NUMERICAL MATHEMATICS
Summer semester 2020/2021
Lectures: Thursday 10:45-12:15, online
Tutorials: Thursday 14:15-15:45, online
Course Schedule - Lectures
Week 1- Introduction, example pdf.
- Fixed point iterations: in R - graphical illustration, in R^{n} - introduction.
- Vector norms - illustration.
- Matrix properties: norms, spectral radius.
- Ilustrations in Matlab:
- Matrices - what are they? Video (I recommend the whole course on linear algebra).
- 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
- Ilustrations in Matlab:
- Powers of a matrix - graphically
- Iterative methods for linear systems
- 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
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
- I. Berg: Comparison of RK methods
- Matlab tutorial - Clarkson University - html
- * M.K.Jain, S.R.K.Iyengar, R.K.Jain: Numerical Methods - Problems and Solution
- * 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