Course detail

Mathematics 5 (V)

FAST-CA004Acad. year: 2022/2023

Errors. Transecndental equations in one or several unknowns solved by iteration methods. Iterational methods for systems of linear algebraic equations.
Interpolation and approximation of functions. Numerical differentiation, numeric integration and applications in solving the boundary--value problems for ODE.

Applications according to the concrete specialization.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

prof. Ing. Jiří Vala, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

The outputs of this course are the skills and the knowledge which enable the graduates understanding of basic numerical problems and of the ideas on which the procedures for their solutions are based. In their future practice they will be able to recognize the applicability of numerical methods for the solution of technical problems and use the existing universal programming systems for the solution of basic types of numerical problems and their future improvements effectively.

Prerequisites

Basics of the theory of functions in one variable (derivative, limit and continuity, elementary functions). Definite integrals and their basic applications.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Errors in numerical calculations, approximation of the solutions of one equation in one real variable by bisection and by iteration
2. Approximation of the solutions of one equation in one real variable by iteration, the Newton method and its modifications
3. Norms of matrices and vectors, calculations of the inverse matrices
4. Solutions of systems of linear equations with speciál matrice and the condition numer of a matrix
5. Solutions of systems of linear equations by iteration
6. Solutions of systems of non—linear equations
7. Lagrange interpolation by polynomials and cubic splines, Hermite interpolation by polynomials and Hermite cubic splines
8. The discrete least squares Metod, numerical differentiation
9. Classical formulation of the boundary—value problem for the ODE of second order and its approximation by the finite diference method
10. Numerical integration. Variational formulation of the boundary—value problem for the ODE of second order
11. Discertization of the variational boundary—value problem for the ODE of second order by the finite element method
12. Classical and variational formulations of the boundary—value problem for the ODE of order four
13. Discertization of the variational boundary—value problem for the ODE of order four by the finite element method

Work placements

Not applicable.

Aims

To understand the basic principles of numeric calculation and the factors influencing nueric calculation. Know how to solve selected basic problems of numeric mathematics. Understand iteration methods used to solve a f(x)=0 equation and systems of linear algebraic equations, practice calculation algorithms. Learn how to interpolate functions. Understand the principles of numerical differentiation to calculate numerical solutions to a boundary value problem for the ordinary differential equations. Learn how to calculate definite integrals.

Specification of controlled education, way of implementation and compensation for absences

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme N-P-C-SI Master's

    branch V , 1 year of study, winter semester, compulsory

  • Programme N-K-C-SI Master's

    branch V , 1 year of study, winter semester, compulsory

  • Programme N-P-E-SI Master's

    branch V , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

1. Errors in numerical calculations, approximation of the solutions of one equation in one real variable by bisection and by iteration 2. Approximation of the solutions of one equation in one real variable by iteration, the Newton method and its modifications 3. Norms of matrices and vectors, calculations of the inverse matrices 4. Solutions of systems of linear equations with speciál matrice and the condition numer of a matrix 5. Solutions of systems of linear equations by iteration 6. Solutions of systems of non—linear equations 7. Lagrange interpolation by polynomials and cubic splines, Hermite interpolation by polynomials and Hermite cubic splines 8. The discrete least squares Metod, numerical differentiation 9. Classical formulation of the boundary—value problem for the ODE of second order and its approximation by the finite diference method 10. Numerical integration. Variational formulation of the boundary—value problem for the ODE of second order 11. Discertization of the variational boundary—value problem for the ODE of second order by the finite element method 12. Classical and variational formulations of the boundary—value problem for the ODE of order four 13. Discertization of the variational boundary—value problem for the ODE of order four by the finite element method

Exercise

13 hod., compulsory

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

Follows directly particular lectures. 1. Errors in numerical calculations, approximation of the solutions of one equation in one real variable by bisection and by iteration 2. Approximation of the solutions of one equation in one real variable by iteration, the Newton method and its modifications 3. Norms of matrices and vectors, calculations of the inverse matrices 4. Solutions of systems of linear equations with speciál matrice and the condition numer of a matrix 5. Solutions of systems of linear equations by iteration 6. Solutions of systems of non—linear equations 7. Lagrange interpolation by polynomials and cubic splines, Hermite interpolation by polynomials and Hermite cubic splines 8. The discrete least squares Metod, numerical differentiation 9. Classical formulation of the boundary—value problem for the ODE of second order and its approximation by the finite diference method 10. Numerical integration. Variational formulation of the boundary—value problem for the ODE of second order 11. Discertization of the variational boundary—value problem for the ODE of second order by the finite element method 12. Classical and variational formulations of the boundary—value problem for the ODE of order four 13. Discertization of the variational boundary—value problem for the ODE of order four by the finite element method