Course detail

Mathematics 5 (R)

FAST-CA006Acad. year: 2024/2025

Errors in numeric calculations, solvig transcendental equations in one and several unknowns using iteration methods. Iteration methods used to solve systems of linear algebraic equations. Interpolating and approximating functions. Numerical differentiation and integration and their application to solving boundary value problems for the ordinary differential equations. Applications given by the 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)

Entry knowledge

Basic notions of the theory of functions in one variable (derivative, limit, continuous functions, graphs of functions). Calculating definite integrals, knowing about their basic applications.

Rules for evaluation and completion of the course

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

Aims

The students should understand the basic principles of numerical calculations and the factors that influence them. They should be able to solve selected basic problems in numerical mathematics, understand the principle of iteration methods for solving the equation f(x)=0 and the systems of linear algebraic equations mastering the calculation algorithms. They should learn how to get the basics of interpolation and approximation of functions to solve practical problems. They should be acquainted with the principles of numerical differentiation to be able to numerically solve boundary value problems for ordinary differential equations. They should be able to evaluate definite integrals numerically.
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

VALA J.: Numerická matematika. FAST VUT v Brně 2021. (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