NUMERICAL MATHEMATICS

Summer semester 2024/2025

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

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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

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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.

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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
  

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