Točnost sumacijskih algoritama

master's thesis

• Author: Ivona Varnica
• Mentor: Nela Bosner (mentor)
• Committee member: Nela Bosner (predsjednik povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-02-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Glavna tema ovog rada je točnost algoritama koje koristimo pri sumiranju brojeva s pomičnom točkom, u aritmetici konačne preciznosti na računalu. Na početku rada smo opisali konačnu aritmetiku računala i neke njene osnovne pojmove na kojima se temelji ostatak rada. Zatim smo definirali tri osnovne metode sumacije. Koristeći analizu grešaka i statističke procjene točnosti, usporedili smo metode. Nakon toga smo uveli još jednu točniju vrstu sumacije, takozvano kompenzirano sumiranje. Uz pomoć Kahanovog i Priestovog algoritma, detaljnije smo analizirali i tu metodu. Između svih tih metoda, kao i nekih dodatnih metoda, nismo mogli izdvojiti onu najbolju, u smislu točnosti za baš sve moguće ulazne podatke. Ipak, numerički eksperimenti u MATLAB-u su pokazali da za bilo koje dvije metode sumacije, moguće je naći podatke za koje je jedna od tih dviju metoda točnija.

Abstract (english)

The main subject of this thesis is accuracy of summation algorithms in finite precision arithmetic on a computer. At the beginning, we have described the finite arithmetic of a computer and some of the basic concepts that the rest of the thesis is based on. Then we define three main summation methods. By using the error analysis and statistical estimates of accuracy, we compare the methods. After that, we introduced a more accurate type of summation, so-called compensated summation. With the help of Kahan’s and Priest’s algorithms, we described in detail that method. Among all these methods and some additional methods, we could not single out the best one, in terms of accuracy for every possible choice of input data. However, numerical experiments in MATLAB show that, given any two summation methods, data can be found for which either method is more accurate than the other.

• Keywords: točnost algortiama; konačna aritmetika računala; Kahanov algoritam; Priestov algoritam; MATLAB
• Keywords (english): summation algorithm accuracy; finite precision arithmetic; Kahan’s algorithm; Priest’s algorithm; MATLAB
• Language: croatian
• URN:NBN: urn:nbn:hr:217:100525
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-10-27 11:49:05