Parallel Algorithm For Solving Large-scale Linear Equations And Applications

Solving large-scale linear equations plays a very important role in Engineering and Computing application field. Parallel algorithm for solving large-scale linear equations on distributed-memory multi-computer is one of the hotspot problems. The work presented in this thesis focuses on parallel algorithm for solving linear equations and the application of electromagnetic computing.The main results of this research are as follows:(1) A parallel algorithm for solving dense matrix linear equations on distributed-memory multi-computer is presented. Convergent velocity of this algorithm is better than parallel Jacobi method's, and parallelism is better than Gauss-Seidel method's. According to theoretical analysis, convergent conditions are given. Moreover numerical experiments indicate that results agree with theory.(2) We construct preconditioner of symmetry and dissymmetry conjugate gradient method based on the splitting matrix of (1), which made the algorithm to suit for parallel computing of dense linear equations, and the numerical experiments show that the algorithm is feasible and effective.(3) Optimization technique of sparse linear equations is discussed. Optimization storage methods for sparse matrix are researched, including: Coordinate Storage Scheme, Diagonal Storage Scheme, Ellpack-Itpack Storage Scheme and Compressed Row Storage Scheme. And the implemented method has been given for distributed-memory multi-computer. Numerical experiments show the advantage of optimization storage technique.(4) Developed a solving software for sparse linear equations based on optimization of (3). The software which run on distributed-memory multi-computer and Linux System can solve sparse matrix by simple operations. It is a try in application of parallel algorithm.(5) Static electromagnetic problem is solved with parallel FEM technique, multisplitting parallel algorithm for sparse matrix is used.