Computernumerik

# Computernumerik Prof. J.M. Melenk

Ein paar Beispiele bei denen schlechte Numerik katastrophale Folgen hatte.

Ein paar Software bugs mit Folgen

# exercises:

• Gruppe 1: MON 10-10:45
• Gruppe 2: TUE 10-10:45
• Gruppe 3: TUE 10-10:45
• Gruppe 4: THU 10-10:45

# Exercise sheets

## codestuecke

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

## Content

• 5.10.21: Chap. 1.1-1.2: introduction to polynomial interpolation: existence and uniqueness
• 6.10.21 (CSE): Chap. 1.3: Horner scheme
• 12.10.21: Chap. 1.4 (Neville scheme), Chap. 1.5 (a simple error estimate for polynomial interpolation) and application to extrapolation of one-sided difference quotients
• 13.10.21 (CSE): Chap. 1.8 (splines)
• 19.10.21: Chap. 1.6 (extrapolation of functions with special structure), Chap. 1.7 (Chebyshev interpolation), Chap 2.0 (introduction to numerical integration)
• 20.10.21 (CSE): Chap. 1.10.1 (trigonometric interpolation)
• 26.10.21 (postponed to 27.10.21): Chap. 2.1 (Newton Cotes formulas, convergence results for trapezoidal and Simpson rules, Romberg extrapolation)
• 27.10.21: Chap. 1.10.2 (FFT)
• 2.11.21 (postponed to 3.11.21): Chap. 2.3 (an adaptive quadrature algorithm), Chap 2.4 (Legendre polynomials and a glance at Gaussian quadrature)
• 3.11.21 (CSE): Chap. 1.10.3+1.10.4 (applications of FFT like fast multiplication of large numbers)
• 9.11.21: Chap. 2.5 (comments on trapezoidal rule), 2.6 (quadrature in 2D), Chap. 3 (conditioning and stability)
• 10.11.21 (CSE) Chap 2.7 (computing Gauss points and weights; Gauss-Jacobi quadrature)
• 16.11.21 Chap 4.1 (lower and upper triangular matrices), Chap 4.2 (classical Gaussian elimination), Chap 4.3 (LU-factorization via Crout),
• 17.11.21 (CSE) Chap 4.6 Householder method for QR factorization
• 23.11.21 Chap 4.4 (Gaussian elimination with pivoting), Chap 4.5 (condition number of a matrix), Chap 5.1 (least squares methods with the method of normal equations)
• 24.11.21 (CSE) Householder QR-factorization with pivoting, QR-factorization with Givens rotations
• 30.11.21 Chap 5.2 (least squares methods with QR-factorization), Chap 5.3 (SVD and its properties),
• 1.12.21 (CSE) Chap 5.3.5 (Moore-Penrose Pseudoinverse)
• 7.12.21 Chap 6.1 (Newton's method in 1d), Chap 6.2 (convergence of fixed point iterations), Chap 6.3 (Newton's method in multi-d), Chap 6.4 (implementational aspects)
• 8.12.21 (CSE) Chap 6.7.1 Broyden's method
• 14.12.21 Chap 6.5.2 (descent methods), Chap 6.5.3 globalized Newton's method, Chap 6.5.4 (Gauss-Newton method)
• 15.12.21 (CSE) Chap 6.8 (unconstrained minimization problems: gradient methods, trust region methods)
• 11.1.22 Chap 7.1 (power method), Chap 7.2 (inverse iteration, iteration with shift, Rayleigh quotient iteration), Chap 7.3 (error estimates, stopping criteria)
• 12.1.22 (CSE) Chap 7.6 (Jacobi method to compute eigenvalues)
• 18.1.22 Chap 7.4 (orthogonal iteration), chap 7.5 (basic QR algorithm)
• 19.1.22 (CSE) Chap 7.8 (QR algorithm with shift)

## Literature

• current version of class notes (will be irregularly updated during the semester)
• further literature:
• W. Dahmen, A. Reusken: Numerik für Ingenieure und Naturwissenschaftler
• 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
• R. Plato: Numerische Mathematik kompakt (Vieweg)