Applied Numerical Methods

Course guarantor:

Jan Pšikal

Lecturer:

Jan Pšikal, Pavel Váchal, Alena Zavadilová

Tutor:

Jan Pšikal, Pavel Váchal, Alena Zavadilová

Supervisor:

Department of Laser Physics and Photonics

Synopsis:

For English version use code 12YNME1.

The basic principles of numerical mathematics important for the numerical solution of physical and technical problems are explained. In addition to fundamental numerical tasks, problems relevant to physicists are also included (e.g. solving ordinary differential equations). The exercises take place in a computer lab, featuring demonstrations of various numerical methods, their properties, and applications.

Requirements:

A condition for successfully completing the course is obtaining a sufficient number of points from the credit test. The evaluation is also influenced by the solutions of the assigned tasks during the teaching period in the semester.

Syllabus of lectures:

For English version use code 12YNME1.

1.Numerical mathematics, truncation error, floating point representation of numbers, roundoff error

2.Correctness of problem, condition number, numerical stability; numerical libraries

3.Solution of linear equation systems - direct methods

4.Sparse matrices, iteration methods for linear equation systems; eigensystems

5.Interpolation and extrapolation, interpolation in more dimensions

6.Chebyshev approximation, Chebyshev polynomials, least square approximation

7.Evaluation of functions; sorting

8.Root finding and nonlinear set of equations

9.Search for extremes of functions

10.Numerical integration of functions

11.Random numbers and Monte Carlo integration

12.Ordinary differential equations - initial problem, stiff equations

13.Ordinary differential equations - boundary value problem

Syllabus of tutorials:

The seminars are held in computer laboratory and Python is used as a principle programming language.

1. Floating point representation of numbers, roundoff error, condition number

2. Solution of linear equation systems - direct methods, condition number of matrix

3. Iteration methods for linear equation systems; eigenvectors and eigenvalues

4. Interpolation and extrapolation, cubic spline, Chebyshev polynomials, least square approximation

5. Root finding and nonlinear set of equations

6. Search for extremes of functions

7. Numerical integration of functions

8 .Ordinary differential equations - initial problem, stiff equations

9. Ordinary differential equations - boundary value problem

Study Objective:

Knowledge:

Basic principles of numerical mathematics important for numerical solving of problems important for physics and technology including also ordinary differential equations.

Skills:

Usage of numerical mathematics for solving of practical problems, ability to avoid most common errors.

Study materials:

Key references:

[1] W.H. Press, B.P. Flannery, S.A. Teukolsky, V. H. Vetterling: Numerical Recipes in Pascal (The art of scientific computation), Cambridge University Press, Cambridge 1989 (also versions for C and Fortran).

Recommended references:

[2] A. Ralston, P. Rabinowicz, A First Course in Numerical Analysis, McGraw-Hill 1965 (reprinted by Dover Publiícations, 2001)

[3] R.W. Hamming, Numerical Methods for Scientists and Engineers, 2nd edition, Dover Publiícations 1986

Equipment:

Computer laboratory with Pascal programming language and Matlab program.

Note:

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikace informatiky v přírodních vědách (compulsory course in the program)
• Aplikované matematicko-stochastické metody (compulsory course in the program)
• Jaderné inženýrství - Aplikovaná fyzika ionizujícího záření (PS)
• Aplikovaná informatika (elective course)
• Jaderná chemie (compulsory course in the program)
• Jaderné inženýrství - Jaderné reaktory (PS)
• Radiologická technika (compulsory course in the program)
• jaderné inženýrství - Radioaktivita v životním prostředí (PS)
• Vyřazování jaderných zařízení z provozu (compulsory course in the program)