NUMERICAL MATHEMATICS
Summer semester 2023/2024
Lectures: Monday 9:00-10:30 (room KN:A-309)
Tutorials: Monday 14:15-15:45 (room KN:A-447)
Course Schedule
Week 1- Introduction.
- Fixed point iterations: in R with graphical illustration, in Rn - introduction.
- Vector norms - illustration.
- Matrices - what are they? Video (I recommend the whole
course on linear algebra).
Graphically: 2x2 matrices as linear transformations - Matrix properties. Graphically: Spectral norm and spectral radius.
- Illustrations in Matlab:
- Fixed point iterations
- Norms and other properties of matrices
- Matrices as 2D transformations - graphically
- Matrix norms - graphically
- Logistic equation - Video
- HW
- fixed point iterations in Rn
- Iterative methods for linear systems:
- graphically: powers of a matrix, Jacobi and Gauss-Seidel iterations
- Solved problems
- Illustrations in Matlab:
- Powers of a matrix - graphically
- Iterative methods for linear systems
- HW
- Newton's method, example.
- Illustration in Matlab.
- Video: Newton's method in R and C, Newton's fractal
- Recapitulation.
- HW
- Substitution of derivatives with finite differences.
- Cauchy problem for first-order ordinary differential equation:
explicit and implicit Euler's method, midpoint (Collatz) method. - Existence and uniqueness of exact solution: Example.
- Illustration in Matlab.
- Video:
Euler's method - the basics
Video: Midpoint method - the basics - first 6:30 minutes
- HW
-
Cauchy problem for systems of ODEs:
explicit and implicit Euler's method, midpoint (Collatz) method. - Explicit Runge-Kutta methods - example.
- One-step methods graphically
- Illustration in Matlab.
- Wiki - Runge-Kutta methods.
- HW
- One-step methods - consistence, stability, convergence.
- Behavior of errors.
- Stability of Euler's methods: Illustration in Matlab.
- Video: Gilbert Strang (the first 25 minutes)
- Video, Gilbert Strang: Lecture I, Lecture II
- Recapitulation.
- HW
- Boundary value problem for ordinary differential equations.
- Illustration in Matlab.
- Video: Gilbert Strang, MIT
- HW
- Classification of the 2-nd order linear partial differential equations of two independent variables.
- Dirichlet problem for Poisson equation, Finite difference method.
- Illustration in Matlab.
- Video: Laplacian
- HW
- Mixed problem for heat equation, Finite difference method.
- Graphical illustration.
- Illustration in Matlab.
- Video: Heat equation
- HW
- Mixed problem for wave equation, Finite difference method.
- Illustration in Matlab.
- HW
- Approximation by polynomials - the least squares method.
- Illustration in Matlab.
- Gradient methods. The steepest descent method.
Illustration in Matlab. - Video: Gilbert Strang (the least squares from approx. 25-th minute)
- HW
- Recapitulation.
- Assessment test: Monday 13-th of May, 9:00 - 10:30, KN:A-309
- Information:
- You should be present 10 minutes before the beginning of the test, so that I can give you some short instructions and you are prepared to start writing at 9:00.
- No electronic devices are allowed (calculator, notebook, watch, ... etc.). You can use only a sheet of paper which you will be given and your ballpoint pen.
Exams: Requirements for exams. At exam, you should expect similar problems to those given as HWs together with theoretical questions like these, see also requirements above. Examples of exam tests.
References
- T. Petersdorff: Fixed Point Iteration and Contraction Mapping Theorem
- Y. Saad: Iterative methods for sparse linear systems ( pdf )
- J. R. Chasnov: Numerical Methods for Engineers
- 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
- I. Berg: Comparison of RK methods
- Joel Feldman: Variable Step Size Methods
- Matlab tutorial - Clarkson University - 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