230-0261/02 – Interpolation and Approximation of Functions (IaAF)
| Gurantor department | Department of Mathematics | Credits | 10 |
| Subject guarantor | doc. RNDr. Pavel Kreml, CSc. | Subject version guarantor | doc. RNDr. Pavel Kreml, CSc. |
| Study level | postgraduate | Requirement | Choice-compulsory |
| Year | | Semester | winter + summer |
| | Study language | English |
| Year of introduction | 2019/2020 | Year of cancellation | |
| Intended for the faculties | FBI, HGF, FS, FEI, FMT, USP, FAST | Intended for study types | Doctoral |
Subject aims expressed by acquired skills and competences
Main study goals:
(i) to be acquainted with actual progress in this mathematical discipline,
(ii) to extend needed theoretical knowledge with emphasized orientation to its applica-bility,
(iii) to increase communication ability of specialists in different branches.
With regard to professional orientation of students learning themes modification is offered to fulfill presented aims.
Teaching methods
Seminars
Individual consultations
Other activities
Summary
These lectures deal with methods for constructing approximating functions for
any set of data by using polynomial interpolation, least-squares approximation
and Chebyshev approximation. There are compared several ways of doing
interpolation and there are contrasted these procedures with several ways for
fitting imprecise data and for drawing smooth curves. It is shown how can help
symbolic algebra computer algebra programs in obtaining interpolating and
least-squares polynomials.
Compulsory literature:
Recommended literature:
http://mdg.vsb.cz/portal/nm/nm.pdf
Way of continuous check of knowledge in the course of semester
Continuous work on project. Using aproximation and interpolation methods.
Examination.
E-learning
http://www.studopory.vsb.cz
http://mdg.vsb.cz/portal/nm/index.php
(in Czech language)
Gerald,F.-Wheatley,P.: Applied Numerical Analysis., Addison Wesley 1994.
Stoer,J. – Bulirsch,R.: Introduction to Numerical Analysis., Springer-Verlag, New York 1993.
Other requirements
Each student must discharge:
a) examples,
b) project.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
Syllabus
1. Kinds of:
- dependence and independence of functions,
- existence and definiteness of approximating functions,
- error of approximation.
2. Polynomial interpolation:
-the error estimate of the interpolation,
- Lagrangian and Mewton polynomials,
- extrapolation, interpolation of rational functions,
- choice of points for fitting.
3. Interpolating with a spline functions :
- interpolating with a cubic spline,
- features of cubic spline,
- B-spline curves,
- Bezier curves.
4. Orthogonal system of functions:
- orthogonal polynomials,
- Chebyshev, Hermitov, Gramov polynomials.
5. Least-squares approximations:
- the best L2- approximation, least-squares method,
- normal equations, solving sets of linear equations,
- algorithm of least-squares method,
- nonlinear data.
6. Chebyshev approximation:
- the best uniform approximation,
- algorithm of the method, maximum error.
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks