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st: Matrix inversion "bug"


From   Mark Schaffer <M.E.Schaffer@hw.ac.uk>
To   Statalist <statalist@hsphsun2.harvard.edu>
Subject   st: Matrix inversion "bug"
Date   Tue, 10 Jun 2003 00:18:44 +0100 (BST)

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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