Course contents *
Linear systems of equations: direct algorithms, perturbation theory and condition, rounding errors. Sparse matrices. Iterative methods: stationary iterations, Krylov space methods and preconditioning.
Eigenvalue problems: Theory, transformation methods and iterative methods.
Singular value decomposition and its applications.
Nonlinear systems of equations and numerical optimization. Model fitting.
Numerical treatment of initial value problems, boundary value problems, and eigenvalue problems for ordinary and partial differential equations. Discretization by finite differences, finite elements, and finite volumes. Convergence, stability and error analysis.
Application oriented computer labs and a project.
Intended learning outcomes *
An overall aim with this course is to give the student knowledge about how to formulate, use, analyse and implement advanced computer oriented numerical methods to solve problems in numerical algebra and differential equations from different application areas.
After completing the course the student should be able to
1) in numerical algebra
- identify algebra computations, linear and nonlinear, in a practical problem
- implement such a computation, estimate computer resource needs and judge the quality of the results.
- implement special numerical algorithms adapted to the properties of the problem
2) in numerical solution of differential equations
- for a given problem, identify problem type within the area of differential equations, ordinary and partial, and suggest an algorithm for the numerical solution
- utilise and analyze the most important algorithms for the kind of problems presented in this course
- utilise those algorithms from other areas of numerical analysis which are necessary for solving differential equations, e.g. large sparse linear systems of equations, Fourier analysis, etc
- set up and explain some fundamental mathematical models in science which are based on differential equations
- implement the algorithms i a programming language suitable for numerical computation, e.g. Matlab
- utilise computer tools for simulation and visualization of differential equation models in science and engineering.