A large unsymmetric linear system problem is transformed into the problem of computing the eigenvector of a large symmetric nonnegative definite matrix associated with the eigenvalue zero, i.e., the computation of the...A large unsymmetric linear system problem is transformed into the problem of computing the eigenvector of a large symmetric nonnegative definite matrix associated with the eigenvalue zero, i.e., the computation of the elgenvector of the cross-product matrix of an augmented matrix associated with the eigenvalue zero. The standard Lanczos method and an improved refined Lanczos method are proposed that compute approximate eigenvectors and return approximate solutions of the linear system. An implicitly restarted Lanczos algorithm and its refined version are developed. Theoretical analysis and numerical experiments show the refined method is better than the standard one. If the large matrix has small eigenvalues, the two new algorithms are much faster than the unpreconditioned restarted GMRES.展开更多
In this paper,we present a kind of pre-symmetrizers for the nonsymmetric linear systems arising from the discretization of nonself-adjoint second order scalar elliptic equation.Based on combination these pre-symmetriz...In this paper,we present a kind of pre-symmetrizers for the nonsymmetric linear systems arising from the discretization of nonself-adjoint second order scalar elliptic equation.Based on combination these pre-symmetrizers with CG method, the new algorithm, LRSCG algorithm, is presented.The numerical results show that the LRSCG algorithm is better than BiCG, CGS, BiCGSTAB, GMRES, QMR and SGMRES methods for thses nonsymmetric linear systems.展开更多
Many applications require the solution of large un-symmetric linear systems with multiple right-hand sides.Instead of applying an iterative method to each of these systems individually,it is often more efficient to us...Many applications require the solution of large un-symmetric linear systems with multiple right-hand sides.Instead of applying an iterative method to each of these systems individually,it is often more efficient to use a block version of the method that generates iterates for all the systems simultaneously.This paper proposes a new adaptive block QMR version based on the incomplete or-thogomalization method(IOM(q))for solving large multi-ple nusymmetric linear systems.How to incorporate de-flation to drop comverged linear systems,and how to delete linearly and almost liearly dependent vectors in the underlying block Krylov sequences are discussed.Nu-merical experiments show that the new adaptive block method has better practical performance and less compu-tational cost and CPU time than block GMRES and other proposed methods for the solution of systems with multi- ple right-hand sides.展开更多
文摘A large unsymmetric linear system problem is transformed into the problem of computing the eigenvector of a large symmetric nonnegative definite matrix associated with the eigenvalue zero, i.e., the computation of the elgenvector of the cross-product matrix of an augmented matrix associated with the eigenvalue zero. The standard Lanczos method and an improved refined Lanczos method are proposed that compute approximate eigenvectors and return approximate solutions of the linear system. An implicitly restarted Lanczos algorithm and its refined version are developed. Theoretical analysis and numerical experiments show the refined method is better than the standard one. If the large matrix has small eigenvalues, the two new algorithms are much faster than the unpreconditioned restarted GMRES.
文摘In this paper,we present a kind of pre-symmetrizers for the nonsymmetric linear systems arising from the discretization of nonself-adjoint second order scalar elliptic equation.Based on combination these pre-symmetrizers with CG method, the new algorithm, LRSCG algorithm, is presented.The numerical results show that the LRSCG algorithm is better than BiCG, CGS, BiCGSTAB, GMRES, QMR and SGMRES methods for thses nonsymmetric linear systems.
基金supported by the National Natural Science Foundation of China(6117030961202098+2 种基金91130024)the Key Project of Development Foundation of Science and Technology of CAEP(2011A0202012: 2012A0202008)the Foundation of National Key Laboratory of Computational Physics
文摘Many applications require the solution of large un-symmetric linear systems with multiple right-hand sides.Instead of applying an iterative method to each of these systems individually,it is often more efficient to use a block version of the method that generates iterates for all the systems simultaneously.This paper proposes a new adaptive block QMR version based on the incomplete or-thogomalization method(IOM(q))for solving large multi-ple nusymmetric linear systems.How to incorporate de-flation to drop comverged linear systems,and how to delete linearly and almost liearly dependent vectors in the underlying block Krylov sequences are discussed.Nu-merical experiments show that the new adaptive block method has better practical performance and less compu-tational cost and CPU time than block GMRES and other proposed methods for the solution of systems with multi- ple right-hand sides.
