Numerical Analysis

Unit aims

To introduce students to the basics of numerical analysis; this is broadly the study of numerical methods for solving mathematical problems.

Unit description

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 numerical techniques for solving these problems theoretically. Computer programming is not required for this unit.

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.

Transferable skills

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

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.

Reading and References

A good text which covers most of the course is: 

  1. R.L. Burden and J.D. Faires, Numerical analysis (Brooks/Cole) (QA297 BUR, available online) 
Other texts that may be helpful:
  1. C.F. Gerald and P.O.Wheatley, Applied numerical analysis (Addison-Wesley) (QA297 GER)
  2. E. Süli and D.F. Mayers, An introduction to numerical analysis (Cambridge) (QA297 SUL)
  3. B. Bradie, A friendly introduction to numerical analysis (Pearson) (QA297 BRA)

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

Unit code: MATH30010
Level of study: H/6 
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturer: Dr Martin Sieber

Pre-requisites

MATH10003 Analysis 1A and MATH10006 Analysis 1B, MATH11007 Calculus 1, MATH11005 Linear Algebra and Geometry.

Co-requisites

None

Methods of teaching

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

Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

  • 100% 2 hour 30 minutes Examination in May/June

NOTE: Calculators of an approved type (non-programmable, no text facility) are allowed.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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