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Numerical Analysis II
Lecturer: Dr M.R. O'Donohoe
No. of lectures: 12
Prerequisite course: Numerical Analysis I
Aims
This course will build on the foundation of Numerical Analysis I
and provide more mathematical analysis to cover a much wider range of
problems and techniques. It will blend theory and practice in order
to provide a usefully broad body of knowledge.
Lectures
- Elementary approximation theory.
Polynomial approximation; Taylor series; interpolation; best
approximations; Chebyshev polynomials; Chebyshev series;
economisation; least-squares approximation; the Gram-Schmidt process;
range reduction; square roots; splines. [3.7 lectures]
- Quadrature.
Riemann sum rules; calculation of weights; error analysis;
Gaussian quadrature; composite rules; singular integrals;
multi-dimensional integration; standard regions; product rules; Monte
Carlo methods. [2.3 lectures]
- Non-linear equations and optimisation.
Non-linear equations in one variable; simple iterative methods;
fixed-point iteration theory; zeros of polynomials; unconstrained
optimisation; Newton methods; steepest descent methods. [2 lectures]
- Numerical linear algebra.
Calculation of eigenvalues and eigenvectors; effective rank of a
matrix; generalised inverse matrices; singular value decomposition;
matrix norms; condition of linear equations. [2 lectures]
- Differential equations.
Initial value problems; Euler's method; Runge-Kutta methods;
multistep methods; predictor-corrector methods; stability theory;
stiff systems. [2 lectures]
Objectives
At the end of the course students should
- have an understanding of basic approximation theory and its
extension to quadrature, fixed-point iteration theory, and elementary
stability theory for numerical solution of initial value ordinary
differential equations
- be able to apply several advanced numerical methods, such as
singular value decomposition
- be able to make an informed choice, in several problem areas,
between diverse methods commonly available in numerical software
libraries
Recommended books
Conte, S.D. & Boor, C. de (1980). Elementary numerical analysis.
McGraw-Hill.
Shampine, L.F., Allen, R.C. Jr & Pruess, S. (1997). Fundamentals of
numerical computing. Wiley.
Next: Easter Term 2004
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Christine Northeast
Thu Sep 4 13:12:26 BST 2003