Computernumerik

# Computernumerik Prof. J.M. Melenk

Ein paar Beispiele bei denen schlechte Numerik katastrophale Folgen hatte.

Ein paar Software bugs mit Folgen

# Exercise sheets

## codestuecke

• gauleg.m produces Gauss points and weights for quadrature on (-1,1)
• my_cg.m the classical CG-method

## Content

• 1.10.19: Chap. 1.1-1.2: introduction to polynomial interpolation: existence and uniqueness, Neville scheme chap1.pdf
• 2.10.19 (CSE): Chap 1.3: Newton representation of interpolating polynomial, Horner scheme
• 7.10.19: Chap 1.4-1.6: error estimates for polynomial interpolation, Romberg extrapolation, extrapolation of one-sided and symmetric difference quotients chap1_vollstaendig.pdf
• 8.10.10 (CSE): Chap 1.8: classical cubic spline
• 15.10.19: Chap 1.7: Chebyshev interpolation and Chap 2.1: Newton-Cotes formulas, convergence of trapezoidal and Simpson rule
• 16.10.19 (CSE): Chap 1.10: trigonometric interpolation chap1a.pdf
• 22.10.19: Chap. 2.3 (adaptivity), Chap 2.4 (Gaussian integration)
• 29.10.19: Chap. 2.2 (Romberg extrapolation of trapezoidal rule), Chap. 2.5 (periodic integrands), Chap. 2.6 (quadrature on triangles/squares)
• 30.10.19: (CSE) FFT (see chap1a.pdf)
• 5.11.19: Chap 3: conditioning and stability
• 6.11.19: (CSE) application of FFT: fast convolution of sequences (see chap1a.pdf)
• 12.11.19: Chap. 4.1-4.2 (Gaussian elimination), Chap. 4.3 (LU-factorization a la Crout method, banded matrices, Cholesky factorization, skyline matrices) chap4.pdf (updated!)
• 13.11.19: (CSE): Chap. 4.6 (QR-factorization using Householder reflections)
• 19.11.19: Chap. 4.4: partial pivoting, Chap 4.5 condition numbers of matrices Chap 5.1: Least Squares Methods with normal equations chap5.1.pdf
• 20.11.19 (CSE): Chap 4.6: QR-factorization with pivoting, QR-factorization with Givens rotations
• 26.11.19: Chap 5.2: least squares methods with QR-factorization, Chap 5.3: SVD, solving underdetermined systems, Chap 6.1: Newton's method in 1D
• 27.11.19 (CSE) Chap 5.3.5: Moore-Penrose pseudoinverse
• 3.12.19: Chap. 6.2 (fixed point iterations), Chap 6.3 (Newton's method in multi-d), Chap 6.4 (error estimators); full chap 6
• 4.12.19 (CSE): Chap 6.6 (Broyden's method)
• 10.12.19: Chap 6.5: globalization of Newton's method by minimizing F^\top F, Chap 7.1 (power method)
• 11.12.19: Chap 6.7 (CSE): descent method for quadratic functions; a sketch of trust region methods
• 16.12.19: Chap 7.2, 7.3 inverse iteration, Rayleigh quotient method, Thm. of Bauer-Fike full chap 7
• 17.12.19: Chap 7.6 (CSE): Jacobi method for eigenvalue computation
• 7.1.20: Chap 7.4, 7.5 orthogonal iteration and basic QR-method
• 8.1.20: Chap 7.7 (CSE) QR-method with shift
• 14.1.20: Chap 8.1 (CG-method); chap8.pdf
• 15.1.20: Chap 8.2 (CSE): GMRES
• 21.1.20: Chap 9.1: explicit and implicit Euler method, convergence; chap9.pdf
• 22.1.20: Chap 9.3 (CSE): multistep methods (Adams and BDF)
• 28.1.20: Chap 9.2: explicit and implicit RK-methods
• 29.1.20: chap 9.2 (CSE): concept of A-stability

## Literature

Class notes of a previous semester can be found here (partly in German) Weitere Literatur ist:
• Skript von G. Schranz-Kirlinger (wird im TUWEL zur Verfuegung gestellt)
• R. Plato: Numerische Mathematik kompakt (Vieweg)
• Numerical Recipes (Sammlung von C-Routinen fuer Numerik) gibt es jetzt auch als on-line Buch! Neben den C-Routinen werden die Algorithmen auch kurz "hergeleitet" und beschrieben