Racionalna aproksimacija funkcij

delo diplomskega seminarja

Žan Kastelic (Author), Jan Grošelj (Mentor)

Abstract

Standardne tehnike polinomske aproksimacije funkcij lahko posplošimo metode na aproksimacije z racionalnimi funkcijami. Za Padejevo aproksimacijo idejo najdemo v Taylorjevi polinomski aproksimaciji. Izkaže se, da dobro aproksimacijo zagotavlja le v izbrani točki in njeni bližnji okolici, z oddaljevanjem od točke pa približki postajajo slabši. Pri Čebiševi aproksimaciji posežemo po drugačni bazi polinomov, za katere je znano, da se obnašajo bolj enakomerno. Izkaže se, da ta baza res vodi k bolj enakomerni aproksimaciji in da so napake vzdolž intervala manjše kot pri Padejevi aproksimaciji. Ob koncu je obravnavana še najboljša enakomerna racionalna aproksimacija, v zvezi s katero je po zgledu najboljše enakomerne polinomske aproksimacije vpeljan Remesov postopek. Ker je neekonomičen, je predstavljen poenostavljen postopek, za katerega je na primerih ugotovljeno, da ne daje optimalnih rešitev, vseeno pa ponuja še en način dobre enakomerne aproksimacije.

Keywords

racionalna aproksimacija;Padejeva aproksimacija;Čebiševa aproksimacija;Remesov postopek;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[Ž. Kastelic]
UDC:	517.9
COBISS:	79160323
Views:	711
Downloads:	65
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Rational function approximation
Secondary abstract:	Standard techniques for polynomial approximation of functions can be generalized to methods for approximation with rational functions. The Pade approximation is based on the same idea as the Taylor polynomial approximation. It turns out that it provides a good approximation only at a selected point and its close vicinity, and that approximations become worse when we move away from the selected point. In Chebyshev approximation we use a different basis of polynomials which are known to behave more uniformly. It turns out that this basis really leads to a more uniform approximation and that the errors along the interval are smaller than those observed in Pade approximation. In the end, the best uniform rational approximation is considered. Following the standard Remes algorithm for best uniform polynomial approximation, its adaptation for rational functions is introduced. Since this algorithm is quite uneconomical, another simplistic procedure is presented, which according to the performed computational experiments does not give optimal solutions but nevertheless provides another method for a good uniform approximation.
Secondary keywords:	rational approximation;Pade approximation;Chebyshev approximation;Remes process;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja
Pages:	33 str.
ID:	13595043