Some Studies On The Parallel Solving For Sparse Linear System Of Equations

The question of partial differential equations in a numerical way is solved by using the numerical discrete method.It is to transform it into a sparse linear equation set.Two common solutions to linear equations are direct method and iterative method.When solving large sparse linear equations,the Krylov subspace method in iteration is the most important method and commonly used.When the Krylov subspace method runs on a parallel computer,the efficiency of solving problems is often very low,so exploring the suitable Krylov subspace method which runs on a parallel computer in parallel algorithm is very necessary.In the distributed memory parallel computers,global communication caused by the way to calculate Inner product effects the effective performing of the parallel computing.So the key-point to Krylov subspace of parallel algorithm is to weaken the impact on global inner product computation communication.Because inner product computation may cause some global communication problems,the variable preconditioning Successive Over Relaxation(SOR)-Bicon-jugate Residual Method(BiCR),and preconditioning of Jacobi Biconjugate Resid-ual Method(JBiCR)of the two methods were studied.The preconditioning of SOR-Biconjugate Residual Method(SOR-BiCR)which is based on the biconjugate residual method using various preconditioning techniques,JBiCR algorithm is a few steps Jacobi constructed by iterative adaptive preprocessing is embedded in the BiCR algorithm,further to make these two kinds of parallel algorithm can ef-ficiently solve the distributed computing environment,computing the order of the transformation of preconditioned SOR-BiCR algorithm and JBiCR algorithm,are suitable for solving large sparse nonsymmetric linear equations parallel variable pre improved SOR-Biconjugate Residual Algorithm(SOR-IBiCR)and the improved preconditioning Jacobi Conjugate Residual Method(PJBiCR),SOR-IBiCR algo-rithm and PJBiCR algorithm are the original algorithm of discrete inner product computation instead of computing for continuous inner product,and the global com-munication number of the original algorithm two times down time,and computing the inner product matrix vector multiplication algorithm independent,reduces data correlation,in order to increase the amount of calculation of small the cost of the parallel efficiency of the algorithm improved.Through the theoretical analysis of parallel time,speedup,scalability and paral?lel performance improvement ratio comparison between SOR-IBiCR algorithm and SOR-BiCR algorithm,the result shows that SOR-IBiCR algorithm is better than the SOR-BiCR algorithm has a better parallel time,less scalability and speed up,SOR-IBiCR algorithm can achieve a speedup of 2 times SOR-BiCR algorithrm,par-allel improvement ratio to achieve 50%performance;through theoretical analysis of parallel time,speedup,scalability and parallel performance improvement ratio com-parison between PJBiCR algorithm and JBiCR algorithm,the PJBiCR algorithm is better than the JBiCR algorithm has less parallel time,better scalability and speedup,PJBiCR algorithm can achieve the speedup of 2 times JBiCR the par-allel algorithm,the improved ratio reached 50%.by parallel numerical experiment results show that consistent with the theoretical analysis of numerical performance.The convergence time and convergence stability of JBiCR and BiCR,CGS and BiCGSTAB algorithm are compared.The results show that JBiCR algorithm has better convergence stability and shorter iteration time than those three algorithms.