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Development of symbolic algorithms for certain algebraic processes

This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general clas...

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Main Authors: Abd. Rahman, Ali, Aris, Nor'aini
Format: Monograph
Language:English
Published: Research Management Center 2007
Subjects:
QA Mathematics
Online Access:http://eprints.utm.my/id/eprint/2417/1/NorainiAris2007_DevelopmentAlgorithmsforAlgebraicProcesses.pdf
http://eprints.utm.my/id/eprint/2417/
http://rmc.utm.my/
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spelling my.utm.24172017-10-10T04:10:15Z http://eprints.utm.my/id/eprint/2417/ Development of symbolic algorithms for certain algebraic processes Abd. Rahman, Ali Aris, Nor'aini QA Mathematics This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested. Research Management Center 2007-04 Monograph NonPeerReviewed application/pdf en http://eprints.utm.my/id/eprint/2417/1/NorainiAris2007_DevelopmentAlgorithmsforAlgebraicProcesses.pdf Abd. Rahman, Ali and Aris, Nor'aini (2007) Development of symbolic algorithms for certain algebraic processes. Project Report. Research Management Center. (Submitted) http://rmc.utm.my/
institution Universiti Teknologi Malaysia
building UTM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Teknologi Malaysia
content_source UTM Institutional Repository
url_provider http://eprints.utm.my/
language English
topic QA Mathematics
spellingShingle QA Mathematics
Abd. Rahman, Ali
Aris, Nor'aini
Development of symbolic algorithms for certain algebraic processes
description This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested.
format Monograph
author Abd. Rahman, Ali
Aris, Nor'aini
author_facet Abd. Rahman, Ali
Aris, Nor'aini
author_sort Abd. Rahman, Ali
title Development of symbolic algorithms for certain algebraic processes
title_short Development of symbolic algorithms for certain algebraic processes
title_full Development of symbolic algorithms for certain algebraic processes
title_fullStr Development of symbolic algorithms for certain algebraic processes
title_full_unstemmed Development of symbolic algorithms for certain algebraic processes
title_sort development of symbolic algorithms for certain algebraic processes
publisher Research Management Center
publishDate 2007
url http://eprints.utm.my/id/eprint/2417/1/NorainiAris2007_DevelopmentAlgorithmsforAlgebraicProcesses.pdf
http://eprints.utm.my/id/eprint/2417/
http://rmc.utm.my/
_version_ 1643643584558137344
score 13.211869

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