In this paper, we develop an implicitly restarted block Arnoldi algorithm in a vector-wise fashion. The vector-wise construction greatly simplifies both the detection of necessary deflation and the actual deflation it...In this paper, we develop an implicitly restarted block Arnoldi algorithm in a vector-wise fashion. The vector-wise construction greatly simplifies both the detection of necessary deflation and the actual deflation itself, so it is preferable to the block-wise construction. The numerical experiment shows that our algorithm is effective.展开更多
Many applications require the solution of large nonsymmetric 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...Many applications require the solution of large nonsymmetric 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. In this paper, we propose a block version of generalized minimum backward (GMBACK) for solving large multiple nonsymmetric linear systems. The new method employs the block Arnoldi process to construct a basis for the Krylov subspace K m(A, R 0) and seeks X m∈X 0+K m(A, R 0) to minimize the norm of the perturbation to the data given in A.展开更多
The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz ...The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz vectors, a modified strategy is proposed such that new approximate eigenvectors are certain combinations of the Ritz vectors and the waSted (m+1) th block basis vector and their corresponding residual norms are minimized in a certain sense. They can be cheaply computed by solving a few small 'dimensional minimization problems. The resulting modified m-step block Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, a modified m-step iterative block Arnoldi algorithm is presented. Numerical experiments are reported to show that the modified m-step algorithm is often considerably more efficient than the standard (m+1)-step iterative one.展开更多
基金This work is supported by National Natural Science Foundation of China No. 10531080.
文摘In this paper, we develop an implicitly restarted block Arnoldi algorithm in a vector-wise fashion. The vector-wise construction greatly simplifies both the detection of necessary deflation and the actual deflation itself, so it is preferable to the block-wise construction. The numerical experiment shows that our algorithm is effective.
基金Supported by the National Natural Science Foundation of China(61473148)the Natural Science Foundation of Jiangsu Province of China(BK20141408)the Fundamental Research Funds for the Central Universities(NZ2014101)
文摘Many applications require the solution of large nonsymmetric 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. In this paper, we propose a block version of generalized minimum backward (GMBACK) for solving large multiple nonsymmetric linear systems. The new method employs the block Arnoldi process to construct a basis for the Krylov subspace K m(A, R 0) and seeks X m∈X 0+K m(A, R 0) to minimize the norm of the perturbation to the data given in A.
文摘The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz vectors, a modified strategy is proposed such that new approximate eigenvectors are certain combinations of the Ritz vectors and the waSted (m+1) th block basis vector and their corresponding residual norms are minimized in a certain sense. They can be cheaply computed by solving a few small 'dimensional minimization problems. The resulting modified m-step block Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, a modified m-step iterative block Arnoldi algorithm is presented. Numerical experiments are reported to show that the modified m-step algorithm is often considerably more efficient than the standard (m+1)-step iterative one.
