A Hybrid Parallel Solving Algorithm on GPU for Quasi-Tridiagonal System of Linear Equations

Abstract

There are some quasi-tridiagonal system of linear equations arising from numerical simulations, and some solving algorithms encounter great challenge on solving quasi-tridiagonal system of linear equations with more than millions of dimensions as the scale of problems increases. We present a solving method which mixes direct and iterative methods, and our method needs less storage space in a computing process. A quasi-tridiagonal matrix is split into a tridiagonal matrix and a sparse matrix using our method and then the tridiagonal equation can be solved by the direct methods in the iteration processes. Because the approximate solutions obtained by the direct methods are closer to the exact solutions, the convergence speed of solving the quasi-tridiagonal system of linear equations can be improved. Furthermore, we present an improved cyclic reduction algorithm using a partition strategy to solve tridiagonal equations on GPU, and the intermediate data in computing are stored in shared memory so as to significantly reduce the latency of memory access. According to our experiments on 10 test cases, the average number of iterations is reduced significantly by using our method compared with Jacobi, GS, GMRES, and BiCG respectively, and close to those of BiCGSTAB, BiCRSTAB, and TFQMR. For parallel mode, the parallel computing efficiency of our method is raised by partition strategy, and the performance using our method is better than those of the commonly used iterative and direct methods because of less amount of calculation in an iteration.