Breadcrumb

Numerical Analysis 2 (MATH 20700)

Academic Year:

Contents of this document:

Administrative Information

  1. Unit number and title: MATH 20700 Numerical Analysis 2
  2. Level: I/5
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: In this form: 2008/09
  6. Unit Organiser: Martin Sieber
  7. Lecturer: Dr Martin Sieber
  8. Teaching block: 2
  9. Prerequisites: First year core units (MATH11006 Analysis 1, MATH 11007 Calculus 1, MATH 11005 Linear Algebra & Geometry)

Unit aims

To introduce students to the basics of numerical analysis; this is broadly the study of how to use computers to solve mathematical problems.

General Description of the Unit

This unit is intended to serve as a first course in numerical analysis. As such the fundamental areas of root finding, numerical differentiation, numerical integration and solving ordinary differential equations will be covered. The emphasis will be to explore practical numerical techniques for solving these problems.

Teaching Methods

Lectures; weekly problems classes; theoretical and computational exercises to be done by students.

Learning Objectives

At the end of this unit, students should be able to

  • solve nonlinear equations
  • numerically differentiate;
  • evaluate complicated integrals and
  • estimate the solutions to ordinary differential equations to any required accuracy.

Assessment Methods

The final assessment mark will be entirely based upon a 2½-hour examination in May/June (details below).

Summer Examination

The examination in May/June consists of a 2 ½-hour written examination consisting of FIVE questions; you should attempt FOUR. If you attempt more than four, your best four answers will be used for assessment. Calculators of an approved type (non-programmable, no text facility) are allowed. From 2012-13 ONLY calculators carrying a 'Faculty of Science approved' sticker will be allowed in the examination room.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

Transferable Skills

Computational techniques; interpretation of computational results; relation of numerical results to mathematical theory.

Texts

A good text which covers most of the course is: 

  1. R.L. Burden and J.D. Faires, Numerical Analysis (PWS-Kent) (QA297 BUR)
Other texts that may be helpful to students looking for an alternative point of view on the material:
  1. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (QA297 STO)
  2. G. Dahlquist, A.Bjorck, and N. Anderson, Numerical Methods (Prentice) (QA297 DAH)
  3. C.F. Gerald and P.O.Wheatley, Applied Numerical Analysis (Addison-Wesley (QA297 GER)

Many other books can be found in the numerical analysis section (books QA297 ***).

Syllabus

  • Root finding. Linear systems: Gaussian elimination and LU decomposition. Nonlinear equations: bisection, fixed point iteration, Newton-Raphson, accelerating convergence. Systems of nonlinear equations, Newton's method, steepest descent.
  • Numerical differentiation and integration. Interpolation polynomial, trapezoidal rule, Simpson's rule, Richardson's extrapolation, Romberg integration, Gaussian quadrature.
  • Ordinary differential equations
    • Initial value problems: Euler's methods, Runge-Kutta methods, multistep methods, stability, time stability, stiffness.
    • Boundary value problems: Shooting, finite difference methods, spectral methods.