Course detail

Mathematics 5 (R)

FAST-NAA020Acad. year: 2022/2023

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

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

Basic courses of mathematics for bachelor students, MATLAB programming (as in the recommended course at MAT FCE).

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. 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.

Work placements

Not applicable.

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.

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

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

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme NPC-SIR Master's 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Mgr. Ing. Miroslav Trcala, Ph.D.

Syllabus

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.

Exercise

13 hod., compulsory

Teacher / Lecturer

RNDr. Oto Přibyl
doc. Mgr. Ing. Miroslav Trcala, Ph.D.

Syllabus

Seminars follow the related lectures: 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.