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]
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.
