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Course, academic year 2017/2018
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Approximation Theory - NMNV543
Czech title: Teorie aproximace
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2017
Semester: winter
E-Credits: 4
Hours per week, examination: winter s.:2/1 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Additional information: http://numapprox.blogspot.cz
Guarantor: doc. RNDr. Petr Tichý, Ph.D.
Class: M Mgr. NVM
M Mgr. NVM > Povinně volitelné
Classification: Mathematics > Numerical Analysis
Incompatibility : NNUM011
Interchangeability : NNUM011
Annotation -
Last update: T_KNM (07.04.2015)

Best approximation in normed linear spaces, best uniform approximation of continuous functions, Remez algorithm, the Jackson theorems, the theorems of Bernstein. Least squares approximation based on orthogonal polynomials, approximation of periodic functions. General questions about convergence, convergence of interpolation polynomials. Basics of Korovkin's theory. Rational approximation (interpolations, best approximation, continued fractions, Pade approximation). The course is suitable for students focused on numerical analysis and matrix computations.
Terms of passing the course -
Last update: doc. RNDr. Petr Tichý, Ph.D. (06.10.2017)

It is not necessary to obtain a course-credit before passing the exam.

The course-credit will be granted for the attendance and for a short presentation given during the semester.

The nature of these requirements does not allow a possibility of some additional attempts to obtain the course-credit.

Literature - Czech
Last update: doc. RNDr. Petr Tichý, Ph.D. (07.04.2015)

M. J. D. Powell, Approximation theory and methods. Cambridge University Press, Cambridge-New York, 1981.

N. L. Trefethen, Approximation Theory and Approximation Practice. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.

E. W. Cheney, Introduction to approximation theory. AMS Chelsea Publishing, Providence, RI, 1982.

R. A. DeVore, G. G. Lorentz, Constructive Approximation, vol. 303 of Grundlehren der Mathematischen Wissenschaften,, Springer-Verlag, Berlin, 1993.

Requirements to the exam -
Last update: doc. RNDr. Petr Tichý, Ph.D. (06.10.2017)

The exam is oral. Requirements for the oral exam correspond to the syllabus of the course, presented at the lectures.

Syllabus -
Last update: doc. RNDr. Petr Tichý, Ph.D. (07.04.2015)

Best approximation in normed linear spaces, approximation operators. Polynomial approximation: Interpolation, minimax, the exchange algorithm, least squares approximation, orthogonal polynomials, periodic functions, uniform convergence, Jackson's theorems. Rational approximation: Best approximation, interpolation, Pade approximation. Practical applications: Chebfun, spectral methods, matrix functions.

 
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