SCQF Level: 10
Availability: Spring 2016, Advanced module (Not Semester 8)
Course Prerequisite: MATU913
Credit Value: 20 (1 module)
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.
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.
Interpolation: Lagrange's formula, error estimates, Neville's algorithm, inverse interpolation, forward differences, Gregory's formula, propagation 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.
Investigative ability, report presentation, experience in Mathematica™.
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.
3 one-hour lectures and a 1.5 hour workshop per week.
1/3 coursework (2 assignments) and 2/3 examination.