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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. 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. |
