Approximation on disjoint intervals and its applicability to matrix preconditioning

Maurice Hasson; Juan M. Restrepo

doi:10.1080/17476930701524222

Approximation on disjoint intervals and its applicability to matrix preconditioning

Maurice Hasson, Juan M. Restrepo

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

A polynomial preconditioner to an invertible matrix is constructed from the (near) best uniform polynomial approximation to the function 1/x on the eigenvalues of the matrix. The preconditioner is developed using symmetric matrices whose eigenvalues are located in the union of two (disjoint) intervals. The full spectrum of the matrix need not be known in order for the strategy to be applicable. All that is essential is an estimate on the bounds of the larger of the two clusters of eigenvalues. The algorithm, based on the strategy, is shown to be numerically stable with respect to the size of the matrix. In fact it yields, at low cost, an approximation to the inverse of the matrix to within a specified tolerance.

Original language	English
Pages (from-to)	757-769
Number of pages	13
Journal	Complex Variables and Elliptic Equations
Volume	52
Issue number	9
DOIs	https://doi.org/10.1080/17476930701524222
State	Published - 2007
Externally published	Yes

Funding

The authors wish to express their appreciation to James Hyman and Rob Indik for very fruitful discussions. JMR is supported by a DOE ‘Early Career’ grant DE-FG02-02ER25533. JMR also thanks T7 at Los Alamos, where much of this work was performed. MH was supported by a NSF/VIGRE fellowship.

Funders	Funder number
VIGRE
National Science Foundation
U.S. Department of Energy	DE-FG02-02ER25533

Keywords

41A05
41A25
41A28
65F10
65F15
AMS Subject Classifications
Matrix preconditioning
Normal matrices
Polynomial approximation

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10.1080/17476930701524222

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Approximation on disjoint intervals and its applicability to matrix preconditioning

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