Numerical linear algebra

Garant předmětu:

Ivana Pultarová

Lecturer:

Ivana Pultarová

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Students are introduced to basic computational methods related to the problems of linear algebra which can be obtained in engineering problems. The following topics are studied. Basics of linear algebra: vectors, matrices, systems of linear equations, solvability. Vector and matrix norms, eigenvalues and eigenvectors. Spectra of matrices. Coordinates with respect to a basis; change of a basis. Schur complement. Symmetric and positive definite matrices. Gauss elimination. LU decomposition. Matrix iterative methods: Jacobi method, Gauss-Seidel method. Gradient methods: method of steepest descent, conjugate gradient method. Convergence criteria and convergence rate. Conditioning of a system of linear equation. Preconditioning methods. Incomplete LU decomposition. Eigenproblems. Gram-Schmidt orthogonalization. Discrete Fourier transformation and its properties. Circulent matrix.

Requirements:

Participating at classes.

Completening homeworks and seminar works.

Syllabus of lectures:

1. Vectors, matrices, systems of linear equations.

2. Norms, inner product.

3. Eigenvalues and eigenvectors. Conditioning.

4. Gauss elimination. Iterative methods.

5. Jacobi method. Gauss-Seidel method. Steepest descent method.

6. Conjugate gradient method.

7. Convergence.

8. Preconditioning.

9. Incomplete LU decomposition.

10. Discrete Fourier transform.

Syllabus of tutorials:

1.-2. Matlab or another solution software.

3. Arrays and matrices. Positive definite matrices.

4.-6. Iterative methods.

7. Convergence

8.-9. Preconditioning. Conjugate gradient method.

10. Matrix decomposition.

Study Objective:

The goal is to improve students§ skill in numerical methods of linear algebra with applications in practical problems.

Study materials:

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans: