Tridiagonal matrix algorithm

Abstract: The paper consideres a parallel implementation of the splitting methods for heat conductivity problem, provides results of the numeric simulation performed on clustered computer system, diagrams of the speedup factors dependencies on a number of elementary machines, and defines dependence of the optimal computer systems size on the size of the model. The goal of the research performed, is a determination of the optimal parameters of the parallel algorithm and computer system that would minimize the programs execution time. A parallel implementation of splitting method for the heat conductivity problem t xx U =U +Uyy, defined in a square area with known initial and boundary conditions, has been selected as a point of research. Numeric model was built upon a mesh that is uniformly set for spatial and time coordinates. The splitting method reduces solution of the chosen problem to a numeric solving of 2TN sets of linear equations with square tridiagonal N  N matrices (here N stands for the size of models mesh on spatial coordinates, T is the mesh size on time coordinate). Thomas algorithm has been applied to solve matrices. A linear array of p elementary machines has been chosen as a special structure of the computer system. The definition domain has been decomposed into p uniform sub-domains along one of the spatial axis. A separate thread of the parallel program calculates data within each sub-domain. The chosen algorithm assumes point-to-point data exchange operations between neighbor threads. Program code has been written in C++. MPICH 1.21 library has been utilized to establish intercommunication between threads of the parallel program. Numeric simulation has carried on a segment of the distributed multi-cluster computer system of CPCT SibSUTIS.

Abstract: The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for m-component reaction-diffusion systems with a tridiagonal symmetric toeplitz matrix of diffusion coefficients and with nonhomogeneous boundary conditions. The proposed technique is based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth.