Course Details

Numerical methods I

DA61 course is part of 22 study plans

Ph.D. full-t. program nD > PST compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nD > FMI compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nD > MGS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nD > VHS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nD > KDS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDK > PST compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDK > KDS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDK > VHS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDK > FMI compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDK > MGS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nDA > PST compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nDA > FMI compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nDA > MGS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nDA > VHS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program nDA > KDS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDKA > PST compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDKA > FMI compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDKA > MGS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDKA > VHS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program nDKA > KDS compulsory-elective Summer Semester 1st year 4 credits

Ph.D. full-t. program I > GAK compulsory-elective Summer Semester 1st year 4 credits

Ph.D. combi. program IK > GAK compulsory-elective Summer Semester 1st year 4 credits

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. Numeric differentiation and integration. Extrapolation to the limit.

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

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.

Planned educational 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.

Objective

Understanding the main priciples of numeric calculation and the factors influencing calculation. Solving 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. Learning how to approximate eigenvalues and eigenvectors of matrices. Learning about the basic problems in interpolation and approximation of functions. Getting acquainted with the principles of numeric differentiation and knowing how to numerically approximate integrals of one- and two-functions.

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

Vymezení kontrolované výuky a způsob jejího provádění stanoví každoročně aktualizovaná vyhláška garanta předmětu.

Lecture

3 hours/week, 13 weeks, elective

Syllabus of lectures

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.