NUMERICAL MATHEMATICS

Summer semester 2024/2025

Lectures: Thursday 12:30-14:00 (room T4:C2-438)
Tutorials: Friday 12:30-14:00 (room T4:A1-405a)

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
HW

Week 2

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

Week 3

Newton's method, example.
Illustration in Matlab.
Video: Newton's method in R and C, Newton's fractal
Recapitulation.
HW

Week 4

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

Week 5

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

Week 6

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

Week 7

Boundary value problem for ordinary differential equations.
Illustration in Matlab.
Video: Gilbert Strang, MIT
HW

Week 8

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

Week 9 ... 21-st of April: Tutorial cancelled (Easter holiday)

Mixed problem for heat equation, Finite difference method.
Graphical illustration.
Illustration in Matlab.
Video: Heat equation

Week 10

Mixed problem for wave equation, Finite difference method.
Illustration in Matlab.
HW from week 9

Week 11 ... 1-st of May: Lecture cancelled (holiday)

HW from week 10

Week 12 ... 8-th of May: Lecture cancelled (holiday)

Recapitulation.

Week 13

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.

Week 14

Assessment test: Thursday 22-nd of May, 12:30 - 14:00, T4:C2-438

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
Logistic equation - Video