Course detail

Numerical methods I

FAST-DA61Acad. year: 2024/2025

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.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Entry knowledge

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.

Rules for evaluation and completion of the course

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

Aims

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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

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

Classification of course in study plans

  • Programme D-K-C-SI (N) Doctoral

    branch VHS , 1 year of study, summer semester, compulsory-optional
    branch MGS , 1 year of study, summer semester, compulsory-optional
    branch PST , 1 year of study, summer semester, compulsory-optional
    branch FMI , 1 year of study, summer semester, compulsory-optional
    branch KDS , 1 year of study, summer semester, compulsory-optional

  • Programme D-K-C-GK Doctoral

    branch GAK , 1 year of study, summer semester, compulsory-optional

  • Programme D-K-E-CE (N) Doctoral

    branch FMI , 1 year of study, summer semester, compulsory-optional
    branch KDS , 1 year of study, summer semester, compulsory-optional
    branch MGS , 1 year of study, summer semester, compulsory-optional
    branch VHS , 1 year of study, summer semester, compulsory-optional
    branch PST , 1 year of study, summer semester, compulsory-optional

Type of course unit

 

Lecture

39 hod., 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.