Course detail

# Mathematics 5 (K)

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.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

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.

Prerequisites

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

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.

Assesment methods and criteria linked to learning outcomes

Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops. The result of the semester examination is given by the sum of maximum of 70 points obtained for a written test and a maximum of 30 points from the seminar.

Language of instruction

Czech

Work placements

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

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.

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.

Classification of course in study plans

• Programme N-P-E-SI (N) Master's

branch K , 1. year of study, winter semester, 4 credits, compulsory

• Programme N-K-C-SI (N) Master's

branch K , 1. year of study, winter semester, 4 credits, compulsory

• Programme N-P-C-SI (N) Master's

branch K , 1. year of study, winter semester, 4 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

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 hours, compulsory

Teacher / Lecturer

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