714-0552/07 – Numerical Methods and Statistics (NMaS)

Gurantor departmentDepartment of Mathematics and Descriptive GeometryCredits5
Subject guarantorprof. RNDr. Radek Kučera, Ph.D.Subject version guarantorprof. RNDr. Radek Kučera, Ph.D.
Study levelundergraduate or graduateRequirementCompulsory
Year1Semesterwinter
Study languageCzech
Year of introduction2004/2005Year of cancellation
Intended for the facultiesHGFIntended for study typesFollow-up Master
Instruction secured by
LoginNameTuitorTeacher giving lectures
KRC23 Mgr. Jitka Krčková, Ph.D.
KUC14 prof. RNDr. Radek Kučera, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2
Combined Credit and Examination 14+0

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.

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. Numerical solution of systems of linear algebraic equations: LU-factorization, iterative methods, condition number of matrix, ill-conditioned matrices. Interpolation and approximation of functions: polynomial interpolation, interpolation by spline functions, least squares approximation. Numerical integration: Trapezoid rule, Simpson’s rule, Richardson extrapolation. Statistical processing data with one or more arguments, empirical characteristics of statistical data, testing of hypotheses. Regression analysis.

Compulsory literature:

1. Boháč, Zdeněk: Numerical Methods and Statistics, VŠB – TUO, Ostrava 2005, ISBN 80-248-0803-X

Recommended 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. 3. Stoer, J., Burlish, R.: Introduction to Numerical Analysis. Springer-Verlag, New York, Berlin, Heidelberg, 1992.

Way of continuous check of knowledge in the course of semester

Tests and credits ================= Exercises --------- Conditions for obtaining credit points (CP): - participation in exercises, 20% can be to apologize - completion of three written tests, 0-15 CP - completion of two programs, 5 CP Exam ---- - written exam 0-60 CP, successful completion at least 25 CP - oral exam 0-20 CP, successful completion at least 5 CP Exam Questions ================= 1st Give examples of discrete and continuous problems. What is it a discretization? 2nd Define the absolute, relative error and their estimates? How does the round-off error influence the arithmetic operations? 3rd Explain what is it meant by stable and unstable computations? What is it the condition number? 4th Derive the computer epsilon. What is its usual value on standard PC? 5th Explain how to separate roots of one nonlinear equation in real domain. 6th Bisection method: formula and stopping criterion. 7th Regula-falsi method: derive the formula, stopping criterion. 8th The Newton method: derive the formula, stopping criterion. 9th Derive an order of the Newton by the Taylor expanssion, the global convergence theorem. 10th Fixed-point method, Brower fixed-point theorem. 11th The convergence analysis of the fixed-point method by contractivity. 12th Gaussian elimination, phases and complexity. 13th LU-decomposition via Gaussian elimination. 14th Using the LU-decomposition to solve linear systems, to calculate the inverse matrices and determinants. 15th Matrix norms and condition number. An example of ill-conditioned matrix. 16th Eigenvalues and eigenvectors. The definition and calculations. 17th Iterative methods for solving systems of linear equations, Jacobi and Gauss-Seidel. 18th The convergence analysis of general linear iterative method. 19th Interpolation polynomials. Existence and uniqueness. 20th Interpolation error for polynomials. The Runge example. 21st Interpolation splines. 22nd Least square method. System of normal equations. Existence and uniqueness of the solution. 23rd The Newton-Cotes formulas for numerical calculation of integrals. 24th Error in numerical integration. 25th The precision control in the numerical calculation of integrals. The Richardson extrapolation. 26th Numerical differentiation formulas. Order. Behavior of error. 27st Statistical processing of data with one or two arguments. 28nd Determination of empirical characteristics of statistical data. 29rd Parameter estimation and hypothesis testing. 30th Regression analysis.

E-learning

Další požadavky na studenta

They are no other requests.

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

Week. Lecture ------------- 1st Course contents, the issue of errors, stability of calculations. 2nd Solution of nonlinear equations, separation of roots, the simplest methods. 3rd Newton's method and fixed-point iterations. 4th Direct methods for solving linear equations, Gaussian elimination and LU-decomposition. 5th Eigenvalues and eigenvectors, numerical calculation. 6th Iterative methods for solving linear equations. 7th Interpolation by polynomials and splines. 8th Least squares approximation. 9th Numerical differentiation and integration. 10th Extrapolation in the calculation of integrals. Gaussian integration formulas. 11th Statistical processing data with one or more arguments. 12th Empirical characteristics of statistical data. 13th Parameter estimation and testing of hypotheses. 14th Regression analysis.

Conditions for subject completion

Combined form (validity from: 1960/1961 Summer semester, validity until: 2009/2010 Winter semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (100) 51
        Exercises evaluation Credit 20 (20) 0
                Other task type Other task type 20  0
        Examination Examination 80 (80) 0
                Written examination Written examination 60  0
                Oral Oral examination 20  0
Mandatory attendence parzicipation:

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2007/2008 (N3646) Geodesy and Cartography (3602T002) Geoinformatics (10) P Czech Ostrava 1 Choice-compulsory study plan
2007/2008 (N3646) Geodesy and Cartography (3602T002) Geoinformatics (10) K Czech Ostrava 1 Choice-compulsory study plan
2006/2007 (N3646) Geodesy and Cartography (3646T001) Mine Surveying K Czech Ostrava 1 Compulsory study plan
2006/2007 (N3646) Geodesy and Cartography (3602T002) Geoinformatics (10) K Czech Ostrava 1 Choice-compulsory study plan
2006/2007 (N2102) Mineral Raw Materials (2102T012) Management of Resource of Building Mineral Raw Materials K Czech Ostrava 1 Compulsory study plan
2006/2007 (N2102) Mineral Raw Materials (2102T012) Management of Resource of Building Mineral Raw Materials K Czech Most 1 Compulsory study plan
2005/2006 (N2102) Mineral Raw Materials (2102T012) Management of Resource of Building Mineral Raw Materials K Czech Ostrava 1 Compulsory study plan
2005/2006 (N3646) Geodesy and Cartography (3646T001) Mine Surveying K Czech Ostrava 1 Compulsory study plan
2004/2005 (N2102) Mineral Raw Materials (2102T012) Management of Resource of Building Mineral Raw Materials K Czech Ostrava 1 Compulsory study plan
2004/2005 (N3646) Geodesy and Cartography (3646T001) Mine Surveying K Czech Ostrava 1 Compulsory study plan

Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner