Numerical Methods 1

Course guarantor:

Pavel Váchal

Lecturer:

Jan Vábek, Pavel Váchal

Tutor:

Martin Jirka, Dominika Jochcová, Jiří Löffelmann, Jan Vábek, Pavel Váchal

Supervisor:

Department of Laser Physics and Photonics

Synopsis:

The course explains the basic principles of numerical mathematics important for numerical solving of problems important for physics and technology. Methods for solution of tasks very important for physicists (ordinary differential equations, random numbers) are included in addition to the basic numerical methods. Integrated computational environment MATLAB is used as a demonstration tool. The seminars are held in computer laboratory.

Requirements:

Initial knowledge:

- Basic knowledge of linear algebra and mathematical analysis (e.g., practical application of Taylor expansion; solution of systems of linear equations; knowledge of quadratic forms, norms and scalar products, eigenvalues and eigenvectors; use of substitution in integration; fundamentals of multivariable analysispartial derivatives, Taylor expansion, ...) to the extent provided by the respective FNSPE courses (any of the mathematical tracks in the first three semesters).

- Elementary knowledge of programming (equivalent to the scope of the course 18ZPRO/18YZPRO).

Programming languages:

Python and MATLAB are used for demonstrations during lectures and tutorials. These languages are also preferred for students to use when solving the tutorial and assessment problems. Standard programming languages in numerics are also allowed (C, C++, Fortran, Java, Julia). Other languages may be used after approval by the teachers of the class.

Requirements for successful completion of the course:

To be awarded the class assessment, you must:

- Individually solve a small project related to the content of the tutorials. A correct solution of the project is then either presented to other students or accompanied by a report. The fulfilment is controlled by a tutor of the class.

- Have no more than two tutorial absences. Two more absences are possible with a second small project (same conditions as in the previous point). More absences are allowed only with a formal medical (or similar) justification and extra work assessed individually.

Exam:

- Oral exam covering the content of the course.

- The exam may be taken conditionally before the end of the exam period without the assessment awarded; the result is validated after the assessment is received. After this date, the exam may be taken only with the assessment.

Syllabus of lectures:

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 PASCAL is used as a principle programming language and system MATLAB is applied for demonstrations.

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

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

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

4.Interpolation and extrapolation, cubic spline

5.Chebyshev approximation, Chebyshev polynomials, least square approximation

6.Evaluation of functions

7.Root finding and nonlinear set of equations

8.Search for extremes of functions

9.Numerical integration of functions

10.Ordinary differential equations - initial problem, stiff equations

11.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 choose routines from numerical libraries and 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 C++ (The art of scientific computing), Cambridge University Press, Cambridge, 3rd edition 2007 (also versions for C, 2nd edition 1993 and Fortran, 2nd edition 1993) (available at http://www.numerical.recipes/oldverswitcher.html).

Recommended references:

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

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

Equipment:

Computer laboratory.

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:

• Physical Engineering - Computational physics (PS)
• Quantum Technologies (compulsory course in the program)
• Nuclear and Particle Physics (compulsory course in the program)
• Physical Engineering - Plasma Physics and Thermonuclear Fusion (PS)
• Physical Engineering - Solid State Engineering (PS)
• Physical Engineering - Laser Technology and Photonics (PS)