Course detail

Numerical methods I

FAST-DA61Acad. 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.
Numeric differentiation and integration. Extrapolation to the limit.

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

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

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 education, way of implementation and compensation for absences

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

Classification of course in study plans

  • Programme D-K-C-SI (N) Doctoral

    branch PST , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch PST , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch PST , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-K-E-SI (N) Doctoral

    branch PST , 1. year of study, summer semester, 4 credits, compulsory-optional
    branch MGS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch MGS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-K-C-SI (N) Doctoral

    branch KDS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch KDS , 1. year of study, summer semester, 4 credits, compulsory-optional
    branch MGS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-K-C-SI (N) Doctoral

    branch MGS , 1. year of study, summer semester, 4 credits, compulsory-optional
    branch FMI , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch FMI , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-K-E-SI (N) Doctoral

    branch FMI , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch FMI , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-K-E-SI (N) Doctoral

    branch KDS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch KDS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-K-E-SI (N) Doctoral

    branch VHS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-K-C-SI (N) Doctoral

    branch VHS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch VHS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch VHS , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-P-C-GK Doctoral

    branch GAK , 1. year of study, summer semester, 4 credits, compulsory-optional

  • Programme D-K-C-GK Doctoral

    branch GAK , 1. year of study, summer semester, 4 credits, compulsory-optional

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.