From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: accuracy of computed matrix inverse
Date: 7 Nov 2000 00:45:46 -0500
Newsgroups: sci.math.num-analysis
Summary: [missing]

In article <fA1N5.1824$6b7.36122@vixen.cso.uiuc.edu>,
John Prussing <prussing@aae.uiuc.edu> wrote:
:One way to test the accuracy of a computed matrix inverse is
:obviously to multiply the computed inverse by the original matrix
:and see how close the product is to the identity matrix.
:
:1) Is this the best way? 
:
 I am not qualified to pass judgment on that. I do have a
(conditional) formula estimating the distance from A^(-1) 
to a guess X. Pick any matrix norm, denote 

 q the smaller of  ||I - A*X|| and  ||I - X*A||,
 and suppose q < 1.

 Then ||inv(A) - X || <= ||X - X*A*X|| / (1-q).

 The proof uses just algebraic identities and comparison with
geometric series. Care must be exercised when A is 
ill-conditioned.

 Incidentally, X - X*A*X  is the "correction" if one uses
Hotelling's iteration : X_new = X + (X - X*A*X) which will
converge quadratically if q<1  (but watch for ill-conditioning
for loss of valid digits).

 Here is a contrived example when ||I-X*A|| is drastically 
different from ||I-A*X||:

A = 

[ 500  911 ]
[ 309  563 ]

X =

[ 563.563  -911.911 ]
[ -309      500     ]

We have

I - X*A =

[ -0.001   0 ]
[  0       0 ]

I - A*X =

[ -281.5     455.5   ]
[ -173.967   281.499 ]

so in the "maximum of absolute row sums" norm,

||I - X*A|| = 0.001
||I - A*X|| = 737

>2) If some additional property exists, say the determinant is equal
>to 1, can this be used to enhance the accuracy of the computed
>inverse?

Not to my knowledge. The accuracy circumstances should not change
with scaling (especially by powers of the base, 10 or 2), which may
change the determinant according to well-known rules. Perhaps some
other properties can be observed and used, such as symmetry.

Finally, users of numerical methods have been urged for some time
to re-phrase problems so that inverse matrices are not used, unless
they are very obvious and "safe" (such as inverses of unitary
matrices...), or unless the entries of the inverse matrix are parts
of the required final answer.
It may be more accurate and faster to use the inverse linear mapping
without representing it as a matrix multiplication (e.g. as a pair of
forward and backward substitution routines after a triangular
factorization).

Hope it helps, ZVK(Slavek)

>--
>=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
>John E. Prussing
>Dept. of Aeronautical & Astronautical Engineering
>University of Illinois at Urbana-Champaign
>http://www.uiuc.edu/~prussing
>=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=