Course detail

Numerical Methods I

FSI-SN1 Acad. year: 2026/2027 Winter semester

Supervisor

doc. Ing. Petr Tomášek, Ph.D.

Department

Institute of Mathematics

Learning outcomes of the course unit

Prerequisites

Differential and integral calculus for functions of one and more variables. Fundamentals of linear algebra. Basic programming skills.

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

Language of instruction

Czech

Aims

Specification of controlled education, way of implementation and compensation for absences

The study programmes with the given course

Programme B-MAI-P: Mathematical Engineering, Bachelor's
branch ---: no specialisation, 4 credits, compulsory

Programme C-AKR-P: , Lifelong learning
branch CZS: , 4 credits, elective

Type of course unit

 

Lecture

26 hours, optionally

Syllabus

1. Introduction to computing: error analysis, computer arithmetic, conditioning of problems, stability of algorithms.
2. Gaussian elimination method. LU decomposition. Pivoting.
3. Solution of special linear systems. Stability and conditioning. Error analysis.
4. Classical iterative methods: Jacobi, Gauss-Seidel, SOR, SSOR.
5. Generalized minimum rezidual method, conjugate gradient method.
6. Lagrange, Newton and Hermite interpolation polynomial. Piecewise linear and piecewise cubic Hermite interpolation.
7. Cubic interpolating spline. Least squares method: data fitting, solving overdetermined systems.
8. Numerical differentiation: basic formulas, Richardson extrapolation.
9. Numerical integration: Newton-Cotes formulas, Romberg's method, Gaussian formulas, adaptive integration.
10. Solving nonlinear equations in one dimension: bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
11. Solving nonlinear systems: Newton's method, fixed point iteration.


12. QR decomposition and singular value decomposition in the least squares method.
13. Orthogonalization methods (Householder transformation, Givens rotations, Gram-Schmidt orthogonalization)

Computer-assisted exercise

26 hours, compulsory

Syllabus

Students create elementary programs in MATLAB related to each subject-matter delivered at lectures and verify how the methods work. Furthermore students individually elaborate semester assignemets.