Numerical Analysis

Abstract

This course covers two different areas of numerical analysis:1) Iterative methods for large sparse linear systems of equations.2) Numerical methods for ordinary differential equations and systems of such equations.

Objective

The students should learn the ideas on which the numerical methods in the two covered areas (iterative methods for linear equations and numerical methods for ordinary differential equations) are based on. Not only the motivation and derivation of algorithms matters, but also their analysis, like proofs of convergence.

Content

1) Iterative methods for large sparse linear systems of equations: Classical methods like Jacobi and Chebyshev iteration; the general notion of Krylov space methods; the conjugate gradient method; the connection with the symmetric Lanczos algorithm; the biconjugate gradient method; the Arnoldi process and the MinRes and GMRes algorithms. 2) Numerical methods for ordinary differential equations and systems of such equations: The classical Euler method and its convergence; Runge-Kutta methods; linear multistep methods; order of convergence and stability; stiff differential equations; A-stability; symplectic methods.

Resources