Course detail

# Mathematics

FAST-DA01Acad. year: 2018/2019

Errors in numerical calculations and numerical methods for one nonlinear equation in one unknown.

Iterative methods. The Banach fixed-point theorem.

Iterative methods for the systems of linear and nonlinear equations.

Direct methods for the systems of linear algebraic equations, matrix inversion, eigenvalues and eigenvectors of matrices.

Interpolation and approximation of functions. Splines.

Numerical differentiation and integration. Extrapolation to the limit.

Supervisor

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

Basics of linear algebra and vector calculus. Basics of the theory of one- and more-functions (limit, continuous functions, graphs of functions, derivative, partial derivative). Basics of the integral calculus of one- and two-functions.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

DALÍK, J.: Numerické metody. CERM Brno 1997

HOROVÁ, I., ZELINKA, J.: Numerické metody. Masarykova univerzita v Brně 2004

MIKA, S.: Numerické metody algebry. SNTL Praha 1982

PŘIKRYL, P., BRANDNER, M.: Numerické metody II. ZČU Plzeň 2000

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.

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. Numerical methods for one nonlinear equation in one unknown

2. Basic principles of iterative methods. The Banach fixed-point theorem.

3. Norms of vectors and of matrices, eigenvalues and eigenvectors of matrices. Iterative methods for systems of linear algebraic equations– part I.

4. Iterative methods for linear algebraic equations– part II. Iterative methods for systems of nonlinear equations.

5. Direct methods for systems of linear algebraic equations, LU-decomposition. Systems of linear algebraic equations with special matrice – part I.

6. Systems of linear algebraic equations with special matrices – part II. The methods based on the minimization of a quadratic form.

7. Computing inverse matrices and determinants, the stability and the condition number of a matrix.

8. Eigenvalues of matrices - the power method. Basic principles of interpolation.

9. Polynomial interpolation.

10. Interpolation by means of splines. Orthogonal polynoms.

11. Approximation by the discrete least squares.

12. Numerical differentiation, Richardson´s extrapolation. Numerical integration of functions in one variables– part I.

13. Numerical integration of functions in one variables– part II. Numerical integration of functions in two variables.

Aims

After the course the students should understand the main priciples of numeric calculation and the factors influencing calculation. They should be able to solve selected basic problems of numerical analysis, using iteration methods to solve the f(x)=0 equation and systems of linear algebraic equations using calculation algorithms. Theu should learnhow to approximate eigenvalues and eigenvectors of matrices. They should learn about the basic problems in interpolation and approximation of functions solving silple practical problems. Theu should be acquainted with tyhe principles of numeric differentiation and know how to numerically approximate integral of one- and two-functions.

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.

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1. Errors in numerical calculations. Numerical methods for one nonlinear equation in one unknown

2. Basic principles of iterative methods. The Banach fixed-point theorem.

3. Norms of vectors and of matrices, eigenvalues and eigenvectors of matrices. Iterative methods for systems of linear algebraic equations– part I.

4. Iterative methods for linear algebraic equations– part II. Iterative methods for systems of nonlinear equations.

5. Direct methods for systems of linear algebraic equations, LU-decomposition. Systems of linear algebraic equations with special matrice – part I.

6. Systems of linear algebraic equations with special matrices – part II. The methods based on the minimization of a quadratic form.

7. Computing inverse matrices and determinants, the stability and the condition number of a matrix.

8. Eigenvalues of matrices - the power method. Basic principles of interpolation.

9. Polynomial interpolation.

10. Interpolation by means of splines. Orthogonal polynoms.

11. Approximation by the discrete least squares.

12. Numerical differentiation, Richardson´s extrapolation. Numerical integration of functions in one variables– part I.

13. Numerical integration of functions in one variables– part II. Numerical integration of functions in two variables.