Small-Sample Statistical Condition Estimates (Kenney and Laub)
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SIAM Journal on Scientific Computing
Volume 15-1, January 1994, pp. 36-61
(C) 1994 by Society for Industrial and Applied Mathematics
All rights reserved
Title: Small-Sample Statistical Condition Estimates for
General Matrix Functions
Author: C. S. Kenney and A. J. Laub
AMS Subject
Classifications: 65F35, 65F30, 15A12
Key words: conditioning, matrix functions
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ABSTRACT
A new condition estimation procedure for general matrix functions
is presented that accurately gauges sensitivity by measuring the
effect of random perturbations at the point of evaluation. In this
procedure the number of extra function evaluations used to evaluate
the condition estimate determines the order of the estimate. That is,
the probability that the estimate is off by a given factor is
inversely proportional to the factor raised to the order of the
method. The ``transpose-free'' nature of this new method allows it to
be applied to a broad range of problems in which the function maps
between spaces of different dimensions. This is in sharp contrast to
the more common power method condition estimation procedure that is
limited, in the usual case where the Fr\'{e}chet derivative is known
only implicitly, to maps between spaces of equal dimension. A group
of examples illustrates the flexibility of the new estimation procedure
in handling a variety of problems and types of sensitivity estimates,
such as mixed and componentwise condition estimates.
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