//
// Algorithms for Rigorous Computation of Higher Transcendental functions.
//

Copyright (C) 2011--2012 University of Bristol, UK
  and     (C) 2011--2012 University of Uppsala, Sweden
---------------------------------------------------------------
    This file is part of archt.

    archt is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.

    archt is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with archt.  If not, see <http://www.gnu.org/licenses/>.


Authors
-------
Andrew R. Booker, Andreas Strombergsson, Holger Then


Purpose
-------
Rigorous multi-precision computation of higher transcendental functions.
  reasonably fast -- All computations are tightly bounded.
  rigorous        -- Multi-precision interval arithmetic is used along with
                     rigorous error bounds.
  easy to use     -- The user specifies the desired precision while the
                     algorithm determines automatically the required bounds.
  fairly general  -- For instance lngamma works for any complex argument,
                     including poles and branch cut.


Version: 1.0.3 (13 September 2012)
----------------------------------
archt.tgz includes the following files (which depend on each other).
  COPYING     -- A copy of the GNU General Public License.
  README      -- Some instructions for the use of archt.
  archt.c     -- The archt header.
  bernoulli.c -- Algorithms for computing and tabulating Bernoulli numbers.
  kbessel.c   -- Algorithm for computing K-Bessel of purely imaginary order.
  lngamma.c   -- Algorithm for computing lngamma and gamma for complex argument.
  primes.c    -- Euler sieve for computing and tabulating primes.


Installing archt (for Linux)
----------------------------
Download archt.tgz into an empty directory and unpack via
  tar -xzf archt.tgz

If not done already, you need to install
  GMP, http://gmplib.org/
  MPFR, http://www.mpfr.org/
  MPFI, https://gforge.inria.fr/projects/mpfi/


Using archt (for Linux)
-----------------------
Here we describe the use of lngamma. The other routines are used similarly.
To test lngamma quickly, type 'sh lngamma.c' into your shell.

You can include lngamma into your own program. Here is a demo.

  #include"lngamma.c"
  int main() {
    unsigned long d=53,n;
    mpfi_ptr f; mpfi_t Re_z,Im_z;
    f=init_lngamma(d); if(f!=(mpfi_ptr)0) {
      mpfi_init2(Re_z,mpfi_get_prec(f));
      mpfi_init2(Im_z,mpfi_get_prec(f));
      for(n=0;n<10;n++) {
        mpfi_set_ui(Re_z,n); mpfi_set_ui(Im_z,0l); lngamma(f,Re_z,Im_z);
        printf("lngamma(%lu+I*0.)=",n); mpfi_out_str(stdout,10,0,f);
        printf("+I*"); mpfi_out_str(stdout,10,0,f+1); printf("\n");
      }
      mpfi_clear(Im_z); mpfi_clear(Re_z); clear_lngamma(f);
    } else printf("init_lngamma(d) failed.\n",d);
    return 0;
  }

For another example, see the main() of lngamma.c.

The compiler options are: -lm -lgmp -lmpfr -lmpfi
For code optimization we encourage to use -O2, but not -O3.


Need I formally cite archt in my paper?
---------------------------------------
If you have used archt in the preparation of a paper, then yes, we would
appreciate that.

When an author formally cites the use of a computer algebra system or package
in the bibliography of a paper, then system developers receive credit in
citation indices. This practice encourages development of systems and packages
and helps provide recognition for this work that is similar to that enjoyed by
those who publish research articles. It also makes your experimental results
reproducible.

Currently, you may cite archt as follows
----------------------------------------
A. R. Booker, A. Strombergsson, and H. Then,
Bounds and algorithms for the K-Bessel function of imaginary order,
accepted for publication in the LMS J. of Comp. and Math.


Feedback
--------
Feedback is most welcome.
Let me also know of any benchmark results.


Discovering a bug
-----------------
Should you discover a bug in archt, please sent a bug report to
holger DOT then AT bristol DOT ac DOT uk.


Acknowledgements
----------------
We thank the authors, developers, and contributors of the following software
packages
  GMP, http://gmplib.org/
  MPFR, http://www.mpfr.org/
  MPFI, https://gforge.inria.fr/projects/mpfi/
Without these software packages, archt would not have seen the light of the day.

Further thanks are due to the authors, developers, and contributors of
  Debian, http://www.debian.org/
  GCC, http://gcc.gnu.org/
  PARI/GP, http://pari.math.u-bordeaux.fr/
These operating system and software packages simplified our life a lot when
writing and testing archt.

H.T. thanks Ce Bian for useful feedback and Jens Marklof for useful advice.