# Mathematics and Statistics Group

## MAT9JB: Numerical Analysis

**SCQF Level: **10

**Availability: **Spring, Advanced module (Not
Semester 8)

**Course Prerequisite: **MAT913

**Credit Value: **22 (1 module)

### Aims

The course aims to introduce students to a wide variety of numerical
techniques which are useful for solving problems in various areas of
application. While the practical aspects of the subject will be
emphasised in the mathematics laboratory, the students will be
encouraged to investigate the theoretical limitations of the various
methods.

### Learning Outcomes

At the end of this module the student should:

be familiar with elementary numerical methods for interpolation,
function approximation, integration, the solution of ordinary
differential equations, the solution of linear equations and matrix
inversion, and the calculation of eigenvalues and eigenvectors;

understand the theory behind these methods;

be able to apply these methods to find numerical approximations and
error estimates in a range of problems.

### Content

Interpolation: Lagrange's formula, error estimates, Neville's algorithm,
inverse interpolation, forward differences, Gregory's formula,
propogation of errors.

Solution of Equations: Secant method, Newton's method.

Function approximation: Taylor polynomial, minimax approximation,
least-square approximation, Chebyshev polynomials, tabulated data,
smoothing, mention of splines.

Integration: Trapezoidal rule, Simpson's rule, Romberg integration,
Gaussian integration, integrands with singular derivatives, singular
integrands, infinite limits.

Differential Equations: Euler's method, Richardson extrapolation,
Runge-Kutta methods, systems of first order differential equations.

Linear Algebra: Matrix factorization, matrix norms, Jacobi method,
diagonally-dominant matrices, Gauss-Seidel method, conditioning, power
method for dominant eigenvalue.

### Transferable Skills

Investigative ability, report presentation, experience in
Mathematica™.

### Bibliography

ATKINSON K. E., "Elementary numerical analysis", Wiley, 1993, ISBN
0-471-60010-5.

PHILLIPS, G. M. AND TAYLOR P. J., "Theory and applications of numerical
analysis", Academic Press, 1973, ISBN 0-12-553556-2.

### Teaching Format

3 one-hour lectures and a 1.5 hour workshop per week.

### Assessment

1/3 coursework (2 assignments) and 2/3 examination.