NUMERICAL MATHEMATICS

Summer semester 2017/2018

Possible dates of exams:

Monday 21. 5. 2018,  10:45  room KN:A-312
Thursday 31. 5. 2018,  8:30  room KN:A-214
Thursday 7. 6. 2018,  8:30  room KN:A-214
Tuesday 26. 6. 2018,  8:30  room KN:A-214
Tuesday 18. 9. 2018,  8:30  room KN:A-214

Detailed information (in Czech)

Course Schedule

Week 1

Introduction, example pdf.
Principle of iterative methods.
Simple iteration method, or fixed point iteration: in R, in Rⁿ.
Ilustration in Matlab: Fixed point iterations
Matrix properties: norms (illustration), 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

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

The lecture and the tutorial on Mon 2-nd of April are cancelled - Easter Monday

Week 8

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

Week 9

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

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

Week 13

Assessment test: Mon 14-th of May 10:45, KN:A-312

Requirements for exams: A level, B level, examples of theoretical problems: A level, B level,
example of an exam test A and B level: at B level you should expect similar problems to those given as HWs plus some theoretical questions, at A level you should expect more theory and wider range of problems, see requirements above.

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

3Blue1Brown channel: Essence of linear algebra
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