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Inversion of extremely ill-conditioned matrices in floating-point
Publikationstyp
Journal Article
Date Issued
2009-10
Sprache
English
Author(s)
Institut
TORE-URI
Volume
26
Issue
2/3
Start Page
249
End Page
277
Citation
Japan Journal of Industrial and Applied Mathematics 2/3 (26): 249-277 (2009)
Publisher DOI
Scopus ID
Let an n × n matrix A of floating-point numbers in some format be given. Denote the relative rounding error unit of the given format by eps. Assume A to be extremely ill-conditioned, that is cond(A) > eps-1. In about 1984 I developed an algorithm to calculate an approximate inverse of A solely using the given floating-point format. The key is a multiplicative correction rather than a Newton-type additive correction. I did not publish it because of lack of analysis. Recently, in [9] a modification of the algorithm was analyzed. The present paper has two purposes. The first is to present reasoning how and why the original algorithm works. The second is to discuss a quite unexpected feature of floating-point computations, namely, that an approximate inverse of an extraordinary ill-conditioned matrix still contains a lot of useful information. We will demonstrate this by inverting a matrix with condition number beyond 10300 solely using double precision. This is a workout of the invited talk at the SCAN meeting 2006 in Duisburg.
Subjects
Accurate dot product
Accurate summation
Condition number
Error-free transformations
Extremely ill-conditioned matrix
Multiplicative correction
DDC Class
004: Informatik
510: Mathematik