New algorithms for optimizing the sizes of dixon and dixon dialytic matrices

Extraneous factors are unwanted parameters in the resulting polynomial which is extracted, in the process of eliminating variables in a symbolic polynomial system. The main aim of this research is to reduce the number extraneous factors via optimising the size of Dixon matrices and its modified vers...

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Main Author:	Karimisangdehi, Seyedmehdi
Format:	Thesis
Language:	English
Published:	2012
Subjects:	QA Mathematics
Online Access:	http://eprints.utm.my/id/eprint/35872/1/SeyedmehdiKarimisangdehiPFS2012.pdf
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spelling	my-utm-ep.358722017-09-20T05:58:25Z New algorithms for optimizing the sizes of dixon and dixon dialytic matrices 2012-12 Karimisangdehi, Seyedmehdi QA Mathematics Extraneous factors are unwanted parameters in the resulting polynomial which is extracted, in the process of eliminating variables in a symbolic polynomial system. The main aim of this research is to reduce the number extraneous factors via optimising the size of Dixon matrices and its modified version, Dixon Dialytic matrices. This enhances the process of computing the resultant which is a tool for solving polynomial equations. An optimisation algorithm has been designed based on the sensitivity of the size of the Dixon matrix to the support set of the associated polynomial system. Some new polynomials introduced in this research are used to replace the original polynomials in the system to suppress the effects of the supports of the polynomials on each other. Moreover, in order to find the best position of the support hulls, in relation to each other, appropriate monomial multipliers were constructed and used to multiply each of the polynomials. The Dixon matrix for a generic mixed polynomial system can be optimised using these multipliers. Furthermore, the optimisation of the size of Dixon Dialytic matrix calls for the computation of an optimal arbitrary parameter in the construction of the matrix and the support set of monomial multipliers so that all the Dixon Dialytic sub-matrices are minimised by considering the relationship between the sizes of the corresponding Dixon matrices. Thus, the monomial multipliers that are obtained during the optimization process of the Dixon matrix are used for minimising of the Dixon Dialytic matrix as well; then the search for optimal monomial is initiated in the intersection region of the associated convex hulls of the polynomial system. Appropriate choices of enables further reduction in the size of the matrix. The constructed optimisation algorithms have been analysed in terms of complexity and found to be at least comparable with the existing competing methods of Chtcherba. The results of the implementation of these methods on standard examples reveal the superiority of the new methods and demonstrate no failure in optimising the size of the Dixon matrices compared to Chtcherba’s. 2012-12 Thesis http://eprints.utm.my/id/eprint/35872/ http://eprints.utm.my/id/eprint/35872/1/SeyedmehdiKarimisangdehiPFS2012.pdf application/pdf en public http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:70002?site_name=Restricted Repository phd doctoral Universiti Teknologi Malaysia, Faculty of Science Faculty of Science
institution	Universiti Teknologi Malaysia
collection	UTM Institutional Repository
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Karimisangdehi, Seyedmehdi New algorithms for optimizing the sizes of dixon and dixon dialytic matrices
description	Extraneous factors are unwanted parameters in the resulting polynomial which is extracted, in the process of eliminating variables in a symbolic polynomial system. The main aim of this research is to reduce the number extraneous factors via optimising the size of Dixon matrices and its modified version, Dixon Dialytic matrices. This enhances the process of computing the resultant which is a tool for solving polynomial equations. An optimisation algorithm has been designed based on the sensitivity of the size of the Dixon matrix to the support set of the associated polynomial system. Some new polynomials introduced in this research are used to replace the original polynomials in the system to suppress the effects of the supports of the polynomials on each other. Moreover, in order to find the best position of the support hulls, in relation to each other, appropriate monomial multipliers were constructed and used to multiply each of the polynomials. The Dixon matrix for a generic mixed polynomial system can be optimised using these multipliers. Furthermore, the optimisation of the size of Dixon Dialytic matrix calls for the computation of an optimal arbitrary parameter in the construction of the matrix and the support set of monomial multipliers so that all the Dixon Dialytic sub-matrices are minimised by considering the relationship between the sizes of the corresponding Dixon matrices. Thus, the monomial multipliers that are obtained during the optimization process of the Dixon matrix are used for minimising of the Dixon Dialytic matrix as well; then the search for optimal monomial is initiated in the intersection region of the associated convex hulls of the polynomial system. Appropriate choices of enables further reduction in the size of the matrix. The constructed optimisation algorithms have been analysed in terms of complexity and found to be at least comparable with the existing competing methods of Chtcherba. The results of the implementation of these methods on standard examples reveal the superiority of the new methods and demonstrate no failure in optimising the size of the Dixon matrices compared to Chtcherba’s.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Karimisangdehi, Seyedmehdi
author_facet	Karimisangdehi, Seyedmehdi
author_sort	Karimisangdehi, Seyedmehdi
title	New algorithms for optimizing the sizes of dixon and dixon dialytic matrices
title_short	New algorithms for optimizing the sizes of dixon and dixon dialytic matrices
title_full	New algorithms for optimizing the sizes of dixon and dixon dialytic matrices
title_fullStr	New algorithms for optimizing the sizes of dixon and dixon dialytic matrices
title_full_unstemmed	New algorithms for optimizing the sizes of dixon and dixon dialytic matrices
title_sort	new algorithms for optimizing the sizes of dixon and dixon dialytic matrices
granting_institution	Universiti Teknologi Malaysia, Faculty of Science
granting_department	Faculty of Science
publishDate	2012
url	http://eprints.utm.my/id/eprint/35872/1/SeyedmehdiKarimisangdehiPFS2012.pdf
_version_	1747816376394842112

New algorithms for optimizing the sizes of dixon and dixon dialytic matrices

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