*C , follow SGECO by SGESL.
    To compute  DETERMINANT(A) , follow SGECO by SGEDI.
    To compute  INVERSE(A) , follow SGECO by SGEDI.
    On Entry
       A       REAL(LDA, N)
               the matrix to be factored.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix and the multipliers
               which were used to obtain it.
               The factorization can be written  A = L*U , where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK SGEFA
    BLAS SAXPY,SDOT,SSCAL,SASUM
    Fortran ABS,AMAX1,SIGN


SGEDI


      SUBROUTINE SGEDI(A,LDA,N,IPVT,DET,WORK,JOB)

    SGEDI computes the determinant and inverse of a matrix
    using the factors computed by SGECO or SGEFA.
    On Entry
       A       REAL(LDA, N)
               the output from SGECO or SGEFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       IPVT    INTEGER(N)
               the pivot vector from SGECO or SGEFA.
       WORK    REAL(N)
               work vector.  Contents destroyed.
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       A       inverse of original matrix if requested.
               Otherwise unchanged.
       DET     REAL(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. ABS(DET(1)) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if SGECO has set RCOND .GT. 0.0 or SGEFA has set
       INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SSCAL,SSWAP
    Fortran ABS,MOD


SGEEV


      SUBROUTINE SGEEV(A,LDA,N,E,V,LDV,WORK,JOB,INFO)

    LICEPACK.    This version dated 08/08/80.
    David Kahaner, Cleve Moler, G. W. Stewart,
      N.B.S.         U.N.M.      N.B.S./U.MD.
    Abstract
     SGEEV computes the eigenvalues and, optionally,
     the eigenvectors of a general real matrix.
    Call Sequence Parameters-
      (The values of parameters marked with * (star) will be changed
        by SGEEV.)
       A*      REAL(LDA,N)
               real nonsymmetric input matrix.
       LDA     INTEGER
               set by the user to
               the leading dimension of the real array A.
       N       INTEGER
               set by the user to
               the order of the matrices A and V, and
               the number of elements in E.
       E*      COMPLEX(N)
               on return from SGEEV, E contains the eigenvalues of A.
               See also INFO below.
       V*      COMPLEX(LDV,N)
               on return from SGEEV, if the user has set JOB
               = 0        V is not referenced.
               = nonzero  the N eigenvectors of A are stored in the
               first N columns of V.  See also INFO below.
               (Note that if the input matrix A is nearly degenerate,
                V may be badly conditioned, i.e., may have nearly
                dependent columns.)
       LDV     INTEGER
               set by the user to
               the leading dimension of the array V if JOB is also
               set nonzero.  In that case, N must be .LE. LDV.
               If JOB is set to zero, LDV is not referenced.
       WORK*   REAL(2N)
               temporary storage vector.  Contents changed by SGEEV.
       JOB     INTEGER
               set by the user to
               = 0        eigenvalues only to be calculated by SGEEV.
                          Neither V nor LDV is referenced.
               = nonzero  eigenvalues and vectors to be calculated.
                          In this case, A & V must be distinct arrays.
                          Also, if LDA .GT. LDV, SGEEV changes all the
                          elements of A thru column N.  If LDA < LDV,
                          SGEEV changes all the elements of V through
                          column N. If LDA = LDV, only A(I,J) and V(I,
                          J) for I,J = 1,...,N are changed by SGEEV.
       INFO*   INTEGER
               on return from SGEEV the value of INFO is
               = 0  normal return, calculation successful.
               = K  if the eigenvalue iteration fails to converge,
                    eigenvalues K+1 through N are correct, but
                    no eigenvectors were computed even if they were
                    requested (JOB nonzero).
     Error Messages
          No. 1  recoverable  N is greater than LDA
          No. 2  recoverable  N is less than one.
          No. 3  recoverable  JOB is nonzero and N is greater than LDV
          No. 4  warning      LDA > LDV, elements of A other than the
                              N by N input elements have been changed.
          No. 5  warning      LDA < LDV, elements of V other than the
                              N x N output elements have been changed.
    Subroutines used
    EISPACK-  BALANC,BALBAK, ORTHES, ORTRAN, HQR, HQR2
    BLAS-  SCOPY, SCOPYM
    SLATEC- XERROR


SGEFA


      SUBROUTINE SGEFA(A,LDA,N,IPVT,INFO)

    SGEFA factors a real matrix by Gaussian elimination.
    SGEFA is usually called by SGECO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (Time for SGECO) = (1 + 9/N)*(Time for SGEFA) .
    On Entry
       A       REAL(LDA, N)
               the matrix to be factored.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix and the multipliers
               which were used to obtain it.
               The factorization can be written  A = L*U , where
               L  is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if  U(K,K) .EQ. 0.0 .  This is not an error
                    condition for this subroutine, but it does
                    indicate that SGESL or SGEDI will divide by zero
                    if called.  Use  RCOND  in SGECO for a reliable
                    indication of singularity.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SSCAL,ISAMAX


SGEFS


      SUBROUTINE SGEFS(A,LDA,N,V,ITASK,IND,WORK,IWORK)

   Subroutine SGEFS solves a general NxN system of single
   precision linear equations using LINPACK subroutines SGECO
   and SGESL.  That is, if A is an NxN real matrix and if X
   and B are real N-vectors, then SGEFS solves the equation
                         A*X=B.
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to find the
   solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, N and IWORK must not have been altered by the user follow-
   ing factorization (ITASK=1).  IND will not be changed by SGEFS
   in this case.
 Argument Description ***
   A      REAL(LDA,N)
            on entry, the doubly subscripted array with dimension
              (LDA,N) which contains the coefficient matrix.
            on return, an upper triangular matrix U and the
              multipliers necessary to construct a matrix L
              so that A=L*U.
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  The first N elements of
            the array A are the elements of the first column of
            the  matrix A.  N must be greater than or equal to 1.
            (terminal error message IND=-2)
   V      REAL(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A and IWORK.
            If ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT. 0  see error message corresponding to IND below.
   WORK   REAL(N)
            a singly subscripted array of dimension at least N.
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   The matrix A is computationally singular.
                        A solution has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              Note-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


SGEIR


      SUBROUTINE SGEIR(A,LDA,N,V,ITASK,IND,WORK,IWORK)

   Subroutine SGEIR solves a general NxN system of single
   precision linear equations using LINPACK subroutines SGEFA and
   SGESL.  One pass of iterative refinement is used only to obtain
   an estimate of the accuracy.  That is, if A is an NxN real
   matrix and if X and B are real N-vectors, then SGEIR solves
   the equation
                         A*X=B.
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to calculate
   the solution, X.  Then the residual vector is found and
   used to calculate an estimate of the relative error, IND.
   IND estimates the accuracy of the solution only when the
   input matrix and the right hand side are represented
   exactly in the computer and does not take into account
   any errors in the input data.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to solve only (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, N, WORK, and IWORK must not have been altered by the
   user following factorization (ITASK=1).  IND will not be
   changed by SGEIR in this case.
 Argument Description ***
   A      REAL(LDA,N)
            the doubly subscripted array with dimension (LDA,N)
            which contains the coefficient matrix.  A is not
            altered by the routine.
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  The first N elements of
            the array A are the elements of the first column of
            matrix A.  N must be greater than or equal to 1.
            (terminal error message IND=-2)
   V      REAL(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A (stored in WORK).
            If ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.  IND=75 means
                    that the solution vector X is zero.
            LT. 0  see error message corresponding to IND below.
   WORK   REAL(N*(N+1))
            a singly subscripted array of dimension at least N*(N+1).
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than one.
   IND=-3  terminal   ITASK is less than one.
   IND=-4  terminal   The matrix A is computationally singular.
                        A solution has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              Note-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


SGESL


      SUBROUTINE SGESL(A,LDA,N,IPVT,B,JOB)

    SGESL solves the real system
    A * X = B  or  TRANS(A) * X = B
    using the factors computed by SGECO or SGEFA.
    On Entry
       A       REAL(LDA, N)
               the output from SGECO or SGEFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       IPVT    INTEGER(N)
               the pivot vector from SGECO or SGEFA.
       B       REAL(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B ,
               = nonzero   to solve  TRANS(A)*X = B  where
                           TRANS(A)  is the transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically, this indicates singularity,
       but it is often caused by improper arguments or improper
       setting of LDA .  It will not occur if the subroutines are
       called correctly and if SGECO has set RCOND .GT. 0.0
       or SGEFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL SGECO(A,LDA,N,IPVT,RCOND,Z)
          IF (RCOND is too small) GO TO ...
          DO 10 J = 1, P
             CALL SGESL(A,LDA,N,IPVT,C(1,J),0)
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SDOT


SGLSS


      SUBROUTINE SGLSS(A,MDA,M,N,B,MDB,NB,RNORM,WORK,LW,IWORK,LIW,INFO)

    SGLSS solves both underdetermined and overdetermined
    LINEAR systems AX = B, where A is an M by N matrix
    and B is an M by NB matrix of right hand sides. If
    M.GE.N, the least squares solution is computed by
    decomposing the matrix A into the product of an
    orthogonal matrix Q and an upper triangular matrix
    R (QR factorization). If M.LT.N, the minimal
    length solution is computed by factoring the
    matrix A into the product of a lower triangular
    matrix L and an orthogonal matrix Q (LQ factor-
    ization). If the matrix A is determined to be rank
    deficient, that is the rank of A is less than
    MIN(M,N), then the minimal length least squares
    solution is computed.
    SGLSS assumes full machine precision in the data.
    If more control over the uncertainty in the data
    is desired, the codes LLSIA and ULSIA are
    recommended.
    SGLSS requires MDA*N + (MDB + 1)*NB + 5*MIN(M,N) dimensioned
    real space and M+N dimensioned integer space.
  ******************************************************************
  *                                                                *
  *         WARNING - All input arrays are changed on exit.        *
  *                                                                *
  ******************************************************************
    SUBROUTINE SGLSS(A,MDA,M,N,B,MDB,NB,RNORM,WORK,LW,IWORK,LIW,INFO)
    Input..
    A(,)          Linear coefficient matrix of AX=B, with MDA the
     MDA,M,N      actual first dimension of A in the calling program.
                  M is the row dimension (no. of EQUATIONS of the
                  problem) and N the col dimension (no. of UNKNOWNS).
    B(,)          Right hand side(s), with MDB the actual first
     MDB,NB       dimension of B in the calling program. NB is the
                  number of M by 1 right hand sides. Must have
                  MDB.GE.MAX(M,N). If NB = 0, B is never accessed.
    RNORM()       Vector of length at least NB.  On input the contents
                  of RNORM are unused.
    WORK()        A real work array dimensioned 5*MIN(M,N).
    LW            Actual dimension of WORK.
    IWORK()       Integer work array dimensioned at least N+M.
    LIW           Actual dimension of IWORK.
    INFO          A flag which provides for the efficient
                  solution of subsequent problems involving the
                  same A but different B.
                  If INFO = 0 original call
                     INFO = 1 subsequent calls
                  On subsequent calls, the user must supply A, INFO,
                  LW, IWORK, LIW, and the first 2*MIN(M,N) locations
                  of WORK as output by the original call to SGLSS.
    Output..
    A(,)          Contains the triangular part of the reduced matrix
                  and the transformation information. It together with
                  the first 2*MIN(M,N) elements of WORK (see below)
                  completely specify the factorization of A.
    B(,)          Contains the N by NB solution matrix X.
    RNORM()       Contains the Euclidean length of the NB residual
                  vectors  B(I)-AX(I), I=1,NB.
    WORK()        The first 2*MIN(M,N) locations of WORK contain value
                  necessary to reproduce the factorization of A.
    IWORK()       The first M+N locations contain the order in
                  which the rows and columns of A were used.
                  If M.GE.N columns then rows. If M.LT.N rows
                  then columns.
    INFO          Flag to indicate status of computation on completion
                 -1   Parameter error(s)
                  0 - Full rank
                  N.GT.0 - Reduced rank  rank=MIN(M,N)-INFO


SGTSL


      SUBROUTINE SGTSL(N,C,D,E,B,INFO)

    SGTSL given a general tridiagonal matrix and a right hand
    side will find the solution.
    On Entry
       N       INTEGER
               is the order of the tridiagonal matrix.
       C       REAL(N)
               is the subdiagonal of the tridiagonal matrix.
               C(2) through C(N) should contain the subdiagonal.
               On output, C is destroyed.
       D       REAL(N)
               is the diagonal of the tridiagonal matrix.
               On output, D is destroyed.
       E       REAL(N)
               is the superdiagonal of the tridiagonal matrix.
               E(1) through E(N-1) should contain the superdiagonal.
               On output, E is destroyed.
       B       REAL(N)
               is the right hand side vector.
    On Return
       B       is the solution vector.
       INFO    INTEGER
               = 0 normal value.
               = K if the K-th element of the diagonal becomes
                   exactly zero.  The subroutine returns when
                   this is detected.
    LINPACK.  This version dated 08/14/78 .
    Jack Dongarra, Argonne National Laboratory.
    No externals
    Fortran ABS


SNBCO


      SUBROUTINE SNBCO(ABE,LDA,N,ML,MU,IPVT,RCOND,Z)

    SNBCO factors a real band matrix by Gaussian
    elimination and estimates the condition of the matrix.
    If RCOND is not needed, SNBFA is slightly faster.
    To solve  A*X = B , follow SNBCO by SNBSL.
    To compute  INVERSE(A)*C , follow SNBCO by SNBSL.
    To compute  DETERMINANT(A) , follow SNBCO by SNBDI.
    On Entry
       ABE     REAL(LDA, NC)
               contains the matrix in band storage.  The rows
               of the original matrix are stored in the rows
               of ABE and the diagonals of the original matrix
               are stored in columns 1 through ML+MU+1 of ABE.
               NC must be .GE. 2*ML+MU+1 .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array ABE.
               LDA must be .GE. N .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT. N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT. N .
               More efficient if ML .LE. MU .
    On Return
       ABE     an upper triangular matrix in band storage
               and the multipliers which were used to obtain it.
               The factorization can be written  A = L*U , where
               L is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                        1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  DO 20 I = 1, N
                     J1 = MAX0(1, I-ML)
                     J2 = MIN0(N, I+MU)
                     DO 10 J = J1, J2
                        K = J - I + ML + 1
                        ABE(I,K) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses columns  1  through  ML+MU+1  of ABE .
          Furthermore,  ML  additional columns are needed in
          ABE  starting with column  ML+MU+2  for elements
          generated during the triangularization.  The total
          number of columns needed in  ABE  is  2*ML+MU+1 .
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
           * 11 12 13  +     , * = not used
          21 22 23 24  +     , + = used for pivoting
          32 33 34 35  +
          43 44 45 46  +
          54 55 56  *  +
          65 66  *  *  +
    SLATEC.  This version dated 07/23/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
    SLATEC SNBFA
     SDOT,SAXPY,SSCAL,SASUM
    Fortran  ABS,AMAX1,MAX0,MIN0,SIGN


SNBDI


      SUBROUTINE SNBDI(ABE,LDA,N,ML,MU,IPVT,DET)

    SNBDI computes the determinant of a band matrix
    using the factors computed by SNBCO or SNBFA.
    If the inverse is needed, use SNBSL  N  times.
    On Entry
       ABE     REAL(LDA, NC)
               the output from SNBCO or SNBFA.
               NC must be .GE. 2*ML+MU+1 .
       LDA     INTEGER
               the leading dimension of the array  ABE .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from SNBCO or SNBFA.
    On Return
       DET     REAL(2)
               determinant of original matrix.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. ABS(DET(1)) .LT. 10.0
               or  DET(1) = 0.0 .
    SLATEC.  This version dated 07/25/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
    Fortran ABS


SNBFA


      SUBROUTINE SNBFA(ABE,LDA,N,ML,MU,IPVT,INFO)

    SNBFA factors a real band matrix by elimination.
    SNBFA is usually called by SNBCO, but it can be called
    directly with a saving in time if RCOND is not needed.
    On Entry
       ABE     REAL(LDA, NC)
               contains the matrix in band storage.  The rows
               of the original matrix are stored in the rows
               of ABE and the diagonals of the original matrix
               are stored in columns 1 through ML+MU+1 of ABE.
               NC must be .GE. 2*ML+MU+1 .
               See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array ABE.
               LDA must be .GE. N .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
               0 .LE. ML .LT. N .
       MU      INTEGER
               number of diagonals above the main diagonal.
               0 .LE. MU .LT. N .
               More efficient if ML .LE. MU .
    On Return
       ABE     an upper triangular matrix in band storage
               and the multipliers which were used to obtain it.
               The factorization can be written  A = L*U , where
               L is a product of permutation and unit lower
               triangular matrices and  U  is upper triangular.
       IPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               =0  normal value
               =K  if  U(K,K) .EQ. 0.0 .  This is not an error
               condition for this subroutine, but it does
               indicate that SNBSL will divide by zero if
               called.  Use RCOND in SNBCO for a reliable
               indication of singularity.
    Band Storage
          If  A  is a band matrix, the following program segment
          will set up the input.
                  ML = (band width below the diagonal)
                  MU = (band width above the diagonal)
                  DO 20 I = 1, N
                     J1 = MAX0(1, I-ML)
                     J2 = MIN0(N, I+MU)
                     DO 10 J = J1, J2
                        K = J - I + ML + 1
                        ABE(I,K) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses columns  1  through  ML+MU+1  of ABE .
          Furthermore,  ML  additional columns are needed in
          ABE  starting with column  ML+MU+2  for elements
          generated during the triangularization.  The total
          number of columns needed in  ABE  is  2*ML+MU+1 .
    Example:  If the original matrix is
          11 12 13  0  0  0
          21 22 23 24  0  0
           0 32 33 34 35  0
           0  0 43 44 45 46
           0  0  0 54 55 56
           0  0  0  0 65 66
     then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
           * 11 12 13  +     , * = not used
          21 22 23 24  +     , + = used for pivoting
          32 33 34 35  +
          43 44 45 46  +
          54 55 56  *  +
          65 66  *  *  +
    SLATEC.  This version dated 06/16/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
     ISAMAX,SAXPY,SSCAL,SSWAP
    Fortran  MIN0


SNBFS


      SUBROUTINE SNBFS(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK)

   Subroutine SNBFS solves a general nonsymmetric banded NxN
   system of single precision real linear equations using
   SLATEC subroutines SNBCO and SNBSL.  These are adaptations
   of the LINPACK subroutines SBGCO and SGBSL, which require
   a different format for storing the matrix elements.  If
   A  is an NxN real matrix and if  X  and  B  are real
   N-vectors, then SNBFS solves the equation
                         A*X=B.
   A band matrix is a matrix whose nonzero elements are all
   fairly near the main diagonal, specifically  A(I,J) = 0
   if  I-J is greater than  ML  or  J-I  is greater than
   MU .  The integers ML and MU are called the lower and upper
   band widths and  M = ML+MU+1  is the total band width.
   SNBFS uses less time and storage than the corresponding
   program for general matrices (SGEFS) if 2*ML+MU .LT. N  .
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to find the
   solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, N and IWORK must not have been altered by the user follow-
   ing factorization (ITASK=1).  IND will not be changed by SNBFS
   in this case.
   Band Storage
         If  A  is a band matrix, the following program segment
         will set up the input.
                 ML = (band width below the diagonal)
                 MU = (band width above the diagonal)
                 DO 20 I = 1, N
                    J1 = MAX0(1, I-ML)
                    J2 = MIN0(N, I+MU)
                    DO 10 J = J1, J2
                       K = J - I + ML + 1
                       ABE(I,K) = A(I,J)
              10    CONTINUE
              20 CONTINUE
         This uses columns  1  through  ML+MU+1  of ABE .
         Furthermore,  ML  additional columns are needed in
         ABE  starting with column  ML+MU+2  for elements
         generated during the triangularization.  The total
         number of columns needed in  ABE  is  2*ML+MU+1 .
   Example:  If the original matrix is
         11 12 13  0  0  0
         21 22 23 24  0  0
          0 32 33 34 35  0
          0  0 43 44 45 46
          0  0  0 54 55 56
          0  0  0  0 65 66
    then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
          * 11 12 13  +     , * = not used
         21 22 23 24  +     , + = used for pivoting
         32 33 34 35  +
         43 44 45 46  +
         54 55 56  *  +
         65 66  *  *  +
 Argument Description ***
   ABE    REAL(LDA,NC)
            on entry, contains the matrix in band storage as
              described above.  NC  must not be less than
              2*ML+MU+1 .  The user is cautioned to specify  NC
              with care since it is not an argument and cannot
              be checked by SNBFS.  The rows of the original
              matrix are stored in the rows of  ABE  and the
              diagonals of the original matrix are stored in
              columns  1  through  ML+MU+1  of  ABE .
            on return, contains an upper triangular matrix U and
              the multipliers necessary to construct a matrix L
              so that A=L*U.
   LDA    INTEGER
            the leading dimension of array ABE.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater
            than or equal to 1 .  (terminal error message IND=-2)
   ML     INTEGER
            the number of diagonals below the main diagonal.
            ML  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-5)
   MU     INTEGER
            the number of diagonals above the main diagonal.
            MU  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-6)
   V      REAL(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A and IWORK.
            If ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT. 0  See error message corresponding to IND below.
   WORK   REAL(N)
            a singly subscripted array of dimension at least N.
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   the matrix A is computationally singular.
                        A solution has not been computed.
   IND=-5  terminal   ML is less than zero or is greater than
                        or equal to N .
   IND=-6  terminal   MU is less than zero or is greater than
                        or equal to N .
   IND=-10 warning    the solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              Note-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


SNBIR


      SUBROUTINE SNBIR(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK)

   Subroutine SNBIR solves a general nonsymmetric banded NxN
   system of single precision real linear equations using
   SLATEC subroutines SNBFA and SNBSL.  These are adaptations
   of the LINPACK subroutines SGBFA and SGBSL, which require
   a different format for storing the matrix elements.
   One pass of iterative refinement is used only to obtain an
   estimate of the accuracy.  If  A  is an NxN real banded
   matrix and if  X  and  B  are real N-vectors, then SNBIR
   solves the equation
                         A*X=B.
   A band matrix is a matrix whose nonzero elements are all
   fairly near the main diagonal, specifically  A(I,J) = 0
   if  I-J is greater than  ML  or  J-I  is greater than
   MU .  The integers ML and MU are called the lower and upper
   band widths and  M = ML+MU+1  is the total band width.
   SNBIR uses less time and storage than the corresponding
   program for general matrices (SGEIR) if 2*ML+MU .LT. N .
   The matrix A is first factored into upper and lower tri-
   angular matrices U and L using partial pivoting.  These
   factors and the pivoting information are used to find the
   solution vector X .  Then the residual vector is found and used
   to calculate an estimate of the relative error, IND .  IND esti-
   mates the accuracy of the solution only when the input matrix
   and the right hand side are represented exactly in the computer
   and does not take into account any errors in the input data.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A, LDA,
   N, work and IWORK must not have been altered by the user follow-
   ing factorization (ITASK=1).  IND will not be changed by SNBIR
   in this case.
   Band Storage
         If  A  is a band matrix, the following program segment
         will set up the input.
                 ML = (band width below the diagonal)
                 MU = (band width above the diagonal)
                 DO 20 I = 1, N
                    J1 = MAX0(1, I-ML)
                    J2 = MIN0(N, I+MU)
                    DO 10 J = J1, J2
                       K = J - I + ML + 1
                       ABE(I,K) = A(I,J)
              10    CONTINUE
              20 CONTINUE
         This uses columns  1  Through  ML+MU+1  of ABE .
   Example:  If the original matrix is
         11 12 13  0  0  0
         21 22 23 24  0  0
          0 32 33 34 35  0
          0  0 43 44 45 46
          0  0  0 54 55 56
          0  0  0  0 65 66
    then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABE should contain
          * 11 12 13        , * = not used
         21 22 23 24
         32 33 34 35
         43 44 45 46
         54 55 56  *
         65 66  *  *
 Argument Description ***
   ABE    REAL(LDA,MM)
            on entry, contains the matrix in band storage as
              described above.  MM  must not be less than  M =
              ML+MU+1 .  The user is cautioned to dimension  ABE
              with care since MM is not an argument and cannot
              be checked by SNBIR.  The rows of the original
              matrix are stored in the rows of  ABE  and the
              diagonals of the original matrix are stored in
              columns  1  through  ML+MU+1  of  ABE .  ABE  is
              not altered by the program.
   LDA    INTEGER
            the leading dimension of array ABE.  LDA must be great-
            er than or equal to N.  (terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater
            than or equal to 1 .  (terminal error message IND=-2)
   ML     INTEGER
            the number of diagonals below the main diagonal.
            ML  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-5)
   MU     INTEGER
            the number of diagonals above the main diagonal.
            MU  must not be less than zero nor greater than or
            equal to  N .  (terminal error message IND=-6)
   V      REAL(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK=1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A and IWORK.
            If ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X .  IND=75 means
                    that the solution vector  X  is zero.
            LT. 0  See error message corresponding to IND below.
   WORK   REAL(N*(NC+1))
            a singly subscripted array of dimension at least
            N*(NC+1)  where  NC = 2*ML+MU+1 .
   IWORK  INTEGER(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  terminal   the matrix A is computationally singular.
                        A solution has not been computed.
   IND=-5  terminal   ML is less than zero or is greater than
                        or equal to N .
   IND=-6  terminal   MU is less than zero or is greater than
                        or equal to N .
   IND=-10 warning    the solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              Note-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


SNBSL


      SUBROUTINE SNBSL(ABE,LDA,N,ML,MU,IPVT,B,JOB)

    SNBSL solves the real band system
    A * X = B  or  TRANS(A) * X = B
    using the factors computed by SNBCO or SNBFA.
    On Entry
       ABE     REAL(LDA, NC)
               the output from SNBCO or SNBFA.
               NC must be .GE. 2*ML+MU+1 .
       LDA     INTEGER
               the leading dimension of the array  ABE .
       N       INTEGER
               the order of the original matrix.
       ML      INTEGER
               number of diagonals below the main diagonal.
       MU      INTEGER
               number of diagonals above the main diagonal.
       IPVT    INTEGER(N)
               the pivot vector from SNBCO or SNBFA.
       B       REAL(N)
               the right hand side vector.
       JOB     INTEGER
               = 0         to solve  A*X = B .
               = nonzero   to solve  TRANS(A)*X = B , where
                           TRANS(A)  is the transpose.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains a
       zero on the diagonal.  Technically, this indicates singularity,
       but it is often caused by improper arguments or improper
       setting of LDA.  It will not occur if the subroutines are
       called correctly and if SNBCO has set RCOND .GT. 0.0
       or SNBFA has set INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL SNBCO(ABE,LDA,N,ML,MU,IPVT,RCOND,Z)
          IF (RCOND is too small) GO TO ...
          DO 10 J = 1, P
            CALL SNBSL(ABE,LDA,N,ML,MU,IPVT,C(1,J),0)
       10 CONTINUE
    SLATEC.  This version dated 07/17/80 .
    E. A. Voorhees, Los Alamos Scientific Laboratory
    Subroutines and Functions
     SAXPY,SDOT
    Fortran  MIN0


SNRM2


      REAL FUNCTION SNRM2(N,SX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
    --Output--
   SNRM2  single precision result (zero if N .LE. 0)
    Euclidean norm of the N-vector stored in SX() with storage
    increment INCX .
    If N .LE. 0, return with result = 0.
    If N .GE. 1, then INCX must be .GE. 1
          C. L. Lawson, 1978 Jan 08
    Four Phase Method     using two built-in constants that are
    hopefully applicable to all machines.
        CUTLO = maximum of  SQRT(U/EPS)  over all known machines.
        CUTHI = minimum of  SQRT(V)      over all known machines.
    where
        EPS = smallest no. such that EPS + 1. .GT. 1.
        U   = smallest positive no.   (underflow limit)
        V   = largest  no.            (overflow  limit)
    Brief Outline of Algorithm..
    Phase 1 scans zero components.
    Move to phase 2 when a component is nonzero and .LE. CUTLO
    Move to phase 3 when a component is .GT. CUTLO
    Move to phase 4 when a component is .GE. CUTHI/M
    where M = N for X() real and M = 2*N for complex.
    Values for CUTLO and CUTHI..
    From the environmental parameters listed in the IMSL converter
    document the limiting values are as follows..
    CUTLO, S.P.   U/EPS = 2**(-102) for  Honeywell.  Close seconds are
                  Univac and DEC at 2**(-103)
                  Thus CUTLO = 2**(-51) = 4.44089E-16
    CUTHI, S.P.   V = 2**127 for Univac, Honeywell, and DEC.
                  Thus CUTHI = 2**(63.5) = 1.30438E19
    CUTLO, D.P.   U/EPS = 2**(-67) for Honeywell and DEC.
                  Thus CUTLO = 2**(-33.5) = 8.23181D-11
    CUTHI, D.P.   same as S.P.  CUTHI = 1.30438D19
    SAVE CUTLO, CUTHI, ZERO, ONE
    DATA CUTLO, CUTHI / 8.232D-11,  1.304D19 /
    DATA CUTLO, CUTHI / 4.441E-16,  1.304E19 /


SPBCO


      SUBROUTINE SPBCO(ABD,LDA,N,M,RCOND,Z,INFO)

    SPBCO factors a real symmetric positive definite matrix
    stored in band form and estimates the condition of the matrix.
    If  RCOND  is not needed, SPBFA is slightly faster.
    To solve  A*X = B , follow SPBCO by SPBSL.
    To compute  INVERSE(A)*C , follow SPBCO by SPBSL.
    To compute  DETERMINANT(A) , follow SPBCO by SPBDI.
    On Entry
       ABD     REAL(LDA, N)
               the matrix to be factored.  The columns of the upper
               triangle are stored in the columns of ABD and the
               diagonals of the upper triangle are stored in the
               rows of ABD .  See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. M + 1 .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
               0 .LE. M .LT. N .
    On Return
       ABD     an upper triangular matrix  R , stored in band
               form, so that  A = TRANS(R)*R .
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  A  is singular to working precision, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    Band Storage
          If  A  is a symmetric positive definite band matrix,
          the following program segment will set up the input.
                  M = (band width above diagonal)
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-M)
                     DO 10 I = I1, J
                        K = I-J+M+1
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
          This uses  M + 1  rows of  A , except for the  M by M
          upper left triangle, which is ignored.
    Example:  If the original matrix is
          11 12 13  0  0  0
          12 22 23 24  0  0
          13 23 33 34 35  0
           0 24 34 44 45 46
           0  0 35 45 55 56
           0  0  0 46 56 66
    then  N = 6 , M = 2  and  ABD  should contain
           *  * 13 24 35 46
           * 12 23 34 45 56
          11 22 33 44 55 66
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK SPBFA
    BLAS SAXPY,SDOT,SSCAL,SASUM
    Fortran ABS,AMAX1,MAX0,MIN0,REAL,SIGN


SPBDI


      SUBROUTINE SPBDI(ABD,LDA,N,M,DET)

    SPBDI computes the determinant
    of a real symmetric positive definite band matrix
    using the factors computed by SPBCO or SPBFA.
    If the inverse is needed, use SPBSL  N  times.
    On Entry
       ABD     REAL(LDA, N)
               the output from SPBCO or SPBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
    On Return
       DET     REAL(2)
               determinant of original matrix in the form
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.


SPBFA


      SUBROUTINE SPBFA(ABD,LDA,N,M,INFO)

    SPBFA factors a real symmetric positive definite matrix
    stored in band form.
    SPBFA is usually called by SPBCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    On Entry
       ABD     REAL(LDA, N)
               the matrix to be factored.  The columns of the upper
               triangle are stored in the columns of ABD and the
               diagonals of the upper triangle are stored in the
               rows of ABD .  See the comments below for details.
       LDA     INTEGER
               the leading dimension of the array  ABD .
               LDA must be .GE. M + 1 .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
               0 .LE. M .LT. N .
    On Return
       ABD     an upper triangular matrix  R , stored in band
               form, so that  A = TRANS(R)*R .
       INFO    INTEGER
               = 0  for normal return.
               = K  if the leading minor of order  K  is not
                    positive definite.
    Band Storage
          If  A  is a symmetric positive definite band matrix,
          the following program segment will set up the input.
                  M = (band width above diagonal)
                  DO 20 J = 1, N
                     I1 = MAX0(1, J-M)
                     DO 10 I = I1, J
                        K = I-J+M+1
                        ABD(K,J) = A(I,J)
               10    CONTINUE
               20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SDOT
    Fortran MAX0,SQRT


SPBSL


      SUBROUTINE SPBSL(ABD,LDA,N,M,B)

    SPBSL solves the real symmetric positive definite band
    system  A*X = B
    using the factors computed by SPBCO or SPBFA.
    On Entry
       ABD     REAL(LDA, N)
               the output from SPBCO or SPBFA.
       LDA     INTEGER
               the leading dimension of the array  ABD .
       N       INTEGER
               the order of the matrix  A .
       M       INTEGER
               the number of diagonals above the main diagonal.
       B       REAL(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically, this indicates
       singularity, but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL SPBCO(ABD,LDA,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL SPBSL(ABD,LDA,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SDOT
    Fortran MIN0


SPELI4


      SUBROUTINE SPELI4(IORDER,A,B,M,MBDCND,BDA,ALPHA,BDB,BETA,C,D,N,



SPELIP


      SUBROUTINE SPELIP(INTL,IORDER,A,B,M,MBDCND,BDA,ALPHA,BDB,BETA,C,



SPOCO


      SUBROUTINE SPOCO(A,LDA,N,RCOND,Z,INFO)

    SPOCO factors a real symmetric positive definite matrix
    and estimates the condition of the matrix.
    If  RCOND  is not needed, SPOFA is slightly faster.
    To solve  A*X = B , follow SPOCO by SPOSL.
    To compute  INVERSE(A)*C , follow SPOCO by SPOSL.
    To compute  DETERMINANT(A) , follow SPOCO by SPODI.
    To compute  INVERSE(A) , follow SPOCO by SPODI.
    On Entry
       A       REAL(LDA, N)
               the symmetric matrix to be factored.  Only the
               diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix  R  so that  A = TRANS(R)*R
               where  TRANS(R)  is the transpose.
               The strict lower triangle is unaltered.
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK SPOFA
    BLAS SAXPY,SDOT,SSCAL,SASUM
    Fortran ABS,AMAX1,REAL,SIGN


SPODI


      SUBROUTINE SPODI(A,LDA,N,DET,JOB)

    SPODI computes the determinant and inverse of a certain
    real symmetric positive definite matrix (see below)
    using the factors computed by SPOCO, SPOFA or SQRDC.
    On Entry
       A       REAL(LDA, N)
               the output  A  from SPOCO or SPOFA
               or the output  X  from SQRDC.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       A       If SPOCO or SPOFA was used to factor  A , then
               SPODI produces the upper half of INVERSE(A) .
               If SQRDC was used to decompose  X , then
               SPODI produces the upper half of INVERSE(TRANS(X)*X),
               where TRANS(X) is the transpose.
               Elements of  A  below the diagonal are unchanged.
               If the units digit of JOB is zero,  A  is unchanged.
       DET     REAL(2)
               determinant of  A  or of  TRANS(X)*X  if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if SPOCO or SPOFA has set INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SSCAL
    Fortran MOD


SPOFA


      SUBROUTINE SPOFA(A,LDA,N,INFO)

    SPOFA factors a real symmetric positive definite matrix.
    SPOFA is usually called by SPOCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (Time for SPOCO) = (1 + 18/N)*(Time for SPOFA) .
    On Entry
       A       REAL(LDA, N)
               the symmetric matrix to be factored.  Only the
               diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       an upper triangular matrix  R  so that  A = TRANS(R)*R
               where  TRANS(R)  is the transpose.
               The strict lower triangle is unaltered.
               If  INFO .NE. 0 , the factorization is not complete.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SDOT
    Fortran SQRT


SPOFS


      SUBROUTINE SPOFS(A,LDA,N,V,ITASK,IND,WORK)

   Subroutine SPOFS solves a real positive definite symmetric
   NxN system of single precision linear equations using
   LINPACK subroutines SPOCO and SPOSL.  That is, if A is an
   NxN real positive definite symmetric matrix and if X and B
   are real N-vectors, then SPOFS solves the equation
                         A*X=B.
   The matrix A is first factored into upper and lower tri-
   angular matrices R and R-TRANSPOSE.  These factors are used to
   find the solution vector X.  An approximate condition number is
   calculated to provide a rough estimate of the number of
   digits of accuracy in the computed solution.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to solve only (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, and N must not have been altered by the user following
   factorization (ITASK=1).  IND will not be changed by SPOFS
   in this case.
 Argument Description ***
   A      REAL(LDA,N)
            on entry, the doubly subscripted array with dimension
              (LDA,N) which contains the coefficient matrix.  Only
              the upper triangle, including the diagonal, of the
              coefficient matrix need be entered and will subse-
              quently be referenced and changed by the routine.
            on return, contains in its upper triangle an upper
              triangular matrix R such that  A = (R-TRANSPOSE) * R .
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (Terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater
            than or equal to 1.  (Terminal error message IND=-2)
   V      REAL(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK = 1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A.
            If ITASK .LT. 1, then terminal error message IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.
            LT. 0  see error message corresponding to IND below.
   WORK   REAL(N)
            a singly subscripted array of dimension at least N.
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than 1.
   IND=-3  terminal   ITASK is less than 1.
   IND=-4  Terminal   The matrix A is computationally singular or
                        is not positive definite.  A solution
                        has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the
                        matrix A may be poorly scaled.
              Note-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


SPOIR


      SUBROUTINE SPOIR(A,LDA,N,V,ITASK,IND,WORK)

   Subroutine SPOIR solves a real positive definite symmetric
   NxN system of single precision linear equations using LINPACK
   subroutines SPOFA and SPOSL.  One pass of iterative refine-
   ment is used only to obtain an estimate of the accuracy.  That
   is, if A is an NxN real positive definite symmetric matrix
   and if X and B are real N-vectors, then SPOIR solves the
   equation
                         A*X=B.
   The matrix A is first factored into upper and lower
   triangular matrices R and R-TRANSPOSE.  These
   factors are used to calculate the solution, X.
   Then the residual vector is found and used
   to calculate an estimate of the relative error, IND.
   IND estimates the accuracy of the solution only when the
   input matrix and the right hand side are represented
   exactly in the computer and does not take into account
   any errors in the input data.
   If the equation A*X=B is to be solved for more than one vector
   B, the factoring of A does not need to be performed again and
   the option to only solve (ITASK .GT. 1) will be faster for
   the succeeding solutions.  In this case, the contents of A,
   LDA, N, and WORK must not have been altered by the user
   following factorization (ITASK=1).  IND will not be changed
   by SPOIR in this case.
 Argument Description ***
   A      REAL(LDA,N)
            the doubly subscripted array with dimension (LDA,N)
            which contains the coefficient matrix.  Only the
            upper triangle, including the diagonal, of the
            coefficient matrix need be entered.  A is not
            altered by the routine.
   LDA    INTEGER
            the leading dimension of the array A.  LDA must be great-
            er than or equal to N.  (Terminal error message IND=-1)
   N      INTEGER
            the order of the matrix A.  N must be greater than
            or equal to one.  (Terminal error message IND=-2)
   V      REAL(N)
            on entry, the singly subscripted array(vector) of di-
              mension N which contains the right hand side B of a
              system of simultaneous linear equations A*X=B.
            on return, V contains the solution vector, X .
   ITASK  INTEGER
            If ITASK = 1, the matrix A is factored and then the
              linear equation is solved.
            If ITASK .GT. 1, the equation is solved using the existing
              factored matrix A (stored in WORK).
            If ITASK .LT. 1, then terminal terminal error IND=-3 is
              printed.
   IND    INTEGER
            GT. 0  IND is a rough estimate of the number of digits
                    of accuracy in the solution, X.  IND=75 means
                    that the solution vector X is zero.
            LT. 0  See error message corresponding to IND below.
   WORK   REAL(N*(N+1))
            a singly subscripted array of dimension at least N*(N+1).
 Error Messages Printed ***
   IND=-1  terminal   N is greater than LDA.
   IND=-2  terminal   N is less than one.
   IND=-3  terminal   ITASK is less than one.
   IND=-4  Terminal   The matrix A is computationally singular
                        or is not positive definite.
                        A solution has not been computed.
   IND=-10 warning    The solution has no apparent significance.
                        The solution may be inaccurate or the matrix
                        A may be poorly scaled.
              Note-  The above terminal(*fatal*) error messages are
                     designed to be handled by XERRWV in which
                     LEVEL=1 (recoverable) and IFLAG=2 .  LEVEL=0
                     for warning error messages from XERROR.  Unless
                     the user provides otherwise, an error message
                     will be printed followed by an abort.


SPOPT


      SUBROUTINE SPOPT(PRGOPT,MRELAS,NVARS,INFO,CSC,IBASIS,ROPT,INTOPT,

    THE EDITING REQUIRED TO CONVERT THIS SUBROUTINE FROM SINGLE TO
    DOUBLE PRECISION INVOLVES THE FOLLOWING CHARACTER STRING CHANGES.
    USE AN EDITING COMMAND (CHANGE) /STRING-1/(TO)STRING-2/.
    /REAL (12 BLANKS)/DOUBLE PRECISION/,/ABS(/DABS(/,/AMAX1/
    DMAX1/,/SQRT/DSQRT/,/R1MACH/D1MACH/,/AMIN1/DMIN1/,/E0/D0/
    REVISED 821122-1045
    REVISED YYMMDD-HHMM
    THIS SUBROUTINE PROCESSES THE OPTION VECTOR, PRGOPT(*),
    AND VALIDATES ANY MODIFIED DATA.


SPOSL


      SUBROUTINE SPOSL(A,LDA,N,B)

    SPOSL solves the real symmetric positive definite system
    A * X = B
    using the factors computed by SPOCO or SPOFA.
    On Entry
       A       REAL(LDA, N)
               the output from SPOCO or SPOFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       B       REAL(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically, this indicates
       singularity, but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL SPOCO(A,LDA,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL SPOSL(A,LDA,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SDOT


SPPCO


      SUBROUTINE SPPCO(AP,N,RCOND,Z,INFO)

    SPPCO factors a real symmetric positive definite matrix
    stored in packed form
    and estimates the condition of the matrix.
    If  RCOND  is not needed, SPPFA is slightly faster.
    To solve  A*X = B , follow SPPCO by SPPSL.
    To compute  INVERSE(A)*C , follow SPPCO by SPPSL.
    To compute  DETERMINANT(A) , follow SPPCO by SPPDI.
    To compute  INVERSE(A) , follow SPPCO by SPPDI.
    On Entry
       AP      REAL (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    On Return
       AP      an upper triangular matrix  R , stored in packed
               form, so that  A = TRANS(R)*R .
               If  INFO .NE. 0 , the factorization is not complete.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.  If INFO .NE. 0 , RCOND is unchanged.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  A  is singular to working precision, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
               If  INFO .NE. 0 , Z  is unchanged.
       INFO    INTEGER
               = 0  for normal return.
               = K  signals an error condition.  The leading minor
                    of order  K  is not positive definite.
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK SPPFA
    BLAS SAXPY,SDOT,SSCAL,SASUM
    Fortran ABS,AMAX1,REAL,SIGN


SPPDI


      SUBROUTINE SPPDI(AP,N,DET,JOB)

    SPPDI computes the determinant and inverse
    of a real symmetric positive definite matrix
    using the factors computed by SPPCO or SPPFA .
    On Entry
       AP      REAL (N*(N+1)/2)
               the output from SPPCO or SPPFA.
       N       INTEGER
               the order of the matrix  A .
       JOB     INTEGER
               = 11   both determinant and inverse.
               = 01   inverse only.
               = 10   determinant only.
    On Return
       AP      the upper triangular half of the inverse .
               The strict lower triangle is unaltered.
       DET     REAL(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. DET(1) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal and the inverse is requested.
       It will not occur if the subroutines are called correctly
       and if SPOCO or SPOFA has set INFO .EQ. 0 .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SSCAL
    Fortran MOD


SPPFA


      SUBROUTINE SPPFA(AP,N,INFO)

    SPPFA factors a real symmetric positive definite matrix
    stored in packed form.
    SPPFA is usually called by SPPCO, but it can be called
    directly with a saving in time if  RCOND  is not needed.
    (Time for SPPCO) = (1 + 18/N)*(Time for SPPFA) .
    On Entry
       AP      REAL (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    On Return
       AP      an upper triangular matrix  R , stored in packed
               form, so that  A = TRANS(R)*R .
       INFO    INTEGER
               = 0  for normal return.
               = K  if the leading minor of order  K  is not
                    positive definite.
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SDOT
    Fortran SQRT


SPPSL


      SUBROUTINE SPPSL(AP,N,B)

    SPPSL solves the real symmetric positive definite system
    A * X = B
    using the factors computed by SPPCO or SPPFA.
    On Entry
       AP      REAL (N*(N+1)/2)
               the output from SPPCO or SPPFA.
       N       INTEGER
               the order of the matrix  A .
       B       REAL(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero will occur if the input factor contains
       a zero on the diagonal.  Technically, this indicates
       singularity, but it is usually caused by improper subroutine
       arguments.  It will not occur if the subroutines are called
       correctly and  INFO .EQ. 0 .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL SPPCO(AP,N,RCOND,Z,INFO)
          IF (RCOND is too small .OR. INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL SPPSL(AP,N,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SDOT


SPTSL


      SUBROUTINE SPTSL(N,D,E,B)

    SPTSL given a positive definite tridiagonal matrix and a right
    hand side will find the solution.
    On Entry
       N        INTEGER
                is the order of the tridiagonal matrix.
       D        REAL(N)
                is the diagonal of the tridiagonal matrix.
                On output, D is destroyed.
       E        REAL(N)
                is the offdiagonal of the tridiagonal matrix.
                E(1) through E(N-1) should contain the
                offdiagonal.
       B        REAL(N)
                is the right hand side vector.
    On Return
       B        contains the solution.
    LINPACK.  This version dated 08/14/78 .
    Jack Dongarra, Argonne National Laboratory.
    No externals
    Fortran MOD


SQRDC


      SUBROUTINE SQRDC(X,LDX,N,P,QRAUX,JPVT,WORK,JOB)

    SQRDC uses Householder transformations to compute the QR
    factorization of an N by P matrix X.  Column pivoting
    based on the 2-norms of the reduced columns may be
    performed at the user's option.
    On Entry
       X       REAL(LDX,P), where LDX .GE. N.
               X contains the matrix whose decomposition is to be
               computed.
       LDX     INTEGER.
               LDX is the leading dimension of the array X.
       N       INTEGER.
               N is the number of rows of the matrix X.
       P       INTEGER.
               P is the number of columns of the matrix X.
       JPVT    INTEGER(P).
               JPVT contains integers that control the selection
               of the pivot columns.  The K-th column X(K) of X
               is placed in one of three classes according to the
               value of JPVT(K).
                  If JPVT(K) .GT. 0, then X(K) is an initial
                                     column.
                  If JPVT(K) .EQ. 0, then X(K) is a free column.
                  If JPVT(K) .LT. 0, then X(K) is a final column.
               Before the decomposition is computed, initial columns
               are moved to the beginning of the array X and final
               columns to the end.  Both initial and final columns
               are frozen in place during the computation and only
               free columns are moved.  At the K-th stage of the
               reduction, if X(K) is occupied by a free column,
               it is interchanged with the free column of largest
               reduced norm.  JPVT is not referenced if
               JOB .EQ. 0.
       WORK    REAL(P).
               WORK is a work array.  WORK is not referenced if
               JOB .EQ. 0.
       JOB     INTEGER.
               JOB is an integer that initiates column pivoting.
               If JOB .EQ. 0, no pivoting is done.
               If JOB .NE. 0, pivoting is done.
    On Return
       X       X contains in its upper triangle the upper
               triangular matrix R of the QR factorization.
               Below its diagonal X contains information from
               which the orthogonal part of the decomposition
               can be recovered.  Note that if pivoting has
               been requested, the decomposition is not that
               of the original matrix X but that of X
               with its columns permuted as described by JPVT.
       QRAUX   REAL(P).
               QRAUX contains further information required to recover
               the orthogonal part of the decomposition.
       JPVT    JPVT(K) contains the index of the column of the
               original matrix that has been interchanged into
               the K-th column, if pivoting was requested.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    SQRDC uses the following functions and subprograms.
    BLAS SAXPY,SDOT,SSCAL,SSWAP,SNRM2
    Fortran ABS,AMAX1,MIN0,SQRT


SQRSL


      SUBROUTINE SQRSL(X,LDX,N,K,QRAUX,Y,QY,QTY,B,RSD,XB,JOB,INFO)

    SQRSL applies the output of SQRDC to compute coordinate
    transformations, projections, and least squares solutions.
    For K .LE. MIN(N,P), let XK be the matrix
           XK = (X(JPVT(1)),X(JPVT(2)), ... ,X(JPVT(K)))
    formed from columnns JPVT(1), ... ,JPVT(K) of the original
    N x P matrix X that was input to SQRDC (if no pivoting was
    done, XK consists of the first K columns of X in their
    original order).  SQRDC produces a factored orthogonal matrix Q
    and an upper triangular matrix R such that
             XK = Q * (R)
                      (0)
    This information is contained in coded form in the arrays
    X and QRAUX.
    On Entry
       X      REAL(LDX,P)
              X contains the output of SQRDC.
       LDX    INTEGER
              LDX is the leading dimension of the array X.
       N      INTEGER
              N is the number of rows of the matrix XK.  It must
              have the same value as N in SQRDC.
       K      INTEGER
              K is the number of columns of the matrix XK.  K
              must not be greater than MIN(N,P), where P is the
              same as in the calling sequence to SQRDC.
       QRAUX  REAL(P)
              QRAUX contains the auxiliary output from SQRDC.
       Y      REAL(N)
              Y contains an N-vector that is to be manipulated
              by SQRSL.
       JOB    INTEGER
              JOB specifies what is to be computed.  JOB has
              the decimal expansion ABCDE, with the following
              meaning.
                   If A .NE. 0, compute QY.
                   If B,C,D, or E .NE. 0, compute QTY.
                   If C .NE. 0, compute B.
                   If D .NE. 0, compute RSD.
                   If E .NE. 0, compute XB.
              Note that a request to compute B, RSD, or XB
              automatically triggers the computation of QTY, for
              which an array must be provided in the calling
              sequence.
    On Return
       QY     REAL(N).
              QY contains Q*Y, if its computation has been
              requested.
       QTY    REAL(N).
              QTY contains TRANS(Q)*Y, if its computation has
              been requested.  Here TRANS(Q) is the
              transpose of the matrix Q.
       B      REAL(K)
              B contains the solution of the least squares problem
                   minimize norm2(Y - XK*B),
              if its computation has been requested.  (Note that
              if pivoting was requested in SQRDC, the J-th
              component of B will be associated with column JPVT(J)
              of the original matrix X that was input into SQRDC.)
       RSD    REAL(N).
              RSD contains the least squares residual Y - XK*B,
              if its computation has been requested.  RSD is
              also the orthogonal projection of Y onto the
              orthogonal complement of the column space of XK.
       XB     REAL(N).
              XB contains the least squares approximation XK*B,
              if its computation has been requested.  XB is also
              the orthogonal projection of Y onto the column space
              of X.
       INFO   INTEGER.
              INFO is zero unless the computation of B has
              been requested and R is exactly singular.  In
              this case, INFO is the index of the first zero
              diagonal element of R and B is left unaltered.
    The parameters QY, QTY, B, RSD, and XB are not referenced
    if their computation is not requested and in this case
    can be replaced by dummy variables in the calling program.
    To save storage, the user may in some cases use the same
    array for different parameters in the calling sequence.  A
    frequently occuring example is when one wishes to compute
    any of B, RSD, or XB and does not need Y or QTY.  In this
    case one may identify Y, QTY, and one of B, RSD, or XB, while
    providing separate arrays for anything else that is to be
    computed.  Thus the calling sequence
         CALL SQRSL(X,LDX,N,K,QRAUX,Y,DUM,Y,B,Y,DUM,110,INFO)
    will result in the computation of B and RSD, with RSD
    overwriting Y.  More generally, each item in the following
    list contains groups of permissible identifications for
    a single callinng sequence.
         1. (Y,QTY,B) (RSD) (XB) (QY)
         2. (Y,QTY,RSD) (B) (XB) (QY)
         3. (Y,QTY,XB) (B) (RSD) (QY)
         4. (Y,QY) (QTY,B) (RSD) (XB)
         5. (Y,QY) (QTY,RSD) (B) (XB)
         6. (Y,QY) (QTY,XB) (B) (RSD)
    In any group the value returned in the array allocated to
    the group corresponds to the last member of the group.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    SQRSL uses the following functions and subprograms.
    BLAS SAXPY,SCOPY,SDOT
    Fortran ABS,MIN0,MOD


SREADP


      SUBROUTINE SREADP(IPAGE,LIST,RLIST,LPAGE,IREC)

    READ RECORD NUMBER IRECN, OF LENGTH LPG, FROM UNIT
    NUMBER IPAGEF INTO THE STORAGE ARRAY, LIST(*).
    READ RECORD  IRECN+1, OF LENGTH LPG, FROM UNIT NUMBER
    IPAGEF INTO THE STORAGE ARRAY RLIST(*).
    TO CONVERT THIS PROGRAM UNIT TO DOUBLE PRECISION CHANGE
    /REAL (12 BLANKS)/ TO /DOUBLE PRECISION/.


SROT


      SUBROUTINE SROT(N,SX,INCX,SY,INCY,SC,SS)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
      SY  single precision vector with N elements
    INCY  storage spacing between elements of SY
      SC  element of rotation matrix
      SS  element of rotation matrix
    --Output--
      SX  rotated vector SX (unchanged if N .LE. 0)
      SY  rotated vector SY (unchanged if N .LE. 0)
    Multiply the 2 x 2 matrix  ( SC SS) times the 2 x N matrix (SX**T)
                               (-SS SC)                        (SY**T)
    where **T indicates transpose.  The elements of SX are in
    SX(LX+I*INCX), I = 0 to N-1, where LX = 1 if INCX .GE. 0, else
    LX = (-INCX)*N, and similarly for SY using LY and INCY.


SROTG


      SUBROUTINE SROTG(SA,SB,SC,SS)

               B L A S  Subprogram
   Description of Parameters
    --Input--
      SA  single precision scalar
      SB  single precision scalar
    --Output--
      SA  single precision result R
      SB  single precision result Z
      SC  single precision result
      SS  single precision result
    Designed by C. L. Lawson, JPL, 1977 Sept 08
    Construct the Givens transformation
        ( SC  SS )
    G = (        ) ,    SC**2 + SS**2 = 1 ,
        (-SS  SC )
    which zeros the second entry of the 2-vector  (SA,SB)**T.
    The quantity R = (+/-)SQRT(SA**2 + SB**2) overwrites SA in
    storage.  The value of SB is overwritten by a value Z which
    allows SC and SS to be recovered by the following algorithm:
          If Z=1  set  SC=0.  and  SS=1.
          If ABS(Z) .LT. 1  set  SC=SQRT(1-Z**2)  and  SS=Z
          If ABS(Z) .GT. 1  set  SC=1/Z  and  SS=SQRT(1-SC**2)
    Normally, the subprogram SROT(N,SX,INCX,SY,INCY,SC,SS) will
    next be called to apply the transformation to a 2 by N matrix.


SROTM


      SUBROUTINE SROTM(N,SX,INCX,SY,INCY,SPARAM)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
      SY  single precision vector with N elements
    INCY  storage spacing between elements of SY
  SPARAM  5-element vector. SPARAM(1) is SFLAG described below.
            Locations 2-5 of SPARAM contain elements of the
             transformation matrix H described below.
    --Output--
      SX  rotated vector (unchanged if N .LE. 0)
      SY  rotated vector (unchanged if N .LE. 0)
    Apply the modified Givens transformation, H, to the 2 by N matrix
    (SX**T) , where **T indicates transpose.  The elements of SX are
    in (SY**T)
    SX(LX+I*INCX), I = 0 to N-1, where LX = 1 if INCX .GE. 0, else
    LX = (-INCX)*N, and similarly for SY using using LY and INCY.
    With SPARAM(1)=SFLAG, H has one of the following forms..
    SFLAG=-1.E0     SFLAG=0.E0        SFLAG=1.E0     SFLAG=-2.E0
      (SH11  SH12)    (1.E0  SH12)    (SH11  1.E0)    (1.E0  0.E0)
    H=(          )    (          )    (          )    (          )
      (SH21  SH22),   (SH21  1.E0),   (-1.E0 SH22),   (0.E0  1.E0).
    See SROTMG for a description of data storage in SPARAM.


SROTMG


      SUBROUTINE SROTMG(SD1,SD2,SX1,SY1,SPARAM)

               B L A S  Subprogram
   Description of Parameters
    --Input--
     SD1  single precision scalar used to define A1 below
     SD2  single precision scalar used to define A2 below
     SB1  single precision scalar defining A1 below
     SB2  single precision scalar defining A2 below
  SPARAM  S.P. 5-vector. SPARAM(1)=SFLAG defined below.
          Locations 2-5 contain the rotation matrix.
    --Output--
    SD1  changed to represent the effect of the transformation
    SD2  changed to represent the effect of the transformation
    SB1  changed to represent the effect of the transformation
    SB2  unchanged
    Construct the modified Givens transformation matrix H which zeros
    the second component of the 2-vector  (SQRT(SD1)*SX1,SQRT(SD2)*
    SY2)**T.
    With SPARAM(1)=SFLAG, H has one of the following forms..
    SFLAG=-1.E0     SFLAG=0.E0        SFLAG=1.E0     SFLAG=-2.E0
      (SH11  SH12)    (1.E0  SH12)    (SH11  1.E0)    (1.E0  0.E0)
    H=(          )    (          )    (          )    (          )
      (SH21  SH22),   (SH21  1.E0),   (-1.E0 SH22),   (0.E0  1.E0).
    Locations 2-5 of SPARAM contain SH11,SH21,SH12, and SH22
    respectively.  (Values of 1.E0, -1.E0, or 0.E0 implied by the
    value of SPARAM(1) are not stored in SPARAM.)


SSCAL


      SUBROUTINE SSCAL(N,SA,SX,INCX)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SA  single precision scale factor
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
    --Output--
      SX  single precision result (unchanged if N .LE. 0)
    Replace single precision SX by single precision SA*SX.
    For I = 0 to N-1, replace SX(1+I*INCX) with  SA * SX(1+I*INCX)


SSICO


      SUBROUTINE SSICO(A,LDA,N,KPVT,RCOND,Z)

    SSICO factors a real symmetric matrix by elimination with
    symmetric pivoting and estimates the condition of the matrix.
    If  RCOND  is not needed, SSIFA is slightly faster.
    To solve  A*X = B , follow SSICO by SSISL.
    To compute  INVERSE(A)*C , follow SSICO by SSISL.
    To compute  INVERSE(A) , follow SSICO by SSIDI.
    To compute  DETERMINANT(A) , follow SSICO by SSIDI.
    To compute  INERTIA(A), follow SSICO by SSIDI.
    On Entry
       A       REAL(LDA, N)
               the symmetric matrix to be factored.
               Only the diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    Output
       A       a block diagonal matrix and the multipliers which
               were used to obtain it.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK SSIFA
    BLAS SAXPY,SDOT,SSCAL,SASUM
    Fortran ABS,AMAX1,IABS,SIGN


SSIDI


      SUBROUTINE SSIDI(A,LDA,N,KPVT,DET,INERT,WORK,JOB)

    SSIDI computes the determinant, inertia and inverse
    of a real symmetric matrix using the factors from SSIFA.
    On Entry
       A       REAL(LDA,N)
               the output from SSIFA.
       LDA     INTEGER
               the leading dimension of the array A.
       N       INTEGER
               the order of the matrix A.
       KPVT    INTEGER(N)
               the pivot vector from SSIFA.
       WORK    REAL(N)
               work vector.  Contents destroyed.
       JOB     INTEGER
               JOB has the decimal expansion  ABC  where
                  If  C .NE. 0, the inverse is computed,
                  If  B .NE. 0, the determinant is computed,
                  If  A .NE. 0, the inertia is computed.
               For example, JOB = 111  gives all three.
    On Return
       Variables not requested by JOB are not used.
       A      contains the upper triangle of the inverse of
              the original matrix.  The strict lower triangle
              is never referenced.
       DET    REAL(2)
              determinant of original matrix.
              Determinant = DET(1) * 10.0**DET(2)
              with 1.0 .LE. ABS(DET(1)) .LT. 10.0
              or DET(1) = 0.0.
       INERT  INTEGER(3)
              the inertia of the original matrix.
              INERT(1)  =  number of positive eigenvalues.
              INERT(2)  =  number of negative eigenvalues.
              INERT(3)  =  number of zero eigenvalues.
    Error Condition
       A division by zero may occur if the inverse is requested
       and  SSICO  has set RCOND .EQ. 0.0
       or  SSIFA  has set  INFO .NE. 0 .
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab
    Subroutines and Functions
    BLAS SAXPY,SCOPY,SDOT,SSWAP
    Fortran ABS,IABS,MOD


SSIEV


      SUBROUTINE SSIEV(A,LDA,N,E,WORK,JOB,INFO)

    LICEPACK.  This version dated 08/08/80.
    David Kahaner, Cleve Moler, Pete Stewart
         N.B.S.       U.N.M.     N.B.S./U.MD.
    Abstract
     SSIEV computes the eigenvalues and, optionally, the eigenvectors
     of a real symmetric matrix.
    Call Sequence Parameters-
      (The values of parameters marked with * (star) will be  changed
        by SSIEV.)
      A*      REAL (LDA,N)
              real symmetric input matrix.
              Only the diagonal and upper triangle of A must be input,
              as SSIEV copies the upper triangle to the lower.
              That is, the user must define A(I,J), I=1,..N, and J=I,.
              ..,N.
              On return from SSIEV, if the user has set JOB
              = 0        the lower triangle of A has been altered.
              = nonzero  the N eigenvectors of A are stored in its
              first N columns.  See also INFO below.
      LDA     INTEGER
              set by the user to
              the leading dimension of the array A.
      N       INTEGER
              set by the user to
              the order of the matrix A and
              the number of elements in E.
      E*      REAL (N)
              on return from SSIEV, E contains the N
              eigenvalues of A.  See also INFO below.
      WORK*   REAL (2*N)
              temporary storage vector.  Contents changed by SSIEV.
      JOB     INTEGER
              set by user on input
              = 0         only calculate eigenvalues of A.
              = nonzero   calculate eigenvalues and eigenvectors of A.
      INFO*   INTEGER
              on return from SSIEV, the value of INFO is
              = 0 for normal return.
              = K if the eigenvalue iteration fails to converge.
                  eigenvalues and vectors 1 through K-1 are correct.
    Error Messages-
         No. 1   recoverable  N is greater than LDA
         No. 2   recoverable  N is less than one
    Subroutines Used
    EISPACK- TRED1, TRED2, TQLRAT, IMTQL2
    SLATEC-  XERROR


SSIFA


      SUBROUTINE SSIFA(A,LDA,N,KPVT,INFO)

    SSIFA factors a real symmetric matrix by elimination
    with symmetric pivoting.
    To solve  A*X = B , follow SSIFA by SSISL.
    To compute  INVERSE(A)*C , follow SSIFA by SSISL.
    To compute  DETERMINANT(A) , follow SSIFA by SSIDI.
    To compute  INERTIA(A) , follow SSIFA by SSIDI.
    To compute  INVERSE(A) , follow SSIFA by SSIDI.
    On Entry
       A       REAL(LDA,N)
               the symmetric matrix to be factored.
               Only the diagonal and upper triangle are used.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
    On Return
       A       a block diagonal matrix and the multipliers which
               were used to obtain it.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if the K-th pivot block is singular.  This is
                    not an error condition for this subroutine,
                    but it does indicate that SSISL or SSIDI may
                    divide by zero if called.
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS SAXPY,SSWAP,ISAMAX
    Fortran ABS,AMAX1,SQRT


SSISL


      SUBROUTINE SSISL(A,LDA,N,KPVT,B)

    SSISL solves the real symmetric system
    A * X = B
    using the factors computed by SSIFA.
    On Entry
       A       REAL(LDA,N)
               the output from SSIFA.
       LDA     INTEGER
               the leading dimension of the array  A .
       N       INTEGER
               the order of the matrix  A .
       KPVT    INTEGER(N)
               the pivot vector from SSIFA.
       B       REAL(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero may occur if  SSICO  has set RCOND .EQ. 0.0
       or  SSIFA  has set INFO .NE. 0  .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL SSIFA(A,LDA,N,KPVT,INFO)
          IF (INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL SSISL(A,LDA,N,KPVT,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS SAXPY,SDOT
    Fortran IABS


SSPCO


      SUBROUTINE SSPCO(AP,N,KPVT,RCOND,Z)

    SSPCO factors a real symmetric matrix stored in packed
    form by elimination with symmetric pivoting and estimates
    the condition of the matrix.
    If  RCOND  is not needed, SSPFA is slightly faster.
    To solve  A*X = B , follow SSPCO by SSPSL.
    To compute  INVERSE(A)*C , follow SSPCO by SSPSL.
    To compute  INVERSE(A) , follow SSPCO by SSPDI.
    To compute  DETERMINANT(A) , follow SSPCO by SSPDI.
    To compute  INERTIA(A), follow SSPCO by SSPDI.
    On Entry
       AP      REAL (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    Output
       AP      a block diagonal matrix and the multipliers which
               were used to obtain it stored in packed form.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KPVT    INTEGER(N)
               an integer vector of pivot indices.
       RCOND   REAL
               an estimate of the reciprocal condition of  A .
               For the system  A*X = B , relative perturbations
               in  A  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  A  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  A  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K) = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    LINPACK SSPFA
    BLAS SAXPY,SDOT,SSCAL,SASUM
    Fortran ABS,AMAX1,IABS,SIGN


SSPDI


      SUBROUTINE SSPDI(AP,N,KPVT,DET,INERT,WORK,JOB)

    SSPDI computes the determinant, inertia and inverse
    of a real symmetric matrix using the factors from SSPFA,
    where the matrix is stored in packed form.
    On Entry
       AP      REAL (N*(N+1)/2)
               the output from SSPFA.
       N       INTEGER
               the order of the matrix A.
       KPVT    INTEGER(N)
               the pivot vector from SSPFA.
       WORK    REAL(N)
               work vector.  Contents ignored.
       JOB     INTEGER
               JOB has the decimal expansion  ABC  where
                  If  C .NE. 0, the inverse is computed,
                  If  B .NE. 0, the determinant is computed,
                  If  A .NE. 0, the inertia is computed.
               For example, JOB = 111  gives all three.
    On Return
       Variables not requested by JOB are not used.
       AP     contains the upper triangle of the inverse of
              the original matrix, stored in packed form.
              The columns of the upper triangle are stored
              sequentially in a one-dimensional array.
       DET    REAL(2)
              determinant of original matrix.
              Determinant = DET(1) * 10.0**DET(2)
              with 1.0 .LE. ABS(DET(1)) .LT. 10.0
              or DET(1) = 0.0.
       INERT  INTEGER(3)
              the inertia of the original matrix.
              INERT(1)  =  number of positive eigenvalues.
              INERT(2)  =  number of negative eigenvalues.
              INERT(3)  =  number of zero eigenvalues.
    Error Condition
       A division by zero will occur if the inverse is requested
       and  SSPCO  has set RCOND .EQ. 0.0
       or  SSPFA  has set  INFO .NE. 0 .
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS SAXPY,SCOPY,SDOT,SSWAP
    Fortran ABS,IABS,MOD


SSPEV


      SUBROUTINE SSPEV(A,N,E,V,LDV,WORK,JOB,INFO)

    LICEPACK.  This version dated 08/08/80.
    David Kahaner, Cleve Moler, Pete Stewart
         N.B.S.       U.N.M.     N.B.S./U.MD.
    Abstract
     SSPEV computes the eigenvalues and, optionally, the eigenvectors
     of a real symmetric matrix stored in packed form.
    Call Sequence Parameters-
      (The values of parameters marked with * (star) will be  changed
        by SSPEV.)
       A*      REAL(N*(N+1)/2)
               real symmetric packed input matrix.  Contains upper
               triangle and diagonal of A, by column (elements
               11, 12, 22, 13, 23, 33, ...).
       N       INTEGER
               set by the user to
               the order of the matrix A.
       E*      REAL(N)
               on return from SSPEV, E contains the eigenvalues of A.
               See also INFO below.
       V*      REAL(LDV,N)
               on return from SSPEV, if the user has set JOB
               = 0        V is not referenced.
               = nonzero  the N eigenvectors of A are stored in the
               first N columns of V.  See also INFO below.
       LDV     INTEGER
               set by the user to
               the leading dimension of the array V if JOB is also
               set nonzero.  In that case, N must be .LE. LDV.
               If JOB is set to zero, LDV is not referenced.
       WORK*   REAL(2N)
               temporary storage vector.  Contents changed by SSPEV.
       JOB     INTEGER
               set by the user to
               = 0        eigenvalues only to be calculated by SSPEV.
                          Neither V nor LDV are referenced.
               = nonzero  eigenvalues and vectors to be calculated.
                          In this case, A & V must be distinct arrays.
                          Also, if LDA .GT. LDV, SSPEV changes all the
                          elements of A thru column N.  If LDA < LDV,
                          SSPEV changes all the elements of V through
                          column N.  If LDA=LDV, only A(I,J) and V(I,
                          J) for I,J = 1,...,N are changed by SSPEV.
      INFO*   INTEGER
              on return from SSPEV, the value of INFO is
              = 0 for normal return.
              = K if the eigenvalue iteration fails to converge.
                  Eigenvalues and vectors 1 through K-1 are correct.
    Error Messages-
         No. 1   recoverable  N is greater than LDV and JOB is nonzero
         No. 2   recoverable  N is less than one
    Subroutines Used
     EISPACK- IMTQL2, TQLRAT, TRBAK3, TRED3
     SLATEC- XERROR


SSPFA


      SUBROUTINE SSPFA(AP,N,KPVT,INFO)

    SSPFA factors a real symmetric matrix stored in
    packed form by elimination with symmetric pivoting.
    To solve  A*X = B , follow SSPFA by SSPSL.
    To compute  INVERSE(A)*C , follow SSPFA by SSPSL.
    To compute  DETERMINANT(A) , follow SSPFA by SSPDI.
    To compute  INERTIA(A) , follow SSPFA by SSPDI.
    To compute  INVERSE(A) , follow SSPFA by SSPDI.
    On Entry
       AP      REAL (N*(N+1)/2)
               the packed form of a symmetric matrix  A .  The
               columns of the upper triangle are stored sequentially
               in a one-dimensional array of length  N*(N+1)/2 .
               See comments below for details.
       N       INTEGER
               the order of the matrix  A .
    Output
       AP      a block diagonal matrix and the multipliers which
               were used to obtain it stored in packed form.
               The factorization can be written  A = U*D*TRANS(U)
               where  U  is a product of permutation and unit
               upper triangular matrices , TRANS(U) is the
               transpose of  U , and  D  is block diagonal
               with 1 by 1 and 2 by 2 blocks.
       KPVT    INTEGER(N)
               an integer vector of pivot indices.
       INFO    INTEGER
               = 0  normal value.
               = K  if the K-th pivot block is singular.  This is
                    not an error condition for this subroutine,
                    but it does indicate that SSPSL or SSPDI may
                    divide by zero if called.
    Packed Storage
         The following program segment will pack the upper
         triangle of a symmetric matrix.
               K = 0
               DO 20 J = 1, N
                  DO 10 I = 1, J
                     K = K + 1
                     AP(K)  = A(I,J)
            10    CONTINUE
            20 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS SAXPY,SSWAP,ISAMAX
    Fortran ABS,AMAX1,SQRT


SSPSL


      SUBROUTINE SSPSL(AP,N,KPVT,B)

    SSISL solves the real symmetric system
    A * X = B
    using the factors computed by SSPFA.
    On Entry
       AP      REAL(N*(N+1)/2)
               the output from SSPFA.
       N       INTEGER
               the order of the matrix  A .
       KPVT    INTEGER(N)
               the pivot vector from SSPFA.
       B       REAL(N)
               the right hand side vector.
    On Return
       B       the solution vector  X .
    Error Condition
       A division by zero may occur if  SSPCO  has set RCOND .EQ. 0.0
       or  SSPFA  has set INFO .NE. 0  .
    To compute  INVERSE(A) * C  where  C  is a matrix
    with  P  columns
          CALL SSPFA(AP,N,KPVT,INFO)
          IF (INFO .NE. 0) GO TO ...
          DO 10 J = 1, P
             CALL SSPSL(AP,N,KPVT,C(1,J))
       10 CONTINUE
    LINPACK.  This version dated 08/14/78 .
    James Bunch, Univ. Calif. San Diego, Argonne Nat. Lab.
    Subroutines and Functions
    BLAS SAXPY,SDOT
    Fortran IABS


SSVDC


      SUBROUTINE SSVDC(X,LDX,N,P,S,E,U,LDU,V,LDV,WORK,JOB,INFO)

    SSVDC is a subroutine to reduce a real NxP matrix X by
    orthogonal transformations U and V to diagonal form.  The
    diagonal elements S(I) are the singular values of X.  The
    columns of U are the corresponding left singular vectors,
    and the columns of V the right singular vectors.
    On Entry
        X         REAL(LDX,P), where LDX .GE. N.
                  X contains the matrix whose singular value
                  decomposition is to be computed.  X is
                  destroyed by SSVDC.
        LDX       INTEGER
                  LDX is the leading dimension of the array X.
        N         INTEGER
                  N is the number of columns of the matrix X.
        P         INTEGER
                  P is the number of rows of the matrix X.
        LDU       INTEGER
                  LDU is the leading dimension of the array U.
                  (See below).
        LDV       INTEGER
                  LDV is the leading dimension of the array V.
                  (See below).
        WORK      REAL(N)
                  work is a scratch array.
        JOB       INTEGER
                  JOB controls the computation of the singular
                  vectors.  It has the decimal expansion AB
                  with the following meaning
                       A .EQ. 0  Do not compute the left singular
                                 vectors.
                       A .EQ. 1  Return the N left singular vectors
                                 in U.
                       A .GE. 2  Return the first MIN(N,P) singular
                                 vectors in U.
                       B .EQ. 0  Do not compute the right singular
                                 vectors.
                       B .EQ. 1  Return the right singular vectors
                                 in V.
    On Return
        S         REAL(MM), where MM=MIN(N+1,P).
                  The first MIN(N,P) entries of S contain the
                  singular values of X arranged in descending
                  order of magnitude.
        E         REAL(P).
                  E ordinarily contains zeros.  However, see the
                  discussion of INFO for exceptions.
        U         REAL(LDU,K), where LDU .GE. N.  If JOBA .EQ. 1, then
                                  K .EQ. N.  If JOBA .GE. 2 , then
                                  K .EQ. MIN(N,P).
                  U contains the matrix of right singular vectors.
                  U is not referenced if JOBA .EQ. 0.  If N .LE. P
                  or if JOBA .EQ. 2, then U may be identified with X
                  in the subroutine call.
        V         REAL(LDV,P), where LDV .GE. P.
                  V contains the matrix of right singular vectors.
                  V is not referenced if JOB .EQ. 0.  If P .LE. N,
                  then V may be identified with X in the
                  subroutine call.
        INFO      INTEGER.
                  the singular values (and their corresponding
                  singular vectors) S(INFO+1),S(INFO+2),...,S(M)
                  are correct (here M=MIN(N,P)).  Thus if
                  INFO .EQ. 0, all the singular values and their
                  vectors are correct.  In any event, the matrix
                  B = TRANS(U)*X*V is the bidiagonal matrix
                  with the elements of S on its diagonal and the
                  elements of E on its super-diagonal (TRANS(U)
                  is the transpose of U).  Thus the singular
                  values of X and B are the same.
    LINPACK.  This version dated 03/19/79 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    ***** Uses the following functions and subprograms.
    External SROT
    BLAS SAXPY,SDOT,SSCAL,SSWAP,SNRM2,SROTG
    Fortran ABS,AMAX1,MAX0,MIN0,MOD,SQRT


SSWAP


      SUBROUTINE SSWAP(N,SX,INCX,SY,INCY)

               B L A S  Subprogram
   Description of Parameters
    --Input--
       N  number of elements in input vector(s)
      SX  single precision vector with N elements
    INCX  storage spacing between elements of SX
      SY  single precision vector with N elements
    INCY  storage spacing between elements of SY
    --Output--
      SX  input vector SY (unchanged if N .LE. 0)
      SY  input vector SX (unchanged if N .LE. 0)
    Interchange single precision SX and single precision SY.
    For I = 0 to N-1, interchange  SX(LX+I*INCX) and SY(LY+I*INCY),
    where LX = 1 if INCX .GE. 0, else LX = (-INCX)*N, and LY is
    defined in a similar way using INCY.


STRCO


      SUBROUTINE STRCO(T,LDT,N,RCOND,Z,JOB)

    STRCO estimates the condition of a real triangular matrix.
    On Entry
       T       REAL(LDT,N)
               T contains the triangular matrix.  The zero
               elements of the matrix are not referenced, and
               the corresponding elements of the array can be
               used to store other information.
       LDT     INTEGER
               LDT is the leading dimension of the array T.
       N       INTEGER
               N is the order of the system.
       JOB     INTEGER
               = 0         T  is lower triangular.
               = nonzero   T  is upper triangular.
    On Return
       RCOND   REAL
               an estimate of the reciprocal condition of  T .
               For the system  T*X = B , relative perturbations
               in  T  and  B  of size  EPSILON  may cause
               relative perturbations in  X  of size  EPSILON/RCOND .
               If  RCOND  is so small that the logical expression
                          1.0 + RCOND .EQ. 1.0
               is true, then  T  may be singular to working
               precision.  In particular,  RCOND  is zero  if
               exact singularity is detected or the estimate
               underflows.
       Z       REAL(N)
               a work vector whose contents are usually unimportant.
               If  T  is close to a singular matrix, then  Z  is
               an approximate null vector in the sense that
               NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SSCAL,SASUM
    Fortran ABS,AMAX1,SIGN


STRDI


      SUBROUTINE STRDI(T,LDT,N,DET,JOB,INFO)

    STRDI computes the determinant and inverse of a real
    triangular matrix.
    On Entry
       T       REAL(LDT,N)
               T contains the triangular matrix.  The zero
               elements of the matrix are not referenced, and
               the corresponding elements of the array can be
               used to store other information.
       LDT     INTEGER
               LDT is the leading dimension of the array T.
       N       INTEGER
               N is the order of the system.
       JOB     INTEGER
               = 010       no det, inverse of lower triangular.
               = 011       no det, inverse of upper triangular.
               = 100       det, no inverse.
               = 110       det, inverse of lower triangular.
               = 111       det, inverse of upper triangular.
    On Return
       T       inverse of original matrix if requested.
               Otherwise unchanged.
       DET     REAL(2)
               determinant of original matrix if requested.
               Otherwise not referenced.
               Determinant = DET(1) * 10.0**DET(2)
               with  1.0 .LE. ABS(DET(1)) .LT. 10.0
               or  DET(1) .EQ. 0.0 .
       INFO    INTEGER
               INFO contains zero if the system is nonsingular
               and the inverse is requested.
               Otherwise INFO contains the index of
               a zero diagonal element of T.
    LINPACK.  This version dated 08/14/78 .
    Cleve Moler, University of New Mexico, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SSCAL
    Fortran ABS,MOD


STRSL


      SUBROUTINE STRSL(T,LDT,N,B,JOB,INFO)

    STRSL solves systems of the form
                  T * X = B
    or
                  TRANS(T) * X = B
    where T is a triangular matrix of order N.  Here TRANS(T)
    denotes the transpose of the matrix T.
    On Entry
        T         REAL(LDT,N)
                  T contains the matrix of the system.  The zero
                  elements of the matrix are not referenced, and
                  the corresponding elements of the array can be
                  used to store other information.
        LDT       INTEGER
                  LDT is the leading dimension of the array T.
        N         INTEGER
                  N is the order of the system.
        B         REAL(N).
                  B contains the right hand side of the system.
        JOB       INTEGER
                  JOB specifies what kind of system is to be solved.
                  If JOB is
                       00   solve T*X=B, T lower triangular,
                       01   solve T*X=B, T upper triangular,
                       10   solve TRANS(T)*X=B, T lower triangular,
                       11   solve TRANS(T)*X=B, T upper triangular.
    On Return
        B         B contains the solution, if INFO .EQ. 0.
                  Otherwise B is unaltered.
        INFO      INTEGER
                  INFO contains zero if the system is nonsingular.
                  Otherwise INFO contains the index of
                  the first zero diagonal element of T.
    LINPACK.  This version dated 08/14/78 .
    G. W. Stewart, University of Maryland, Argonne National Lab.
    Subroutines and Functions
    BLAS SAXPY,SDOT
    Fortran MOD


STWAY


      SUBROUTINE STWAY(U,V,YHP,INOUT,STOWA)



SUDS


      SUBROUTINE SUDS(A,X,B,NEQ,NUK,NRDA,IFLAG,MLSO,WORK,IWORK)



SVCO


      SUBROUTINE SVCO(RSAV,ISAV)



SVD


      SUBROUTINE SVD(NM,M,N,A,W,MATU,U,MATV,V,IERR,RV1)



SVECS


      SUBROUTINE SVECS(NCOMP,LNFC,YHP,WORK,IWORK,INHOMO,IFLAG)



SVOUT


      SUBROUTINE SVOUT(N,SX,IFMT,IDIGIT)

SVOUT    REVISED FEB. 27, 1981.
    SINGLE PRECISION VECTOR OUTPUT ROUTINE.
 INPUT..
 N,SX(*) PRINT THE SINGLE PRECISION ARRAY SX(I),I=1,...,N, ON
         OUTPUT UNIT LOUT. THE HEADING IN THE FORTRAN FORMAT
         STATEMENT IFMT(*), DESCRIBED BELOW, IS PRINTED AS A FIRST
         STEP. THE COMPONENTS SX(I) ARE INDEXED, ON OUTPUT,
         IN A PLEASANT FORMAT.
 IFMT(*) A FORTRAN FORMAT STATEMENT. THIS IS PRINTED ON OUTPUT
         UNIT LOUT WITH THE VARIABLE FORMAT FORTRAN STATEMENT
               WRITE(LOUT,IFMT)
 IDIGIT  PRINT AT LEAST IABS(IDIGIT) DECIMAL DIGITS PER NUMBER.
         THE SUBPROGRAM WILL CHOOSE THAT INTEGER 4,6,10 OR 14
         WHICH WILL PRINT AT LEAST IABS(IDIGIT) NUMBER OF
         PLACES.  IF IDIGIT.LT.0, 72 PRINTING COLUMNS ARE UTILIZED
         TO WRITE EACH LINE OF OUTPUT OF THE ARRAY SX(*). (THIS
         CAN BE USED ON MOST TIME-SHARING TERMINALS). IF
         IDIGIT.GE.0, 133 PRINTING COLUMNS ARE UTILIZED. (THIS CAN
         BE USED ON MOST LINE PRINTERS).
 EXAMPLE..
 PRINT AN ARRAY CALLED (COSTS OF PURCHASES) OF LENGTH 100 SHOWING
 6 DECIMAL DIGITS PER NUMBER. THE USER IS RUNNING ON A TIME-SHARING
 SYSTEM WITH A 72 COLUMN OUTPUT DEVICE.
    DIMENSION COSTS(100)
    N = 100
    IDIGIT = -6
    CALL SVOUT(N,COSTS,'(''1COSTS OF PURCHASES'')',IDIGIT)
    AUTHORS    JOHN A. WISNIEWSKI   SANDIA LABS ALBUQUERQUE.
               RICHARD J. HANSON    SANDIA LABS ALBUQUERQUE.
    DATE       JULY 27,1978.


SWRITP


      SUBROUTINE SWRITP(IPAGE,LIST,RLIST,LPAGE,IREC)

    WRITE RECORD NUMBER IRECN, OF LENGTH LPG, FROM STORAGE
    ARRAY LIST(*) ONTO UNIT NUMBER IPAGEF.
    WRITE RECORD NUMBER IRECN+1, OF LENGTH LPG, ONTO UNIT
    NUMBER IPAGEF FROM THE STORAGE ARRAY RLIST(*).
    TO CHANGE THIS PROGRAM UNIT TO DOUBLE PRECISION CHANGE
    /REAL (12 BLANKS)/ TO /DOUBLE PRECISION/.


TEVLC


      SUBROUTINE TEVLC(N,D,E2,IERR)



TEVLS


      SUBROUTINE TEVLS(N,D,E2,IERR)



TINVIT


      SUBROUTINE TINVIT(NM,N,D,E,E2,M,W,IND,Z,IERR,RV1,RV2,RV3,RV4,RV6)

    This subroutine is a translation of the inverse iteration tech-
    nique in the ALGOL procedure TRISTURM by Peters and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 418-439(1971).
    This subroutine finds those eigenvectors of a TRIDIAGONAL
    SYMMETRIC matrix corresponding to specified eigenvalues,
    using inverse iteration.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
       E2 contains the squares of the corresponding elements of E,
         with zeros corresponding to negligible elements of E.
         E(I) is considered negligible if it is not larger than
         the product of the relative machine precision and the sum
         of the magnitudes of D(I) and D(I-1).  E2(1) must contain
         0.0e0 if the eigenvalues are in ascending order, or 2.0e0
         if the eigenvalues are in descending order.  If  BISECT,
         TRIDIB, or  IMTQLV  has been used to find the eigenvalues,
         their output E2 array is exactly what is expected here.
       M is the number of specified eigenvalues.
       W CONTAINS the M eigenvalues in ascending or descending order.
       IND contains in its first M positions the submatrix indices
         associated with the corresponding eigenvalues in W --
         1 for eigenvalues belonging to the first submatrix from
         the top, 2 for those belonging to the second submatrix, etc.
    On Output
      ** All input arrays are unaltered.**
       Z contains the associated set of orthonormal eigenvectors.
         any vector which fails to converge is set to zero.
       IERR is set to
         Zero       for normal return,
         -R         if the eigenvector corresponding to the R-th
                    eigenvalue fails to converge in 5 iterations.
       RV1, RV2, RV3, RV4, and RV6 are temporary storage arrays.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TQL1


      SUBROUTINE TQL1(N,D,E,IERR)

    This subroutine is a translation of the ALGOL procedure TQL1,
    NUM. MATH. 11, 293-306(1968) by Bowdler, Martin, Reinsch, and
    Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 227-240(1971).
    This subroutine finds the eigenvalues of a SYMMETRIC
    TRIDIAGONAL matrix by the QL method.
    On Input
       N is the order of the matrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
     On Output
       D contains the eigenvalues in ascending order.  If an
         error exit is made, the eigenvalues are correct and
         ordered for indices 1,2,...IERR-1, but may not be
         the smallest eigenvalues.
       E has been destroyed.
       IERR is set to
         Zero       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after 30 iterations.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TQL2


      SUBROUTINE TQL2(NM,N,D,E,Z,IERR)

    This subroutine is a translation of the ALGOL procedure TQL2,
    NUM. MATH. 11, 293-306(1968) by Bowdler, Martin, Reinsch, and
    Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 227-240(1971).
    This subroutine finds the eigenvalues and eigenvectors
    of a SYMMETRIC TRIDIAGONAL matrix by the QL method.
    The eigenvectors of a FULL SYMMETRIC matrix can also
    be found if  TRED2  has been used to reduce this
    full matrix to tridiagonal form.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
       Z contains the transformation matrix produced in the
         reduction by  TRED2, if performed.  If the eigenvectors
         of the tridiagonal matrix are desired, Z must contain
         the identity matrix.
     On Output
       D contains the eigenvalues in ascending order.  If an
         error exit is made, the eigenvalues are correct but
         unordered for indices 1,2,...,IERR-1.
       E has been destroyed.
       Z contains orthonormal eigenvectors of the symmetric
         tridiagonal (or full) matrix.  If an error exit is made,
         Z contains the eigenvectors associated with the stored
         eigenvalues.
       IERR is set to
         Zero       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after 30 iterations.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TQLRAT


      SUBROUTINE TQLRAT(N,D,E2,IERR)

    This subroutine is a translation of the ALGOL procedure TQLRAT,
    ALGORITHM 464, COMM. ACM 16, 689(1973) by Reinsch.
    This subroutine finds the eigenvalues of a SYMMETRIC
    TRIDIAGONAL matrix by the rational QL method.
    On Input
       N is the order of the matrix.
       D contains the diagonal elements of the input matrix.
       E2 contains the squares of the subdiagonal elements of the
         input matrix in its last N-1 positions.  E2(1) is arbitrary.
     On Output
       D contains the eigenvalues in ascending order.  If an
         error exit is made, the eigenvalues are correct and
         ordered for indices 1,2,...IERR-1, but may not be
         the smallest eigenvalues.
       E2 has been destroyed.
       IERR is set to
         Zero       for normal return,
         J          if the J-th eigenvalue has not been
                    determined after 30 iterations.
    Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TRBAK1


      SUBROUTINE TRBAK1(NM,N,A,E,M,Z)

    This subroutine is a translation of the ALGOL procedure TRBAK1,
    NUM. MATH. 11, 181-195(1968) by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine forms the eigenvectors of a REAL SYMMETRIC
    matrix by back transforming those of the corresponding
    symmetric tridiagonal matrix determined by  TRED1.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       A contains information about the orthogonal trans-
         formations used in the reduction by  TRED1
         in its strict lower triangle.
       E contains the subdiagonal elements of the tridiagonal
         matrix in its last N-1 positions.  E(1) is arbitrary.
       M is the number of eigenvectors to be back transformed.
       Z contains the eigenvectors to be back transformed
         in its first M columns.
    On Output
       Z contains the transformed eigenvectors
         in its first M columns.
    Note that TRBAK1 preserves vector Euclidean norms.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TRBAK3


      SUBROUTINE TRBAK3(NM,N,NV,A,M,Z)

    This subroutine is a translation of the ALGOL procedure TRBAK3,
    NUM. MATH. 11, 181-195(1968) by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine forms the eigenvectors of a REAL SYMMETRIC
    matrix by back transforming those of the corresponding
    symmetric tridiagonal matrix determined by  TRED3.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       NV must be set to the dimension of the array parameter A
         as declared in the calling program dimension statement.
       A contains information about the orthogonal transformations
         used in the reduction by  TRED3  in its first
         N*(N+1)/2 positions.
       M is the number of eigenvectors to be back transformed.
       Z contains the eigenvectors to be back transformed
         in its first M columns.
    On Output
       Z contains the transformed eigenvectors
         in its first M columns.
    Note that TRBAK3 preserves vector Euclidean norms.
    Questions and comments should be directed to b. s. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TRED1


      SUBROUTINE TRED1(NM,N,A,D,E,E2)

    This subroutine is a translation of the ALGOL procedure TRED1,
    NUM. MATH. 11, 181-195(1968) by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine reduces a REAL SYMMETRIC matrix
    to a symmetric tridiagonal matrix using
    orthogonal similarity transformations.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       A contains the real symmetric input matrix.  Only the
         lower triangle of the matrix need be supplied.
    On Output
       A contains information about the orthogonal trans-
         formations used in the reduction in its strict lower
         triangle.  The full upper triangle of A is unaltered.
       D contains the diagonal elements of the tridiagonal matrix.
       E contains the subdiagonal elements of the tridiagonal
         matrix in its last N-1 positions.  E(1) is set to zero.
       E2 contains the squares of the corresponding elements of E.
         E2 may coincide with E if the squares are not needed.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TRED2


      SUBROUTINE TRED2(NM,N,A,D,E,Z)

    This subroutine is a translation of the ALGOL procedure TRED2,
    NUM. MATH. 11, 181-195(1968) by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine reduces a REAL SYMMETRIC matrix to a
    symmetric tridiagonal matrix using and accumulating
    orthogonal similarity transformations.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       A contains the real symmetric input matrix.  Only the
         lower triangle of the matrix need be supplied.
    On Output
       D contains the diagonal elements of the tridiagonal matrix.
       E contains the subdiagonal elements of the tridiagonal
         matrix in its last N-1 positions.  E(1) is set to zero.
       Z contains the orthogonal transformation matrix
         produced in the reduction.
       A and Z may coincide.  If distinct, A is unaltered.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TRED3


      SUBROUTINE TRED3(N,NV,A,D,E,E2)

    This subroutine is a translation of the ALGOL procedure TRED3,
    NUM. MATH. 11, 181-195(1968) by Martin, Reinsch, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    This subroutine reduces a REAL SYMMETRIC matrix, stored as
    a one-dimensional array, to a symmetric tridiagonal matrix
    using orthogonal similarity transformations.
    On Input
       n is the order of the matrix.
       NV must be set to the dimension of the array parameter A
         as declared in the calling program dimension statement.
       A contains the lower triangle of the real symmetric
         input matrix, stored row-wise as a one-dimensional
         array, in its first N*(N+1)/2 positions.
    On Output
       A contains information about the orthogonal
         transformations used in the reduction.
       D contains the diagonal elements of the tridiagonal matrix.
       E contains the subdiagonal elements of the tridiagonal
         matrix in its last N-1 positions.  E(1) is set to zero.
       E2 contains the squares of the corresponding elements of E.
         E2 may coincide with E if the squares are not needed.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TRI3


      SUBROUTINE TRI3(M,A,B,C,K,Y1,Y2,Y3,TCOS,D,W1,W2,W3)



TRID


      SUBROUTINE TRID(MR,A,B,C,Y,D)



TRIDIB


      SUBROUTINE TRIDIB(N,EPS1,D,E,E2,LB,UB,M11,M,W,IND,IERR,RV4,RV5)

    This subroutine is a translation of the ALGOL procedure BISECT,
    NUM. MATH. 9, 386-393(1967) by Barth, Martin, and Wilkinson.
    HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 249-256(1971).
    This subroutine finds those eigenvalues of a TRIDIAGONAL
    SYMMETRIC matrix between specified boundary indices,
    using bisection.
    On Input
       N is the order of the matrix.
       EPS1 is an absolute error tolerance for the computed
         eigenvalues.  If the input EPS1 is non-positive,
         it is reset for each submatrix to a default value,
         namely, minus the product of the relative machine
         precision and the 1-norm of the submatrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
       E2 contains the squares of the corresponding elements of E.
         E2(1) is arbitrary.
       M11 specifies the lower boundary index for the desired
         eigenvalues.
       M specifies the number of eigenvalues desired.  The upper
         boundary index M22 is then obtained as M22=M11+M-1.
    On Output
       EPS1 is unaltered unless it has been reset to its
         (last) default value.
       D and E are unaltered.
       ELEMENTS of E2, corresponding to elements of E regarded
         as negligible, have been replaced by zero causing the
         matrix to split into a direct sum of submatrices.
         E2(1) is also set to zero.
       LB and UB DEFINE an interval containing exactly the desired
         eigenvalues.
       W contains, in its first M positions, the eigenvalues
         between indices M11 and M22 in ascending order.
       IND contains in its first M positions the submatrix indices
         associated with the corresponding eigenvalues in W --
         1 for eigenvalues belonging to the first submatrix from
         the top, 2 for those belonging to the second submatrix, etc.
       IERR is set to
         Zero       for normal return,
         3*N+1      if multiple eigenvalues at index M11 make
                    unique selection impossible,
         3*N+2      if multiple eigenvalues at index M22 make
                    unique selection impossible.
       RV4 and RV5 are temporary storage arrays.
    Note that subroutine TQL1, IMTQL1, or TQLRAT is generally faster
    than TRIDIB, if more than N/4 eigenvalues are to be found.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



TRIS4


      SUBROUTINE TRIS4(N,A,B,C,D,U,Z)



TRISP


      SUBROUTINE TRISP(N,A,B,C,D,U,Z)



TRIX


      SUBROUTINE TRIX(IDEGBR,IDEGCR,M,A,B,C,Y,TCOS,D,W)



TSTURM


      SUBROUTINE TSTURM(NM,N,EPS1,D,E,E2,LB,UB,MM,M,W,Z,IERR,RV1,RV2,

    This subroutine finds those eigenvalues of a TRIDIAGONAL
    SYMMETRIC matrix which lie in a specified interval and their
    associated eigenvectors, using bisection and inverse iteration.
    On Input
       NM must be set to the row dimension of two-dimensional
         array parameters as declared in the calling program
         dimension statement.
       N is the order of the matrix.
       EPS1 is an absolute error tolerance for the computed
         eigenvalues.  It should be chosen commensurate with
         relative perturbations in the matrix elements of the
         order of the relative machine precision.  If the
         input EPS1 is non-positive, it is reset for each
         submatrix to a default value, namely, minus the
         product of the relative machine precision and the
         1-norm of the submatrix.
       D contains the diagonal elements of the input matrix.
       E contains the subdiagonal elements of the input matrix
         in its last N-1 positions.  E(1) is arbitrary.
       E2 contains the squares of the corresponding elements of E.
         E2(1) is arbitrary.
       LB and UB define the interval to be searched for eigenvalues.
         If LB is not less than UB, no eigenvalues will be found.
       MM should be set to an upper bound for the number of
         eigenvalues in the interval.  WARNING. If more than
         MM eigenvalues are determined to lie in the interval,
         an error return is made with no values or vectors found.
    On Output
       EPS1 is unaltered unless it has been reset to its
         (last) default value.
       D and E are unaltered.
       ELEMENTS of E2, corresponding to elements of E regarded
         as negligible, have been replaced by zero causing the
         matrix to split into a direct sum of submatrices.
         E2(1) is also set to zero.
       M is the number of eigenvalues determined to lie in (LB,UB).
       W contains the M eigenvalues in ascending order if the matrix
         does not split.  If the matrix splits, the eigenvalues are
         in ascending order for each submatrix.  If a vector error
         exit is made, W contains those values already found.
       Z contains the associated set of orthonormal eigenvectors.
         If an error exit is made, Z contains those vectors
         already found.
       IERR is set to
         Zero       for normal return,
         3*N+1      if M exceeds MM.
         4*N+R      if the eigenvector corresponding to the R-th
                    eigenvalue fails to converge in 5 iterations.
       RV1, RV2, RV3, RV4, RV5, and RV6 are temporary storage arrays.
    The ALGOL procedure STURMCNT contained in TRISTURM
    appears in TSTURM in-line.
    Questions and comments should be directed to B. S. Garbow,
    APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY



U11LS


      SUBROUTINE U11LS(A,MDA,M,N,UB,DB,MODE,NP,KRANK,KSURE,H,W,EB,IC,

      This routine performs a QR factorization of A
      using Householder transformations. Row and
      column pivots are chosen to reduce the growth
      of round-off and to help detect possible rank
      deficiency.


U11US


      SUBROUTINE U11US(A,MDA,M,N,UB,DB,MODE,NP,KRANK,KSURE,H,W,EB,IR,

      This routine performs an LQ factorization of the
      matrix A using Householder transformations. Row
      and column pivots are chosen to reduce the growth
      of round-off and to help detect possible rank
      deficiency.


U12LS


      SUBROUTINE U12LS(A,MDA,M,N,B,MDB,NB,MODE,KRANK,RNORM,H,W,IC,IR)

       Given the Householder QR factorization of A, this
       subroutine solves the system AX=B. If the system
       is of reduced rank, this routine returns a solution
       according to the selected mode.
      Note - If MODE.NE.2, W is never accessed.


U12US


      SUBROUTINE U12US(A,MDA,M,N,B,MDB,NB,MODE,KRANK,RNORM,H,W,IR,IC)

       Given the Householder LQ factorization of A, this
       subroutine solves the system AX=B. If the system
       is of reduced rank, this routine returns a solution
       according to the selected mode.
      Note - If MODE.NE.2, W is never accessed.


ULSIA


      SUBROUTINE ULSIA(A,MDA,M,N,B,MDB,NB,RE,AE,KEY,MODE,NP,KRANK,

    ULSIA computes the minimal length solution(s) to the problem AX=B
    where A is an M by N matrix with M.LE.N and B is the M by NB
    matrix of right hand sides.  User input bounds on the uncertainty
    in the elements of A are used to detect numerical rank deficiency.
    The algorithm employs a row and column pivot strategy to
    minimize the growth of uncertainty and round-off errors.
    ULSIA requires (MDA+1)*N + (MDB+1)*NB + 6*M dimensioned space
  ******************************************************************
  *                                                                *
  *         WARNING - All input arrays are changed on exit.        *
  *                                                                *
  ******************************************************************
    Input..
    A(,)          Linear coefficient matrix of AX=B, with MDA the
     MDA,M,N      actual first dimension of A in the calling program.
                  M is the row dimension (no. of EQUATIONS of the
                  problem) and N the col dimension (no. of UNKNOWNS).
                  Must have MDA.GE.M and M.LE.N.
    B(,)          Right hand side(s), with MDB the actual first
     MDB,NB       dimension of B in the calling program. NB is the
                  number of M by 1 right hand sides. Since the solutio
                  is returned in B, must have MDB.GE.N. If NB = 0, B i
                  never accessed.
  ******************************************************************
  *                                                                *
  *         Note - Use of RE and AE are what make this             *
  *                code significantly different from               *
  *                other linear least squares solvers.             *
  *                However, the inexperienced user is              *
  *                advised to set RE=0.,AE=0.,KEY=0.               *
  *                                                                *
  ******************************************************************
    RE(),AE(),KEY
    RE()          RE() is a vector of length N such that RE(I) is
                  the maximum relative uncertainty in row I of
                  the matrix A. The values of RE() must be between
                  0 and 1. A minimum of 10*machine precision will
                  be enforced.
    AE()          AE() is a vector of length N such that AE(I) is
                  the maximum absolute uncertainty in row I of
                  the matrix A. The values of AE() must be greater
                  than or equal to 0.
    KEY           For ease of use, RE and AE may be input as either
                  vectors or scalars. If a scalar is input, the algo-
                  rithm will use that value for each column of A.
                  The parameter KEY indicates whether scalars or
                  vectors are being input.
                       KEY=0     RE scalar  AE scalar
                       KEY=1     RE vector  AE scalar
                       KEY=2     RE scalar  AE vector
                       KEY=3     RE vector  AE vector
    MODE          The integer MODE indicates how the routine
                  is to react if rank deficiency is detected.
                  If MODE = 0 return immediately, no solution
                            1 compute truncated solution
                            2 compute minimal length least squares sol
                  The inexperienced user is advised to set MODE=0
    NP            The first NP rows of A will not be interchanged
                  with other rows even though the pivot strategy
                  would suggest otherwise.
                  The inexperienced user is advised to set NP=0.
    WORK()        A real work array dimensioned 5*M. However,if
                  RE or AE have been specified as vectors, dimension
                  WORK 4*M. If both RE and AE have been specified
                  as vectors, dimension WORK 3*M.
    LW            Actual dimension of WORK
    IWORK()       Integer work array dimensioned at least N+M.
    LIW           Actual dimension of IWORK.
    INFO          Is a flag which provides for the efficient
                  solution of subsequent problems involving the
                  same A but different B.
                  If INFO = 0 original call
                     INFO = 1 subsequent calls
                  On subsequent calls, the user must supply A, KRANK,
                  LW, IWORK, LIW, and the first 2*M locations of WORK
                  as output by the original call to ULSIA. MODE must
                  be equal to the value of MODE in the original call.
                  If MODE.LT.2, only the first N locations of WORK
                  are accessed. AE, RE, KEY, and NP are not accessed.
    Output..
    A(,)          Contains the lower triangular part of the reduced
                  matrix and the transformation information. It togeth
                  with the first M elements of WORK (see below)
                  completely specify the LQ factorization of A.
    B(,)          Contains the N by NB solution matrix for X.
    KRANK,KSURE   The numerical rank of A,  based upon the relative
                  and absolute bounds on uncertainty, is bounded
                  above by KRANK and below by KSURE. The algorithm
                  returns a solution based on KRANK. KSURE provides
                  an indication of the precision of the rank.
    RNORM()       Contains the Euclidean length of the NB residual
                  vectors  B(I)-AX(I), I=1,NB. If the matrix A is of
                  full rank, then RNORM=0.0.
    WORK()        The first M locations of WORK contain values
                  necessary to reproduce the Householder transformatio
    IWORK()       The first N locations contain the order in
                  which the columns of A were used. The next
                  M locations contain the order in which the
                  rows of A were used.
    INFO          Flag to indicate status of computation on completion
                 -1   Parameter error(s)
                  0 - Rank deficient, no solution
                  1 - Rank deficient, truncated solution
                  2 - Rank deficient, minimal length least squares sol
                  3 - Numerical rank 0, zero solution
                  4 - Rank .LT. NP
                  5 - Full rank


USRMAT


      SUBROUTINE USRMAT(I,J,AIJ,INDCAT,PRGOPT,DATTRV,IFLAG)

  The user may supply this code


VNORM


      FUNCTION VNORM(V,NCOMP)



VNWRMS


      REAL FUNCTION VNWRMS(N,V,W)



WNLIT


      SUBROUTINE WNLIT(W,MDW,M,N,L,IPIVOT,ITYPE,H,SCALE,RNORM,IDOPE,



WNLSM


      SUBROUTINE WNLSM(W,MDW,MME,MA,N,L,PRGOPT,X,RNORM,MODE,IPIVOT,