Računanje elementarnih funkcija u računalu

master's thesis

• Title (english): Calculation of elementary functions in a computer
• Author: Vlado Bilogrević
• Mentor: Tina Bosner (mentor)
• Committee member: Tina Bosner (predsjednik povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu su opisane metode i algoritmi za aproksimaciju elementarnih funkcija. Prvo se daju neke osnovne stvari i pojmovi o aritmetici pomične točke. Glavni fokus je na izgradnji različith vrsta polinomnih i racionalnih aproksimacija. Započinjemo prvo s dvije vrste polinomnih aproksimacija, aproksimacija najmanjih kvadrata i minimaks aproksimacija. Pokazujemo jedan iterativni algoritam (Remezov algoritam) koji konstruira minimaks aproksimaciju primjenom Čebiševljevog teorema za karakterizaciju minimaks aproksimacije funkcije. Nakon toga slijede racionalne aproksimacije koje su zapravo nadogradnja na polinomne aproksimacije (polinom je racionalna funkcija s nazivnikom 1). Zatim se govori o metodama za evaluaciju polinoma i greške koje nastaju prilikom aproksimacije gdje se najčešće koristi Hornerova shema. Konačno, bavimo se metodama temeljenih na tablicama koje koristimo kada trebamo aproksimirati funkciju na velikom intervalu. Razmatraju se dvije klase metoda, jedna koristi standarnu tablicu i polinomnu ili racionalnu aproksimaciju, a druga metoda točne tablice.

Abstract (english)

This thesis describes methods and algorithms for the approximation of the elementary functions. It begins by providing some fundamental concepts and principles of floating-point arithmetic. The main focus is on the construction of various types of polynomial and rational approximations. We start with two types of polynomial approximations: least squares approximation and minimax approximation. An iterative algorithm (Remez algorithm) is demonstrated, which constructs a minimax approximation using Chebyshev’s characterization theorem of the minimax approximation of a function. Subsequently, rational approximations are discussed, which are essentially extensions of the polynomial approximations (a polynomial is a rational function with a denominator of 1). The thesis then explores methods for polynomial evaluation and the errors that arise during approximation, with Horner’s scheme being commonly used. Finally, it delves into table-based methods that are employed when approximating a function over a large interval. Two classes of methods are considered, one using a standard table and a polynomial or rational approximation, and the other utilizing accurate tables.

• Keywords: metode za aproksimaciju elementarnih funkcija; algoritmi za aproksimaciju elementarnih funkcija; aritmetika pomične točke; polinomnie aproksimacije; aproksimacija najmanjih kvadrata; minimaks aproksimacija
• Keywords (english): methods for approximation of elementary functions; algorithms for approximation of elementary functions; floating point arithmetic; polynomial approximations; least squares approximation; minimax approximation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:863767
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar računarstva i matematike (sveučilišni magistar računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-01-26 13:58:54