A Parallel Solver for Certain Toeplitz Tridiagonal Systems on Distributed-Memory Multicomputers

TL;DR: The analysis of complexity and numerical experiments show that the algorithm's speedup satisfy: S p(n)→p(n→+∞) .

Abstract: The tridiagonal Toeplitz linear systems occur repeatedly in the solution of the implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the Transport equation, under a variety of boundary conditions. Interest in fast direct methods for solving these kind of linear systems has long been a hot spot of research. A parallel algorithm for certain tridiagonal Toeplitz linear systems on distributed memory multicomputers is presented. Derivation of the algorithm is introduced. The algorithm is based on the factorization of the coefficient matrix and the principle of ‘divide and conquer’ in designing parallel algorithms. Authors make full use of the special structure of the coefficient matrix. By using the customary nesting procedure, Horner's formula, authors avoid the necessary of quantities α i, (-α) i(i=2,3,…,m ) and (α m) i, (-α m) i(i=2,3,…,p-1 ). This reduces the algorithm's redundancy computation caused by parallelization. The complexity of the algorithm is analyzed using Log P model. Its communication mechanism is very simple. The communication complexity is only related to p , the number of processors, and not related to n , the size of the matrix. The algorithm's parallel efficiency is high. The analysis of complexity and numerical experiments show that the algorithm's speedup satisfy: S p(n)→p(n→+∞) . This is the best a parallel algorithm can reach. The algorithm has been implemented on parallel computers. The results of numerical experiments about the algorithm on a distributed memory multicomputer are presented in this paper.