Course detail

Mathematics 5 (K)

FAST-CA002Acad. year: 2024/2025

Numerical solutions of nonlinear equations for one and more ariables, approximations of eigenvalues and eigenvectors of symmetric matrices, iterational methods for systems of linear algebraic equations. Interpolation and approximation of functions, numerical differentiation and integration, numerical methods for the ordinary differential equations of order two, numerical modelling of the heat-flow and of beam-bending in dimension one.

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)

Entry knowledge

Elementary functions, differential calculus of functions in one variable and integral calculus of functions in one variable.

Rules for evaluation and completion of the course

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

Aims

To understand basic principles of numerical computations and learn essential factors affecting numerical calculations. Be able to solve basic elementary problems of numerical mathematics. Iteration methods for the equation f(x)= 0, finite and iterative methods for the soulution of systems of linear equations, interpolation and approximation of functions, numerical differentiation and numerical integration, numerical methods for boundary-value differential problems.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

LITCHMANNOVÁ M.: Úvod do statistiky. VŠB-TU Ostrava 2011. (CS)

Recommended reading

Not applicable.

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