Numerical methods

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Course Unit Code714-0087/01
Number of ECTS Credits Allocated5 ECTS credits
Type of Course Unit *Choice-compulsory
Level of Course Unit *Second Cycle
Year of Study *First Year
Semester when the Course Unit is deliveredSummer Semester
Mode of DeliveryFace-to-face
Language of InstructionCzech
Prerequisites and Co-Requisites Course succeeds to compulsory courses of previous semester
Name of Lecturer(s)Personal IDName
CER365doc. Ing. Martin Čermák, Ph.D.
Summary
Basic problems of the numerical mathematics, errors in computations. Solving of
equation f(x)=0: bisection method, regula-falsi, iterative method, Newton´s
iteration, roots of polynomials. Numerical solution of systems of linear
algebraic equations: LU-factorization, iterative methods, condition number of
matrix, ill-conditioned matrices. Numerical solution of systems of nonlinear
equations: Fixed-point iteration, Newton’s method. Interpolation and
approximation of functions: Polynomial interpolation, interpolation by cubic
spline functions, least squares approximation. Numerical integration: Trapezoid
rule, Simpson’s rule, Richardson extrapolation, Monte Carlo method.
Learning Outcomes of the Course Unit
The aim of this course is to acquaint students with the numerical solution of mathematical problems that arise in the other courses of their study and in the technical practice. The main accent lays in explanations of fundamental principles of numerical methods with emphases their general properties. It should lead to the ability in concrete situations to decide whether a numerical procedure is a suitable tool for solving a particular problem. An important ingredient of the course consists in the algorithmic implementation and in the utilization of existing computer programs specialized for numerical computations.

The graduate of this course should know:
- to recognize problems suitable for solving by numerical procedures and to find
an appropriate numerical method;
- to decide whether the computed solution is sufficiently accurate and, in case
of need, to assess reasons of inaccuracies;
- to propose an algorithmic procedure for solving the problem and to choice
a suitable computer environment for its realization.
Course Contents
1. Course contents, the issue of errors, stability of calculations.
2. Solution of nonlinear equations, separation of roots, bisection method, regula-falsi method..
3. Newton's method and fixed-point iterations.
4. Direct methods for solving linear equations, Gaussian elimination and LU-decomposition.
5. Eigenvalues and eigenvectors, numerical calculation.
6. Iterative methods for solving linear equations.
7. Iterative methods for solving nonlinear equations.
8. Interpolation by polynomials.
9. Interpolation by splines.
10. Least squares approximation.
11. Numerical differentiation and integration, Newton-Cotes formulae.
12. Extrapolation in the calculation of integrals. Gaussian integration formulas.
13. Numerical integration by Monte-Carlo method.
14. Reserve.
Recommended or Required Reading
Required Reading:
[1] Forsythe, G., E., Malcolm, M.,A., Moler, B., C.: Computer Methods for
Mathematical Computations. Prentice –Hall, Inc., Englewood Clifs,
N.J. 07632 1977.
[2] Buchanan, J., L., Turner, P., R.: Numerical Method and Analysis.
McGraw-Hill, Inc., New York 1992. ISBN 0-07-112922-7


[1] Boháč,Z., Častová,N.: Základní numerické metody. Skriptum VŠB-TUO,
Ostrava 1997. ISBN 80-7078-975-1
[2] Demidovič,B.,P.,Maron,J.,A.: Základy numerické matematiky. SNTL,
Praha 1966.
[3] Ralston,A.: Základy numerické matematiky. Academia, Praha, 1978.
[4] http://homel.vsb.cz/~kuc14/teach_NM.html
Recommended Reading:
[1] Stoer, J., Burlish, R.: Introduction to Numerical Analysis.
Springer-Verlag, New York, Berlin, Heidelberg 1992. ISBN 0-387-97878-X
[1] Vitásek,E.: Numerické metody.SNTL, Praha 1987
Planned learning activities and teaching methods
Lectures, Individual consultations, Tutorials, Other activities
Assesment methods and criteria
Task TitleTask TypeMaximum Number of Points
(Act. for Subtasks)
Minimum Number of Points for Task Passing
Exercises evaluation and ExaminationCredit and Examination100 (100)51
        Exercises evaluationCredit20 (20)10
                Jiný typ úlohyOther task type20 10
        ExaminationExamination80 (80)30
                Písemná zkouškaWritten examination60 25
                Ústní zkouškaOral examination20 5