Course Details

Mathematics

Academic Year 2022/23

DA01 course is part of 3 study plans

D-P-C-GK Summer Semester 1st year

D-K-C-GK Summer Semester 1st year

stud.semestr.null 1st year

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.

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Objective

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.

Knowledge

The findings concerning the typical problems solved within the numerical analysis,
understanding of the elementary numerical procedures for their approximate solutions and their essential properties and
the ability to solve elementary typical concrete problems numerically.

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.

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.

Language of instruction

Czech

Credits

no credit

Semester

summer

Forms and criteria of assessment

no final classification

Specification of controlled instruction, the form of instruction, and the form of compensation of the absences

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

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/255820

Lecture

13 weeks, 3 hours/week, elective

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.