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 R^{n}.
- Ilustration in Matlab: Fixed point iterations
- Matrix properties: norms (illustration), spectral radius, symmetry, ...
- Ilustration in Matlab: Matrix properties
- HW
- Iterative methods for linear systems:
- fixed point iteration
- Jacobi and Gauss-Seidel iterations, graphical ilustration - Solved problems
- Ilustration in Matlab
- HW
- Approximation by polynomials - the least squares method.
Ilustration in Matlab. - Gradient methods. The steepest descent method.
Ilustration in Matlab.. - HW
- 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
- 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
- The lecture and the tutorial on Mon 2-nd of April are cancelled - Easter Monday
- Boundary value problem for ordinary differential equations: Finite differences in 1D.
- Ilustration in Matlab.
- HW
- Substitution of derivatives with finite differences in 2D.
- Dirichlet problem for Poisson equation, Finite difference method.
- Ilustration in Matlab.
- HW
- Mixed problem for heat equation, Finite difference method.
- Ilustration in Matlab.
- HW
- Mixed problem for wave equation, Finite difference method.
- Ilustration in Matlab.
- HW
- Classification of the 2-nd order linear partial differential equations of two independent variables.
- Recapitulation.
- 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