714-0781/02 – Numerical methods and statistics (NMS)

Gurantor departmentDepartment of Mathematics and Descriptive GeometryCredits6
Subject guarantorprof. RNDr. Radek Kučera, Ph.D.Subject version guarantorprof. RNDr. Radek Kučera, Ph.D.
Study levelundergraduate or graduateRequirementCompulsory
Year2Semestersummer
Study languageCzech
Year of introduction2013/2014Year of cancellation
Intended for the facultiesUSPIntended for study typesBachelor
Instruction secured by
LoginNameTuitorTeacher giving lectures
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 7+7

Subject aims expressed by acquired skills and competences

The aim of this course is to acquaint students with the numerical solution of mathematical problems and with the basic methods of analysing statistical data. The main accent lays in explanations of fundamental principles of methods so that the students should know to choose appropriate methods for problems arising in the other courses of the study or in the technical practice. An important ingredient of the course consists in the algorithmic implementation of methods and in the utilization of existing computer programms for numerical computations and statistical analyses. 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

The course is devoted to basic numerical methods of the linear algebra and mathematical analysis and to basic methods of analysing statistical data. The following themes will be presented: iterative methods for solving of nonlinear equations, direct and iterative methods for solving of linear systems, eigenvalue problems, interpolation and approximation of functions, numerical computation of derivatives and integrals, solving of ordinary differential equations, estimations of statistical parameters and testing of hypotheses. The programming system Matlab is used during the course.

Compulsory literature:

1. Qaurteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics. Springer, 2007.

Recommended literature:

1. Süli, E., Mayers, D.: An introduction to Numerical Analysis. Cambridge University Press, 2003. 2. Van Loan, C. F.: Introduction to scientific computing. Prentice Hall, Upper Saddle River, NJ 07459, 1999.

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. 27th The Cauchy problem for ODE. Existence and uniqueness of the solution. The Euler method. 28th One-step methods.Local and global error, order. 29th Multi-step methods. Explicit and implicit formulas. 30th Build predictor-corrector algorithm. 31st Statistical processing of data with one or two arguments. 32nd Determination of empirical characteristics of statistical data. 33rd Parameter estimation and hypothesis testing. 34th Regression analysis.

E-learning

Další požadavky na studenta

They are no other requests for students.

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

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 One-step methods for solving initial value problems for ordinary differential equations. 12th Multi-step methods. 13th Statistical data processing, empirical characteristics. 14th Parameter estimation and testing of hypotheses.

Conditions for subject completion

Full-time form (validity from: 2013/2014 Summer 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  5
        Examination Examination 80 (80) 30
                Písemka Written test 60  25
                Ústní zkouška Oral examination 20  5
Mandatory attendence parzicipation:

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Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty
2019/2020 (B3942) Nanotechnology (3942R001) Nanotechnology P Czech Ostrava 2 Compulsory study plan
2018/2019 (B3942) Nanotechnology (3942R001) Nanotechnology P Czech Ostrava 2 Compulsory study plan
2017/2018 (B3942) Nanotechnology (3942R001) Nanotechnology P Czech Ostrava 2 Compulsory study plan
2016/2017 (B3942) Nanotechnology (3942R001) Nanotechnology P Czech Ostrava 2 Compulsory study plan
2015/2016 (B3942) Nanotechnology P Czech Ostrava 2 Compulsory study plan
2015/2016 (B3942) Nanotechnology (3942R001) Nanotechnology P Czech Ostrava 2 Compulsory study plan
2014/2015 (B3942) Nanotechnology P Czech Ostrava 2 Compulsory study plan
2013/2014 (B3942) Nanotechnology P Czech Ostrava 2 Compulsory study plan

Occurrence in special blocks

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