Lecture plan

Week Themes Curriculum Notes
2 Introduction to MATLAB (exercise sheet), ill-conditioned problems and rounding errors, floating point numbers, loss of significance (cancellation errors), Taylor series and O(h) notation. Ch. 1.1, 1.2, 1.3, 1.4
3 Linear systems, Gaussian elimination, the necessity of row exchanges and scaled partial pivoting, analysis of the numerical complexity. Ch. 2.1, 2.2
4 Gaussian elimination for tridiagonal linear systems, matrix factorizations: connection between LU factorization and Gaussian elimination, LDLT factorization for symmetric matrices, Cholesky factorization. Ch. 2.2, 2.3 (not pentadiagonal systems), 8.1
5 Cholesky factorization. Non-linear equations: bisection method, method of false position, fixed-point iterations. Ch. 8.1, 3.1 Cholesky
6 Fixed-point iterations for the solution of non-linear equations, a short excursion to matrix norms, Newton's method, the secant method. Ch. 3.2, 3.3 Fixed Points
7 Eigenvalues, power method (including inverse and shifted power methods), iterative solvers for linear systems (meaning: (damped) Richardson, Jacobi, Gauss-Seidel, and SOR), a short discussion on their convergence properties. Ch. 8.2 (only first 4 pages), 8.3, 8.4 (not conjugate gradient)
8 Polynomial interpolation (Lagrangian interpolation, Newton interpolation and divided differences), interpolation errors. Ch. 4.1, 4.2
9 Minimization of interpolation errors: Chebyshev points, estimation of derivatives, Richardson extrapolation. Ch. 4.2, 4.3
10 Numerical integration: Trapezoid rule, Romberg integration, Simpson rule. Ch. 5.1, 5.2, 5.3
11 Numerical integration: Newton-Cotes rules, approximation errors and convergence rates, Gaussian quadrature. Ch. 5.3, 5.4
12 Numerical integration: Gaussian quadrature & Legendre polynomials. Splines: linear splines, natural cubic splines Ch. 5.4, 6.1, 6.2
13 Least squares, linear regression, and normal equations. Initial value problems: Euler's method. Ch. 9.1, 9.3, 7.1
14 Initial value problems: Runge-Kutta methods, multi-step methods. Ch. 7.2, 7.3, 7.5
15 Implicit methods and stiff ODEs. Summary and questions. Ch. 7.5
2014-04-08, Markus Grasmair