Course Details
Mathematics
Academic Year 2023/24
DA01 course is part of 1 study plan
D-K-C-GK / GAK Summer Semester 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.
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
Institute
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.
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.
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
0 credits
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
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.