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

Abstract

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.