Numerical Methods

Orthogonal polynomials
Linear multistep methods for ODEs
- Explicit and implicit Adams’ methods
- Local error and stability
- Convergence
- Variable step size multistep methods
- General linear multistep methods
Iterative methods for sparse linear systems
- General projection methods
- GG and GMRES
- Preconditioning strategies
- Multigrid methods
Iterative methods for eigenvalue problems
- Power method
- Perron-Frobenius theorem
- Subspace iteration method and QR method
- Krylov method for Ritz values

Homeworks and labs:

Most of the exercises proposed below are taken from E Hairer, G Wanner, S P Nørsett, Solving Ordinary Differential Equations whereas Matlab labs problems are based on the Educational Matlab database of TU Munich.

Link	Summary	Type
H1	Generate orthogonal polynomials using the three term recurrence formula	Matlab
H2	Exercises on explicit and implicit Adams’ methods	Homework
H2	Implement explicit and implicit Adams’ methods for arbitrary number of steps	Matlab
H4	Implement PEC implicit Adams with 6 steps as a solver for the Lorenz attractor system of ODE, coupled with an implicit 6 steps Runge-Kutta method	Matlab
H5	Exercises on stability, local error, order	Homework
H6	Implicit Adams method with adaptive step size	Matlab
H7	Exercises on CG and GMRES	Homework
H8	Precondintioned Conjugate Gradient method	Matlab
H9	Power method and Perron-Frobenius theorem	Matlab

Books of reference:

E Hairer, G Wanner, S P Nørsett – Solving Ordinary Differential Equations I
Y Saad – Iterative methods for Sparse Linear Systems – Free Online Version
Y Saad – Numerical methods for large eigenvalue problems – Free Online Version