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The Main Effects of Rounding Errors in Krylov Solvers for Symmetric Linear Systems
, 1997
"... 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 linear systems, by solving the reduced system in one way or another. This leads to wellknown ..."
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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 linear systems, by solving the reduced system in one way or another. This leads to wellknown methods: MINRES (GMRES), CG, CR, and SYMMLQ. We will discuss in what way and to what extent the various approaches are sensitive to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods (except CR), and we will not consider the errors in the Lanczos process itself. These errors may lead to large perturbations with respect to the exact process, but convergence takes still place. Our attention is focussed to what happens in the solution phase. We will show that the way of solution may lead, under circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are...
Parallel Preconditioning with Sparse Approximate Inverses
 SIAM J. Sci. Comput
, 1996
"... A parallel preconditioner is presented for the solution of general sparse linear systems of equations. A sparse approximate inverse is computed explicitly, and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its applicati ..."
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A parallel preconditioner is presented for the solution of general sparse linear systems of equations. A sparse approximate inverse is computed explicitly, and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its application only requires a matrixvector product. The sparsity pattern of the approximate inverse is not imposed a priori but captured automatically. This keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum. Rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system, and the proximity of the approximate to the true inverse is estimated. An extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach. 1 Introduction We consider the linear system of equations Ax = b; x; b 2 IR n : (1) The work of M. Grote was...
Numerical Algorithms 7(1994)75109 75 BiCGstab(l) and other hybrid BiCG methods
, 1993
"... It is wellknown that BiCG can be adapted so that the operations with A T can be avoided, and hybrid methods can be constructed in which it is attempted to further improve the convergence behaviour. Examples of this are CGS, BiCGSTAB, and the more general BiCGstab(l) method. In this paper it is s ..."
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It is wellknown that BiCG can be adapted so that the operations with A T can be avoided, and hybrid methods can be constructed in which it is attempted to further improve the convergence behaviour. Examples of this are CGS, BiCGSTAB, and the more general BiCGstab(l) method. In this paper it is shown that BiCGstab(l) can be implemented in different ways. Each of the suggested approaches has its own advantages and disadvantages. Our implementations allow for combinations of BiCG with arbitrary polynomial methods. The choice for a specific implementation can also be made for reasons of numerical stability. This aspect receives much attention. Various effects have been illustrated by numerical examples. Keywords: BiConjugate gradients, nonsymmetric linear systems, CGS, BiCGSTAB, iterative solvers, ORTHODIR, Krylov subspace. AMS subject classification: 65F10.