# 714-0287/02 – Numerical Methods and Statistics (NMaS)

 Gurantor department Department of Mathematics and Descriptive Geometry Credits 4 Subject guarantor RNDr. Jana Staňková, Ph.D. Subject version guarantor RNDr. Jana Staňková, Ph.D. Study level undergraduate or graduate Requirement Compulsory Year 1 Semester winter Study language Czech Year of introduction 2006/2007 Year of cancellation 2019/2020 Intended for the faculties FAST Intended for study types Follow-up Master
Instruction secured by
STA50 RNDr. Jana Staňková, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Part-time Credit and Examination 14+0

### Subject aims expressed by acquired skills and competences

The first part of this course is dedicated to finding numerical solutions of mathematical problems. These problems can arise from other courses as well as from practice. The main emphasis lays in explanation of fundamental principles of numerical methods and of their general properties. The students learn how to decide which numerical procedure is a suitable tool for solving a specific problem. An important ingredient of the course is algorithmic implementation of the learned numerical methods. The students learn how to use existing software specialized for numerical computations, too. The second part of the course deals with basic probabilistic notions and with the way in which to understand these notions from both theoretical and practical points of view. The students learn the statistical way of thinking as a means of understanding real-life processes. The basic methods of collecting and analyzing statistical data are introduced. The students are taught how to use these general methods to solve the problems arising from other courses of their study and from practice. The graduate of this course should be able: * to recognize problems solvable by numerical procedures and to find an appropriate numerical method; * to decide whether the obtained numerical solution is accurate enough and, if it is not the case, to assess the reasons of inaccuracies; * to propose an algorithmic procedure to solving a problem and to choose a suitable software for its realization; * to understand and to use basic notions from combinatorics and probability theory; * to formulate questions, which can be answered using the given data; to learn the principles of collecting data, processing data, and presenting relevant values and results for this purpose; * to choose and use suitable statistical methods for data analysis; * to draw conclusions and predictions on the basis of the obtained data.

### Teaching methods

Lectures
Individual consultations
Tutorials
Other activities

### Summary

The first part of this course deals with selected issues in numerical computations (including sources and types of numerical errors, conditionality of certain problems and algorithms), with methods for solving algebraic and transcendent equations, with solving systems of linear equations, with interpolation and approximation of functions, with numerical computations of integrals, and with Cauchy problems for ordinary differential equations. The second part of this course is devoted to the processing of a statistical data set with one argument (characteristics of a statistical data set, processing of large statistical data sets, estimates of parameters of a data set). and to testing of statistical hypotheses.

### Compulsory literature:

Kučera, R.: Numerické metody. VŠB-TU Ostrava 2007, na www.studopory.vsb.cz, mdg.vsb.cz/M,ISBN 80-248-1198-7. Otipka, P.,Šmajstrla V.: Pravděpodobnost a statistika VŠB-TU Ostrava 2007, na www.studopory.vsb.cz, mdg.vsb.cz/M,ISBN 80-248-1198-7.

### Recommended literature:

Harshbarger, Ronald; Reynolds, James: Calculus with Applications, D.C. Heath and Company 1990, ISBN 0-669-21145-1

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

Soubor otázek: 1. Problematika numerických výpočtů . Zdroje a typy chyb. 2. Podmíněnost úloh a algoritmů. 3. Metody řešení algebraických a transcendentních rovnic. 4. Metoda půlení intervalu. 5. Iterační metoda řešení rovnic. 6. Metoda Newtonova, 7. Metoda regula-falsi. 8. Kombinovaná metoda. 9. Přímé metody řešení. soustav lineárních rovnic 10. Iterační metody (prostá, Seidelova). řešení. soustav lineárních rovnic 11. Norma matice. 12. Interpolace a aproximace funkcí. 13. Aproximace - metoda nejmenších čtverců. 14. Lagrangeův interpolační polynom,. 15. Newtonův interpolační polynom. 16. Interpolace spline - funkcemi. 17. Numerický výpočet integrálu. 18. Newton-Cotesovy kvadraturní vzorce. 19. Složené kvadraturní vzorce. Odhad chyby. 20. Richardsonova extrapolace.. 21. Počáteční úlohy pro obyčejné dif. rovnice. 22. Jednokrokové metody. 23. Eulerova metoda. 24. Odhad chyby metodou polovičního kroku. 25. Metody Rungova-Kuttova typu. Odhad chyby aproximace. 26. Charakteristiky statistického souboru s jedním argumentem. 27. Zpracování rozsáhlého statistického souboru. 28. Odhady parametrů základního souboru. 29. Základní soubor, náhodný výběr. 30. Bodové odhady parametrů základního souboru. 31. Intervalové odhady parametrů základního souboru. 32. Testy dobré shody. 33. Pearsonův test 2 dobré shody. 34. Kolmogorův test pro jeden výběr. 35. Kolmogorovův – Smirnovův test pro dva výběry. Základní informace o předmětu: Bodové hodnocení: Získané body známka 86 - 100 výborně 66 - 85 velmi dobře 51 - 65 dobře 0 - 50 nevyhověl Rozdělení bodů: cvičení max. 20 písemná zkouška max. 60 teoretická zkouška max. 20 V průběhu semestru budou napsány tři kontrolní práce a každý student dostane zadány ke zpracování dva programy. a) témata kontrolních prací: • separace a aproximace kořenů rovnice f(x)=0. • řešení soustav rovnic, interpolace funkcí. • počáteční úlohy pro obyčejné diferenciální rovnice, zpracování statistického souboru. Celkem lze získat maximálně 14 bodů, každou kontrolní práci je možno po dohodě s vyučujícím jednou opravit. b) témata programů • řešení algebraických a transc. rovnic • numerická kvadratura, aproximace Celkem lze získat maximálně 6 bodů. Při odevzdání programu po stanoveném termínu o 1 bod méně za každý týden prodlení! c) podmínky pro udělení zápočtu jsou následující: • odevzdání všech programů • účast na cvičeních je povinná, maximální rozsah omluvené neúčasti je 20% d) zkouška je složena z praktické a teoretické části. Ke zpracovaní praktické části je možno využívat počítač. U praktické části lze získat max. 60 bodů, u teoretické části max. 20 bodů.

### Other requirements

There are no other requirements on students.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

Problematics of numerical computing . Sources and types of errors. Conditionality of problems and algorithms. Methods for solving algebraic and transcendental equations. The bisection method, the iterative method for solving equations. The Newton method, the Regula-Falsi (False-Position) method, the combined method. Solving systems of linear equations. Direct solution methods. Iterative methods (the Jacobi method, the Seidel method). Matrix norms. Interpolation and approximation of functions. Approximation – the least-square method. Lagrange interpolation polynomials. Newton interpolation polynomials. Spline-function interpolation. Numerical integration. Newton-Cotes quadrature formulas. Composed quadrature formulas. Error estimation. The Richardson extrapolation. Initial value problems for ordinary differential equations. One-step methods. The Euler method. Error estimation using the half-step method. The Runge-Kutta methods. Estimation of the approximation error. Processing statistical data sets with one argument. Characteristics of statistical data sets, processing extensive statistical data sets. Parameter estimation for basic data sets. Basic data set, random sampling, point and interval parameter estimates of the basic data set. The goodness of fit tests. The Pearson χ2 test of the goodness of fit. The one- sample Kolmogorov-Smirnov test. The two-sample Kolmogorov-Smirnov test.

### Conditions for subject completion

Part-time form (validity from: 1960/1961 Summer semester, validity until: 2009/2010 Winter semester)
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (100) 51
Exercises evaluation Credit 20 (20) 0
Project Project 5  0
Written exam Written test 15  0
Examination Examination 80 (80) 0
Written examination Written examination 60  0
Oral Oral examination 20  0
Mandatory attendence parzicipation:

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### Occurrence in study plans

Academic yearProgrammeField of studySpec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2014/2015 (N3607) Civil Engineering (3607T036) Transport Constructions K Czech Ostrava 1 Compulsory study plan
2014/2015 (N3607) Civil Engineering (3607T013) Municipal Engineering and Construction K Czech Ostrava 1 Compulsory study plan
2014/2015 (N3607) Civil Engineering (3607T021) Building Materials and Diagnostics of Structures K Czech Ostrava 1 Compulsory study plan
2013/2014 (N3607) Civil Engineering (3607T036) Transport Constructions K Czech Ostrava 1 Compulsory study plan
2013/2014 (N3607) Civil Engineering (3607T013) Municipal Engineering and Construction K Czech Ostrava 1 Compulsory study plan
2013/2014 (N3607) Civil Engineering (3607T021) Building Materials and Diagnostics of Structures K Czech Ostrava 1 Compulsory study plan
2012/2013 (N3607) Civil Engineering (3607T036) Transport Constructions K Czech Ostrava 1 Compulsory study plan
2012/2013 (N3607) Civil Engineering (3607T013) Municipal Engineering and Construction K Czech Ostrava 1 Compulsory study plan
2012/2013 (N3607) Civil Engineering (3607T040) Building Environment K Czech Ostrava 1 Compulsory study plan
2012/2013 (N3607) Civil Engineering (3607T021) Building Materials and Diagnostics of Structures K Czech Ostrava 1 Compulsory study plan
2011/2012 (N3607) Civil Engineering (3607T021) Building Materials and Diagnostics of Structures K Czech Ostrava 1 Compulsory study plan
2011/2012 (N3607) Civil Engineering (3607T013) Municipal Engineering and Construction K Czech Ostrava 1 Compulsory study plan
2011/2012 (N3607) Civil Engineering (3607T036) Transport Constructions K Czech Ostrava 1 Compulsory study plan
2011/2012 (N3607) Civil Engineering (3607T040) Building Environment K Czech Ostrava 1 Compulsory study plan
2010/2011 (N3607) Civil Engineering (3607T013) Municipal Engineering and Construction K Czech Ostrava 1 Compulsory study plan
2010/2011 (N3607) Civil Engineering (3607T021) Building Materials and Diagnostics of Structures K Czech Ostrava 1 Compulsory study plan
2010/2011 (N3607) Civil Engineering (3607T036) Transport Constructions K Czech Ostrava 1 Compulsory study plan
2010/2011 (N3607) Civil Engineering (3607T040) Building Environment K Czech Ostrava 1 Compulsory study plan
2009/2010 (N3607) Civil Engineering (3607T013) Municipal Engineering and Construction K Czech Ostrava 1 Compulsory study plan
2009/2010 (N3607) Civil Engineering (3607T036) Transport Constructions K Czech Ostrava 1 Compulsory study plan
2009/2010 (N3607) Civil Engineering (3607T037) Building Constructions K Czech Ostrava 1 Compulsory study plan
2009/2010 (N3607) Civil Engineering (3607T040) Building Environment K Czech Ostrava 1 Compulsory study plan
2009/2010 (N3607) Civil Engineering (3607T021) Building Materials and Diagnostics of Structures K Czech Ostrava 1 Compulsory study plan
2008/2009 (N3607) Civil Engineering (3607T013) Municipal Engineering and Construction K Czech Ostrava 1 Compulsory study plan
2008/2009 (N3607) Civil Engineering (3607T021) Building Materials and Diagnostics of Structures K Czech Ostrava 1 Compulsory study plan
2008/2009 (N3607) Civil Engineering (3607T036) Transport Constructions K Czech Ostrava 1 Compulsory study plan
2008/2009 (N3607) Civil Engineering (3607T037) Building Constructions K Czech Ostrava 1 Compulsory study plan
2008/2009 (N3607) Civil Engineering (3607T040) Building Environment K Czech Ostrava 1 Compulsory study plan
2007/2008 (N3607) Civil Engineering (3607T013) Municipal Engineering and Construction K Czech Ostrava 1 Compulsory study plan
2007/2008 (N3607) Civil Engineering (3607T021) Building Materials and Diagnostics of Structures K Czech Ostrava 1 Compulsory study plan
2007/2008 (N3607) Civil Engineering (3607T036) Transport Constructions K Czech Ostrava 1 Compulsory study plan
2007/2008 (N3607) Civil Engineering (3607T037) Building Constructions K Czech Ostrava 1 Compulsory study plan
2007/2008 (N3607) Civil Engineering (3607T040) Building Environment K Czech Ostrava 1 Compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner