Course detail

# Numerical and applied mathematics

Introduction to numerical mathematics, namely interpolation and approximations of functions, numerical differentiation and quadrature, analysis of algebraic and differential equations and their systems.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

Basic courses Mathematics 1-4, MATLAB programming (as in the recommended course at MAT FCE).

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Errors in numerical calculations. Linear spaces and operators, fixed point theorems. Iterative methods for the analysis of nonlinear algebraic and selected further equations.
2. Iterative and coupled methods for the analysis of linear algebraic equations, relaxation methods, method of conjugated gradients.
3. Multiplicative decomposition of matrices. Numerical evaluation of eigenvalues and eigenvectors of matrices and of inverse matrices, algorithms for special matrices.
4. Condition numbers of systems of linear equations. Least squares method, pseudoinverse matrices.
5. Generalizations of methods from 3. and 4. to the analysis of systems of nonlinear equations.
6. Lagrange and Hermite interpolation of functions of 1 variable, namely polynoms and splines.
7. Approximation of functions of 1 variable using the least squares methos: linear and nonlinear approach.
8. Approximation of function of more variables.
9. Numerical differentiation. Finite difference method for the analysis of selected initial and boundary problems for ordinary differential equations.
10. Numerical quadrature. Finite element method for the analysis of selected initial and boundary problems for ordinary differential equations.
11. Time-dependent problems. Time discretization: Euler methods, Cranka-Nicholson method, Runge-Kutta methods, Newmark method.
12. Generalization of 9. and 10. for pro partial differential equations of evolution, e.g. heat transfer equations, fluid flow equations and equations of dynamics of building structures.
13. Sensitivity and inverse problems. Identification of uncertain material parameters from known measurement results. Selected engineering application, due to other courses.

Aims

Not applicable.

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 NPC-EVB Master's, 1. year of study, winter semester, 3 credits, compulsory-optional

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Exercise

13 hours, compulsory

Teacher / Lecturer