NUMERICAL MATHEMATICS

Summer semester 2016/2017

Possible dates of exams:

Thursday 18. 5. 2017,  14:15  room KN:A-447
Thursday 25. 5. 2017,  14:15  room KN:A-447
Thursday 8. 6. 2017,  9:00  room KN:A-214
Monday 19. 6. 2017,  9:00  room KN:A-214
Tuesday 27. 6. 2017,  9:00  room KN:A-214
Thursday 14. 9. 2017,  9:00  room KN:A-214

- if you plan to attend, please register at least 2 days in advance by e-mail (if no student is registered for a given date, the exam is cancelled)

Lectures: Thursday 14:15 (room KN:A-447)
Tutorials: Friday 12:30 (room KN:A-447)

Detailed information (in Czech)

Course Schedule

Week 1

Introduction, example pdf.
Principle of iterative methods. Simple iteration method ( fixed point iterations).
Ilustration in Matlab: Fixed point iterations
Matrix properties: norms, spectral radius, symmetry, ...
Ilustration in Matlab: Matrix properties
HW

Week 2

Iterative methods for linear systems:
- fixed point iteration
- Jacobi and Gauss-Seidel iterations, graphical ilustration
Solved problems
Ilustration in Matlab
HW

Week 3

Gradient methods. The steepest descent method.
Ilustration in Matlab..
Approximation by polynomials - the least squares method.
Ilustration in Matlab.
HW

Week 4

Week 5

Substitution of derivatives with finite differences.
Cauchy problem for ordinary differential equations, explicit and implicit Euler's method, Collatz's method.
Ilustration in Matlab.
HW

Week 6

Cauchy problem for ordinary differential equations: one-step methods - local and global discretization errors, consistence, stability, convergence.
Ilustration in Matlab.
Explicit Runge-Kutta methods.
Ilustration in Matlab.
HW

Week 7

Boundary value problem for ordinary differential equations: Finite differences in 1D.
Ilustration in Matlab.
HW

Week 8 (the lecture on Fri 14-th of April is cancelled - Good Friday)

Recapitulation.

Week 9 (the lecture on Thu 20-th of April is cancelled)

Substitution of derivatives with finite differences in 2D.
Dirichlet problem for Poisson equation, Finite difference method.
Ilustration in Matlab.
HW

Week 10

Mixed problem for heat equation, Finite difference method.
Ilustration in Matlab.
HW

Week 11

Mixed problem for wave equation, Finite difference method.
Ilustration in Matlab.
HW

Week 12 (the lecture on Thu 11-th of May is cancelled)

Classification of the 2-nd order linear partial differential equations of two independent variables.
Recapitulation.

Week 13

Assessment test Thu 18-th of May
the results will be available on Fri 19-th of May

Requirements for exams: A level, B level, Examples of theoretical problems: A level, B level

References

T. Petersdorff: Fixed Point Iteration and Contraction Mapping Theorem
Y. Saad: Iterative methods for sparse linear systems ( pdf )
G. Strang: Computational Science and Engineering, selected chapters
C. T. Kelley: Iterative Methods for Linear and Nonlinear Equations, SIAM 1995
T. Petersdorff: Errors for Linear Systems
M. Zeltkevic: Forward and Backward Euler Methods
E. Cheever: Fourth Order Runge-Kutta
D. N. Arnold: Stability, consistency, and convergence of numerical discretizations
Matlab tutorial - Clarkson University - html
A short introduction to Matlab - html
* K. B. Petersen, M. S. Pedersen: The Matrix Cookbook - pdf

Video Lectures

Gilbert Strang: Linear algebra, Unit II: Least Squares, Determinants and Eigenvalues
Gilbert Strang: Computational Science and Engineering I, 2008
Gilbert Strang: Computational Science and Engineering II, 2006