NUMERICAL MATHEMATICS
Summer semester 2019/2020
Exam terms:
- Wed 20-th of May, from 15:00 (Czech time)
- Tue 26-th of May, from 15:00
- Mon 7-th of September, from 9:00
Student must have a valid assessment from tutorials registered in the electronic system KOS and student has to register in the KOS system for the chosen date and level of the exam.
Course Schedule
Week 1- Introduction, example pdf.
- Principle of iterative methods.
- Simple iteration method, or fixed point iterations, in R.
- Fixed point iterations in R^{n} - introduction.
- Matrix properties: norms (illustration), spectral radius.
- Ilustration in Matlab: Fixed point iterations
- Ilustration in Matlab: Matrix properties
- Iterative methods for linear systems:
- fixed point iterations in R^{n}
- Jacobi and Gauss-Seidel iterations, graphical ilustration - Matrix properties: symmetry, positive definitness, diagonal dominance.
- Solved problems
- Ilustration in Matlab
- Approximation by polynomials - the least squares method.
- Ilustration in Matlab.
- Gradient methods. The steepest descent method.
Ilustration in Matlab.. - Video: Gilbert Strang (the least squares from approx. 25-th minute)
- Newton's method.
- Ilustration in Matlab.
- Recapitulation.
- Substitution of derivatives with finite differences.
- Cauchy problem for ordinary differential equation, explicit and implicit Euler's method.
- Cauchy problem for systems of ordinary differential equations.
- Ilustration in Matlab.
- Video: Euler's method - the basics
- Video: Gilbert Strang, MIT
- Order of methods, Collatz (midpoint) method.
- Explicit Runge-Kutta methods - example.
- Ilustration in Matlab.
- Video: Midpoint method - the basics - first 6:30 minutes
- Video: Local vs. Global Truncation Errors - from approx. 5:50 minute on
- Video: Gilbert Strang
- Wiki - Runge-Kutta methods.
- One-step methods - consistence, stability, convergence.
- Stability of Euler's methods: Ilustration in Matlab.
- Video, Gilbert Strang: Lecture I, Lecture II (A level)
- Boundary value problem for ordinary differential equations.
- Ilustration in Matlab.
- Dirichlet problem for Poisson equation, Finite difference method.
- Ilustration in Matlab.
- Video: Laplacian
- Mixed problem for heat equation, Finite difference method.
- Ilustration in Matlab.
- Video: Heat equation
- Mixed problem for wave equation, Finite difference method.
- Ilustration in Matlab.
- Classification of the 2-nd order linear partial differential equations of two independent variables.
- Recapitulation.
- Assessment test: Wed 6-th of May, 15:00 online
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
- Linear transformations and matrices (video)
- 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
- 3BLUE1BROWN SERIES: Differential equations, studying the unsolvable | DE1