Course content

Programming

Functional programming and vector notation in Matlab, recursion

Matrix algebra

Solvability of linear systems, norms, eigenvalues and eigenvectors, spectral radius

Eigenvalue problems

How shift, inversion, powers and similarity transforms changes spectrum, diagonalisation, triangularisation, Gershgorin discs, QR factorisation using Householder transformations, the QR method, the power method and invers iteration

Ordinary differential equations

Finite difference and finite element methods for boundary value problems

Partial differential equations

Classification, well-posedness, characteristics, finite difference methods, consistency, stability (CFL condition, Fourier method, energy method, maximum principle), convergence

Iterative solvers for linear systems

Fixed point iteration, the Jacobi and Gauss-Seidel methods

Least squares problems

Polynomial fitting, normal equations, orthogonal projection is the best approximation, Gram-Schmidt, overdetermined systems and QR factorisation

Fast Fourier transform

Discrete Fourier transform, the FFT algorithm, matrix and operator notation