Matrix Iterations: The Six Gaps Between Potential Theory and Convergence

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Abstract

The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, ...) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor rho less than 1 can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in reducing the actual computation to this scalar estimate. These six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.

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1996-06
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Cornell University
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theory center
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http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.tc/96-245
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technical report
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