Small-Sample Statistical Condition Estimates (Kenney and Laub)

======================================================================
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

----

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. 


======================================================================

SIAM
3600 University City Science Center
Philadelphia, PA  19104-2688, USA

Phone:  215-382-9800, 800-447-7426 (USA only)
Fax:  215-386-7999
E-mail:  journals@siam.org