Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems

S Serra - BIT Numerical Mathematics, 1994 - Springer
BIT Numerical Mathematics, 1994Springer
A particular class of preconditioners for the conjugate gradient method and other iterative
methods is proposed for the solution of linear systems A n, mx= b, where A n, m is an n× n
positive definite block Toeplitz matrix with m× m Toeplitz blocks. In particular we propose a
sparse preconditioner P n, m such that the condition number of the preconditioned matrix
turns out to be less than a suitable constant independent of both n and m, even if the
condition number of A n, m tends to∞. This leads to iterative methods which require a …
Abstract
A particular class of preconditioners for the conjugate gradient method and other iterative methods is proposed for the solution of linear systemsA n,mx=b, whereA n,m is ann×n positive definite block Toeplitz matrix withm×m Toeplitz blocks. In particular we propose a sparse preconditionerP n,m such that the condition number of the preconditioned matrix turns out to be less than a suitable constant independent of bothn andm, even if the condition number ofA n,m tends to ∞. This leads to iterative methods which require a number of steps independent ofm andn in order to reduce the error by a given factor.
Springer
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