# Numerical Mathematics

## Lecture Topics

• Introduction:
Computer Arithmetic, Error Propagation, General Aims of Numerical Mathematics,
More about Computer Arithmetics: Floating Point Operations, Rounding Errors,
Condition of a Problem, Numerical Stability
Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
• Direct Methods for Linear Algebraic Systems:
Linear Systems, Gaussian Elimination, LU Decomposition, Examples,
Programming Technique, Pivoting, Complexity, Cholesky Decomposition;
Vector and Matrix Norms, Condition Number of a Matrix, Basic Error Estimates,
Orthogonal Matrices, Matrices with a Special Structure, Gram-Schmidt Orthogonalisation
QR Factorization (Householder, Givens), LU Decomposition for Tridiagonal Matrices
Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
• Iterative Methods for Linear Algebraic Systems:
A First Example: The Dirichlet Problem, Convergence of One-Step Methods,
Jacobi and Gauss-Seidel Iteration, Error Estimations, Relaxation Methods,
Iterative Schemes at Work, Acceleration of Convergence, Richardson Iteration,
Krylov-Space Methods, Richardson Iteration, Pre-Conditioning, Generalising,
CG Methods (CGNR, CGNE, GMRES)
Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
• Interpolation:
Introduction, Examples, Problems, Lagrange and Newton Interpolation
The Choice of node points, Chebychev Polynomials
Hermite Interpolation, Spline Interpolation: Piecewise Polynomial Interpolation, Natural and B-Splines,
Bernstein Polynomials and Bezier Curves,
Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
Examples, Newton-Cotes Formulas,
Richardson Extrapolation: Romberg Integration
Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
• Nonlinear Equations:
Root-Finding for Nonlinear Equations, Conditioning, Geometric Approach
(Bisection, Chord and Secant Method, Newton's Method), Fixpoint Iteration, Convergence,
Systems of nonlinear equations, Newton's method
Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
• Approximation:
Gauss Approximation, Gram-Schmidt Algorithmus, Legendre Polynomials

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Stand / letzte Änderung: Wed Jan 8 09:42:12 CET 2003