Lecture Plan

This page will cover details on what material was covered in the specific weeks and lectures.

The current topics entered are a rough time plan for the semester. It might still be object to change. The Material refers to Chapers in the Quarteroni, Sacco, Saleri book.

Week Lecture Topic Material
34 24.08. Introduction Chapter 1
27.08. Preliminaries Chapter 2
35 31.08. Numerical Linear Algebra.
Tridiagonal and Banded Systems, LU, Cholesky and QR decomposition via Gram Schmidt and Givens Rotations, Jacobi, Gauss-Seidel and their relaxations
QSS Ch. 3.2, 3.3
02.09. Ch. 3.5, 3.4, 3.7
36 07.09. Ch. 5.6.1, 4.1, 4.2
09.09. Nonlinear Systems and Numerical Optimisation
Nonlinear Systems of Equations, Jacobian matrix – optimisation, gradient and Hessian matrix; optimality conditions, stationary points, and relation to nonlinear systems of equations; contractive mappings and convergence, bisection and secant method in 1D; Newtons Method and its convergence rate; Modified Newton, especially Broydens Method; Newton, BFGS (incl. Limited Memory), Gradient descent, Conjugate Gradient descent. Armijo line search, Wolfe condition, outlook.

Additional/Alternative Literature: Nocedal & Wright: Numerical Optimisation (Ch. 2.2, 3.1-3.3, 11.1, maybe further 6.1 and even 7.2)
Ch. 7.1, Ch. 7.2
37 14.09.
38 21.09.
39 28.09.
01.10. Interpolation & Splines
Polynomial Interpolation, Lagrange & Newton bases for polynomial vector spaces, interpolation, interpolation error, Newton Form of Interpolation, Neville Algorithm, B splines
Ch. 8.1, 8.2, 8.3, 8.7
40 05.10.
41 12.10. Numerical Integration (or quadrature)
Basic quadrature (Trapezoidal rule, mid point, Simpson, error formulae in self study), Newton-Cotes formulae, Error in Newton Cotes, Composite Newton-Cotes, Extrapolation method, Richardson extrapolation, Romberg Method, Singular and Improper Integrals, unbounded Integration Intervals.
Ch. 9.1-9.3, 9.6, 9.8
42 19.10.
22.10. Orthogonal Polynomials and Applications
recursive construction, Chebyshev polynomials, application to polynomial interpolation. Orthogonal polynomials, Chebyshev, Legendre polynomials and their application to Gaussian integration, Fourier Trigonometric Polynomials and the fast Fourier transform.
Ch. 10.1-10.4, 10.9
43 26.10.
44 02.11. Numerical Solution to Ordinary Differential Equations
Analysis of one step methods, consistency, local error, convergence, absolute stability
Linear multistep methods, local error and order of multistep methods: consistency, error constant, difference equations, zero stability
Predictor-Corrector methods, and if time permits stiff differential equations, Solving nonlinear system in implicit methods
Ch. 11.1-11.3, 11.6-11.8, (11.10)
45 09.11.
46 16.11.
47 23.11. Summary
2021-11-17, Ronny Bergmann