Numerical Analysis II

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

No. of lectures: 12

Prerequisite course: Numerical Analysis I  

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.

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.

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.

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.

Differential equations.

Initial value problems; Euler's method; Runge-Kutta methods; multistep methods; predictor-corrector methods; stability theory; stiff systems.

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.

Powell, M.J.D. (1981). Approximation Theory and Methods. Cambridge University Press.