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st: RE: Matrix inversion "bug"


From   "Ines Butler" <ibutler@ieral.org>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: Matrix inversion "bug"
Date   Tue, 10 Jun 2003 11:45:13 -0300

Mark
I think I am encountering with a similar problem to the one you mention
about symentric matrices.

I am usingmat function 'mat accum' with the options noconstant and
deviations to construct a matrix from data which should return a positive
definite matrix (because divided by N-1 is the covariance matrix). However,
when I try to use cholesky factorization, an error message appears saying
that the matrix is not positive definite.
Do you have any idea on how to cope with this problem?
Thank you very much!

Inés Butler
IERAL Fundación Mediterránea



-----Mensaje original-----
De: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu]En nombre de Mark Schaffer
Enviado el: Monday, June 09, 2003 8:19 PM
Para: Statalist
CC: Mark Schaffer
Asunto: st: Matrix inversion "bug"


Dear Statalisters:

Every now and then I encounter a matrix inversion "bug" in Stata.  It
cropped up yet again last week when an ivreg2 user wrote to me, and I
finally decided to see what my fellow Statalisters think of it.

Say we have a matrix M which is calculated as X*S*X'.  S is symmetric and
so is M.  We want to invert M and so the best thing to do is to use
-syminv-.  The Stata programming manual tells us to use -syminv- instead
of -inv- wherever possible.

The problem is this.  Although M is guaranteed to be symmetric in
principle, it is not guaranteed to be symmetric in practice.  Tiny rounding
errors can arise when Stata multiplies matrices.  Once in a while, M will
be very slightly non-symmetric, but when it is, -syminv- will exit with an
error.

When I first encountered this, I wrote to Stata Technical Support, and I
received a very helpful and simple fix: prior to the call to syminv, simply
do the following:

mat M = (M+M')/2

This makes the matrix symmetric.  Works, no problem.  But...

Arguably, this is just a fix for what is really a Stata bug.  Ought it be
possible for Stata's code to recognise when the result of a matrix
multiplication is supposed to be a symmetric matrix?

Whether or not this should be called a "bug", it may still be the case that
it's not possible for our friends at Stata Corp to fix it.  If so, then
there are several possibilities:

(1) Prior to using -syminv-, we should always (?) symmetrize the matrix we
want to invert using the code fragment above.

(2) Any (?) time we call -syminv-, we should use -capture-.  If the
inversion using -syminv- fails, we should call -inv-.

(3) Neither of the above is very pretty.  An alternative is to ask our
Stata Corp friends to, say, add a variant of the -syminv- function, say
-syminv2-.  This function would automatically symmetrize a square matrix
before inverting it.

What do you think?

--Mark

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