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Differences in the effects of rounding errors in Krylov solvers for symmetric indefinite linear systems
, 1999
"... The 3term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving the reduced system in one way or another. Thi ..."
Abstract

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The 3term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving the reduced system in one way or another. This leads to wellknown methods: MINRES, GMRES, and SYMMLQ. We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are supported and illustrated by numerical examples. 1 Introduction We will consider iterative methods for the construction of approximate solutions, starting with...
Implementation Aspects
"... e inner products, vector updates and matrix vector product are easily parallelized and vectorized. The more successful preconditionings, i.e, based upon incomplete LU decomposition, are not easily parallelizable. For that reason one is often satisfied with the use of only diagonal scaling as a preco ..."
Abstract
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e inner products, vector updates and matrix vector product are easily parallelized and vectorized. The more successful preconditionings, i.e, based upon incomplete LU decomposition, are not easily parallelizable. For that reason one is often satisfied with the use of only diagonal scaling as a preconditioner on highly parallel computers, such as the CM2 [24]. On distributed memory computers we need large grained parallelism in order to reduce synchronization overhead. This can be achieved by combining the work required for a successive number of iteration steps. The idea is to construct first in parallel a straight forward Krylov basis for the search subspace in which an update for the current solution will be determined. Once this basis has been computed, the vectors are orthogonalized, as is done in Krylov subspace methods. The construction as well as the orthogonalization can be done with large grained parallelism, and has su#cient degree of parallelism in it. This approach has be