*Lecturer: Dr M.R. O'Donohoe*
(`mro2@cam.ac.uk`)

*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.

Isaacson, E. & Keller, H.B. (1966). *Analysis of Numerical
Methods*. Wiley.