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Approximate inverses of almost singular matrices still contain useful information
Citation Link: https://doi.org/10.15480/882.319
Publikationstyp
Technical Report
Date Issued
1990
Sprache
English
Author(s)
Institut
TORE-DOI
First published in
Number in series
90.1
Citation
Berichte des Forschungsschwerpunktes Informations- und Kommunikationstechnik 90.1: (1990)
Publisher
Techn. Univ. Hamburg-Harburg
It is well-known that, roughly spoken, a matrix inversion on a computer working in base B with t digits precision in the mantissa applied to a matrix of condition Bk produces approximately t-k correct digits of the inverse. For condition >> Bt one might conclude that an approximate inverse contains virtually useless information.
In this note we will show that the latter is not true. An approximate inverse may still be useful, e.g. as a preconditioner. An extended set of examples show that preconditioning a matrix using an approximate inverse (computed in t digits precision) lowers the condition number by a factor Bt. As an example we develop an algorithm for solving systems of linear equations up to condition B2t strictly using t digits precision for all calculations and only allowing for double precision accumulation of inner products.
In this note we will show that the latter is not true. An approximate inverse may still be useful, e.g. as a preconditioner. An extended set of examples show that preconditioning a matrix using an approximate inverse (computed in t digits precision) lowers the condition number by a factor Bt. As an example we develop an algorithm for solving systems of linear equations up to condition B2t strictly using t digits precision for all calculations and only allowing for double precision accumulation of inner products.
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