Criterion for SVD Truncation

[Junoexpress]

Hi,

I have a linear system of eqns I am solving of the form:
Ax = b
A is a 32 x 32 matrix with complex entries, having full rank.
It is not a normal matrix, but has, of course, the SVD A = U L V*.
A is somewhat strange in that N of the (2N)^2 entries have a random
component to them.
Also, half of the components of b also have a random component to
them.
Thus, to get a handle on the behavior of the solutions, I've run a lot
of Monte-Carlo simulations.

What I find is that 31 of the 32 singular values are close to unity
( w/i one order of magnitude). The smallest eigenvalue, however, can
range from 0.0001 to a little under 0.1. When I solve for x using the
pseudo-inverse of A, this small eigenvalue can of course wreak havoc
on the stability of my solutions (since it comes into the solution
vector as its reciprocal, which will be large).

Is there a way I can justify dropping this one signular value in
constructing the solution? (In other words, I can use the pseudo-
inverse of A, where in the last diagonal entry of L, I simply put a
zero instead of the reciprocal of the smallest eval.)

Thanks for any help or references you can provide,

Matt

[Ilya Zakharevich]

[A complimentary Cc of this posting was sent to
Junoexpress
<MTBre...@gmail.com>], who wrote in article <gb8fe3$m76$1...@news.acm.uiuc.edu>:

> I have a linear system of eqns I am solving of the form:
> Ax = b

*Why* do you solve it?

> Is there a way I can justify dropping this one signular value in
> constructing the solution?

Hope this helps,
Ilya

[Robert Israel]

Junoexpress <MTBre...@gmail.com> writes:

The usual justification is that the result you get, while not having
Ax = b, has Ax very close to b, or Bx very close to b for some B very
close to A. If the elements of A and/or b are only approximations of some
(possibly unknown) exact values, e.g. come from physical measurements, that is
the best that you can hope for.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[Ross]

On Sep 24, 4:00 pm, Robert Israel
<isr...@math.MyUniversitysInitials.ca> wrote:

There is a discussion of this in "Numerical Recipes", which is
available online. As they point out, it is not often you get to set
infinity equal to zero.