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An Implementation of MacMahon’s Partition Analysis in Ordering the Lower Bound of Processing Elements for the algorithm of LU- Decomposition

By Halil Snopce, Ilir Spahiu and Lavdrim Elmazi

Abstract

the solution of large systems of linear equations of the form Ax � b in an effective manner. LU-Decomposition offers good choices for solving this problem. Our approach is to find the lower bound of processing elements needed for this purpose. Here is used the so called “Omega calculus”, as a computational method for solving problems via their corresponding Diophantine relation. From the corresponding algorithm is formed a system of linear diophantine equalities using the domain of computation which is given by the set of lattice points inside the polyhedron. Then is run the Mathematica program DiophantineGF.m. This program calculates the generating function from which is possible to find the number of solutions to the system of Diophantine equalities, which in fact gives the lower bound for the number of processors needed for the corresponding algorithm. There is given a mathematical explanation of the problem as well. Keywords—generating function, lattice points in polyhedron, lower bound of processor elements, system of Diophantine equations and � calculus. I

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.309.573
Provided by: CiteSeerX
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