MATH3474 Course Overview for Session 2011-12
This 10-credit course has the
following content. All supporting material, i.e. examples sheets, printed
handouts and Maple worksheets, can be obtained here.
This course is the bridge between MATH2600 Numerical Analysis and MATH3475 Modern Numerical Methods and MATH5476M Advanced Modern
Numerical Methods (next given in
2011-12).
The course has the following broad content:
a tabular résumé of lecture times and titles, including detailed lecture-by-lecture information,
can be downloaded in either PostScript or PDF format. There are 22 lectures and 5
examples classes
i. Approximation Theory (11 lectures): polynomial interpolation; Lagrange interpolation; Newton divided differences; interpolation errors; Lp norms; Weierstrass' theorem; minimax approximations; Chebyshev equioscillation theorem; de la Vallée-Poussin theorem; Chebyshev polynomials for least-squares, near-minimax, interpolation and forced-oscillation approximations; spectrally accurate evaluation of Fourier coefficients.
ii.
Numerical
Differentiation (5 lectures): 1-D
finite differences of arbitrary order and accuracy; FD operators; implicit FD
formulae; regular and irregular meshes; molecules and stencils; 2-D FD
formulae; first- and higher-order approximations to Laplacian;
Poisson equation and Mehrstellenverfahren;
higher-order multidimensional derivatives.
iii.
Numerical
linear algebra (6 lectures): matrix
and vector norms; spectral radius; diagonal dominance; Gerschgorin's
and Brauer's theorems; sparse systems; tridiagonal systems and Cholesky
factorisation; Jacobi, Gauss-seidel and SOR
iteration; theoretical convergence estimates; optimum over-relaxation;
theoretical optimum for 2-cyclic matrices; solution of elliptic Dirichlet and Neumann BVPs;
chessboard enumeration; Richardson extrapolation.
