Course detail
Numerical Methods I
FSI-SN1 Acad. year: 2025/2026 Winter semester
The course represents the first systematic explanation of selected basic methods of numerical mathematics. Passing this course, students obtain basic knowledge necessary for further study of more specialised areas of numerical mathematics.
Main topics: Direct and iterative methods for linear systems. Interpolation. Least squares method. Numerical differentiation and integration. Nonlinear equations. The students will demonstrate the acquainted knowledge by elaborating the semester assignment.
Language of instruction
Czech
Number of ECTS credits
4
Supervisor
Department
Entry knowledge
Differential and integral calculus for functions of one and more variables. Fundamentals of linear algebra. Programming in MATLAB.
Rules for evaluation and completion of the course
COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in practicals. Elaboration of tasks, where the students prove their knowledge acquired. At least half of all possible 30 points in a credit test using also own programs.
FORM OF EXAMINATIONS: The exam is of test (max. 75 pts.) and oral part (max 25 pts.). As a result of the exam students will obtain 0--100 points.
FINAL COURSE CLASSIFICATION is based on the exam point classification: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.
Attendance at lectures is recommended, attendance at seminars is required. Lessons are planned according to the week schedules. Absence from lessons may be compensated for by the agreement with the teacher supervising the seminars.
Aims
The aim of the course is to familiarise students with some basic numerical methods. Substantial emphasis is also put on a computer implementation of individual methods. Students ought to understand the essence of particular methods and to realise their advantages and drawbacks. Attention is also paid to the issues of stability and conditionality of individual numerical problems. An important part of the course is independent work on assigned projects.
Students will be made familiar with the basic collection of numerical methods, namely with direct and iterative methods for systems of linear equations, with interpolation, least squares, numerical derivation and integration and with methods for nonlinear equations. Students will verify and deepen the acquired knowledge by processing several projects.
The study programmes with the given course
Programme B-MAI-P: Mathematical Engineering, Bachelor's, compulsory
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. QR decomposition and singular value decomposition in the least squares method.
9. Orthogonalization methods (Householder transformation, Givens rotations, Gram-Schmidt orthogonalization)
10. Numerical differentiation: basic formulas, Richardson extrapolation.
11. Numerical integration: Newton-Cotes formulas, Romberg's method, Gaussian formulas, adaptive integration.
12. Solving nonlinear equations in one dimension: bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
13. Solving nonlinear systems: Newton's method, fixed point iteration.
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.