# 714-0087/01 – Numerical methods (NM)

 Gurantor department Department of Mathematics and Descriptive Geometry Credits 5 Subject guarantor doc. Ing. Martin Čermák, Ph.D. Subject version guarantor doc. Ing. Martin Čermák, Ph.D. Study level undergraduate or graduate Requirement Choice-compulsory Year 1 Semester summer Study language Czech Year of introduction 2006/2007 Year of cancellation 2019/2020 Intended for the faculties FBI Intended for study types Follow-up Master
Instruction secured by
CER365 doc. Ing. Martin Čermák, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2

### Subject aims expressed by acquired skills and competences

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.

### Teaching methods

Lectures
Individual consultations
Tutorials
Other activities

### 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.

### Compulsory literature:

[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

### Recommended literature:

[1] Stoer, J., Burlish, R.: Introduction to Numerical Analysis. Springer-Verlag, New York, Berlin, Heidelberg 1992. ISBN 0-387-97878-X

### Way of continuous check of knowledge in the course of semester

Podmínky absolvování předmětu Podmínky pro udělení zápočtu (prezenční studium): - účast ve cvičení, 20 % neúčasti lze omluvit, - odevzdání programů zadaných vedoucím cvičení v předepsané úpravě, - absolvování písemných testů, každý test je možno jednou opravit. Za splnění podmínek získá student 5 bodů. Za testy může získat student 0 - 15 bodů. (Student, který získá zápočet, bude hodnocen 5 - 20 bodů). Podmínky pro udělení zápočtu (kombinované studium): Za účast na konzultacích může student získat 5 - 20 bodů, v případě neúčasti může student získat 5 bodů za zpracování zadaného programu. Požadavky ke zkoušce: Podmínkou pro účast na zkoušce je zapsaný zápočet z příslušného předmětu. Písemná část zkoušky bude hodnocena 0 - 60 body, za její úspěšné absolvování bude považován zisk 25 bodů. Ústní část zkoušky bude hodnocena 0 - 20 body, za její úspěšné absolvování bude považován zisk 5 bodů. Po sečtení bodů získaných za zápočet, písemnou a ústní část zkoušky bude student hodnocen výborně, velmi dobře, dobře a nevyhověl, podle tabulky studijního a zkušebního řádu VŠB - TUO. Pro zapsání zkoušky podle tabulky musí student úspěšně absolvovat obě části kombinované zkoušky a dosáhnout potřebného počtu bodů. Bodové hodnocení: 86 - 100 výborně 66 - 85 velmi dobře 51 - 65 dobře 0 - 50 nevyhověl Soubor otázek k teoretické části zkoušky: I. Problematika numerických výpočtů 1. Vznik a klasifikace chyb. Zobrazení čísel v počítači. Aproximace čísel. Absolutní, relativní chyby, jejich odhady. 2. Chyby aritmetických operací. 3. Podmíněnost úloh. Numerická stabilita algoritmů. II. Řešení algebraických a transcendentních rovnic 4. Separace kořenů rovnice f(x) = 0. Věty o kořenech algebraických rovnic. 5. Numerické metody řešení (půlení intervalu, regula-falsi, Newtonova metoda, iterační metoda). III. Řešení soustav lineárních rovnic 6. Přímé metody (LU rozklad). 7. Normy matic, podmíněnost matic. Vlastní čísla, vlastní vektory matice. Spektrální poloměr. Posloupnosti vektorů, jejich konvergence. 8. Iterační metody. Prostá a Seidelova iterační metoda. IV. Interpolace a aproximace funkcí 9. Interpolace funkcí. Interpolace algebraickými polynomy, splajnová interpolace. 10. Metoda nejmenších čtverců. 11. Čebyševovy aproximace. V. Výpočet určitého integrálu 12. Lichoběžníková formule. Simpsonova formule. Složené formule. 13. Odhad chyby Richardsonovou extrapolací. Rombergova integrace. 14. Stochastické početní metody. Pseudonáhodná čísla a jejich generátory. Výpočet integrálů metodou Monte Carlo. VI. Řešení soustav nelineárních rovnic 15. Metoda postupných aproximací, podmínky konvergence. 16. Newtonova metoda.

### E-learning

www.studopory.vsb.cz mdg.vsb.cz

### Other requirements

No more requirements are put on the student.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

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.

### Conditions for subject completion

Full-time form (validity from: 2014/2015 Winter semester, validity until: 2019/2020 Summer semester)
Min. number of pointsMax. počet pokusů
Exercises evaluation and Examination Credit and Examination 100 (100) 51
Exercises evaluation Credit 20 (20) 10
Jiný typ úlohy Other task type 20  10
Examination Examination 80 (80) 30 3
Písemná zkouška Written examination 60  25
Ústní zkouška Oral examination 20  5
Mandatory attendence participation:

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Conditions for subject completion and attendance at the exercises within ISP:

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### Occurrence in study plans

Academic yearProgrammeBranch/spec.Spec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2019/2020 (N3908) Fire Protection and Industrial Safety (3908T005) Engineering Safety of Persons and Property P Czech Ostrava 1 Choice-compulsory study plan
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2018/2019 (N3908) Fire Protection and Industrial Safety (3908T005) Engineering Safety of Persons and Property P Czech Ostrava 1 Choice-compulsory study plan
2018/2019 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2018/2019 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2017/2018 (N3908) Fire Protection and Industrial Safety (3908T005) Engineering Safety of Persons and Property P Czech Ostrava 1 Choice-compulsory study plan
2017/2018 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2017/2018 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2016/2017 (N3908) Fire Protection and Industrial Safety (3908T005) Engineering Safety of Persons and Property P Czech Ostrava 1 Choice-compulsory study plan
2016/2017 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2016/2017 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2015/2016 (N3908) Fire Protection and Industrial Safety (3908T005) Engineering Safety of Persons and Property P Czech Ostrava 1 Choice-compulsory study plan
2015/2016 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2015/2016 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2014/2015 (N3908) Fire Protection and Industrial Safety (3908T005) Engineering Safety of Persons and Property P Czech Ostrava 1 Choice-compulsory study plan
2014/2015 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2014/2015 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2013/2014 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2013/2014 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2012/2013 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2011/2012 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2011/2012 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2010/2011 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2010/2011 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2010/2011 (N3908) Fire Protection and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2009/2010 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2009/2010 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2009/2010 (N3908) Fire Protection and Industrial Safety (3908T002) Safety Engineering P Czech Ostrava 1 Choice-compulsory study plan
2008/2009 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2008/2009 (N3908) Fire Protection and Industrial Safety (3908T002) Safety Engineering P Czech Ostrava 1 Choice-compulsory study plan
2008/2009 (N3908) Fire Protection and Industrial Safety (3908T007) Safety and Security Planning P Czech Ostrava 1 Choice-compulsory study plan
2007/2008 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety (30) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan
2007/2008 (N3908) Fire Protection and Industrial Safety (3908T002) Safety Engineering P Czech Ostrava 1 Choice-compulsory study plan
2006/2007 (N3908) Fire Protection and Industrial Safety (3908T006) Fire Protection Engineering and Industrial Safety (30) Fire Protection Engineering and Industrial Safety P Czech Ostrava 1 Choice-compulsory study plan

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