Programski paket Chebfun

PANGERČIČ, ANDREJ (author), Jaklič, Gašper (mentor)

Skoraj vsak matematični problem, ki ga ne moremo več rešiti na roko zaradi naše omejene hitrosti reševanja, smo primorani rešiti z računalnikom. Za začetek moramo predstaviti problem v računalniški obliki in nato nad njim izvršiti neko matematično operacijo v obliki algoritma. Pri tem si želimo imeti algoritme, ki se izvajajo v realnem času. Funkcije največkrat aproksimiramo z interpolacijskimi polinomi, s katerimi najlažje in hitro računamo. Kaj kmalu ugotovimo, da osnovna uporaba interpolacije ni primerna, saj povzroča prevelike napake. V tem diplomskem delu se bomo posvetili Čebiševi vrsti, ki v praksi daje zelo dobre aproksimacije za lepe funkcije. Pogledali si bomo, kako je takšna aproksimacija implementirana v odprtokodnem programskem orodju Chebfun. Na koncu bomo pokazali nekaj uporabnih primerov, ki jih največkrat rešujemo na računalniku in njihovo reševanje s pomočjo programskega okolja.

• Language: Slovenian
• Keywords: polinom, aproksimacija, interpolacija, Čebiševa vrsta, Matlab, Chebfun.
• Work type: Bachelor thesis/paper
• Organization: FRI - Faculty of Computer and Information Science
• Year: 2014
• PID: 20.500.12556/RUL-29567
• Publication date in RUL: 24.09.2014

Secondary language

• Language: English
• Title: Software package Chebfun.

Abstract

Almost every mathematical problem that can not be solved manually due to our limited speed of calculating, ​we need to solve with the computer. We first present the problem in an electronic form and then execute a mathematical operation in a form of an algorithm that runs in real time. Usually we can approximate functions by interpolation with polynomials, because ​the polynoms are the easiest and fastest to compute. A significant drawback of this approach is in that large errors can arise thereby. To overcome this problem we in this thesis investigate Chebyshev series which in practice give a very good approximation for smooth functions. In particular we examine an approximation implemented in the open source software tool Chebfun. ​At the end we present a couple of examples used commonly in problem solving with the computer and solve them with the software tool.

• Keywords: polynom, approximation, interpolation, Chebyshev series, Matlab, Chebfun.