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Re: st: Hilbert matrices Mata testing


From   wgould@stata.com (William Gould, Stata)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Hilbert matrices Mata testing
Date   Fri, 04 May 2007 10:09:31 -0500

Marcos <M.Delprato@westminster.ac.uk> writes, 

> I found in the Mata manual Hilbert matrices. They are defined as a matrix H
> with elements H[i,j]=1/(i+j-1). As it says in the help (type help
> mf_hilbert):  Hilbert matrices are notoriously ill conditioned, with near
> zero determinants.
>
> The help also adds Hilbert(n) and invHilbert(n) are used in testing Mata.  I
> think that is intuitive as since they are ill matrices with very low
> determinants and nearly singular you could use them to test the quality of,
> for instance, Mata's solver (e.g., qrsolve, lusolve, qrinv) and different
> inverters (e.g., cholinv, invsym).
>
> Am I right?  Could anybody supply an example of how Hilbert matrices are
> used to test Mata?

We at StataCorp have do-files used for testing Stata.  One of the do-files
simply serves to call all othe others.  Currently, there are 3,103 do-files
totalling 568,032 lines.

The part that tests Mata includes 325 files comprising 45,626 lines.
Here are a few lines from one of the files:

        ...
        assert(reldif(1/det(Hilbert(2)), 12)<1e-16)
        assert(reldif(1/det(Hilbert(3)), 2160)<1e-14)
        assert(reldif(1/det(Hilbert(4)), 6048000)<1e-13)
        assert(reldif(1/det(Hilbert(5)), 266716800000)<1e-12)
        ...
	assert(mreldif(pinv(pinv(Hilbert(4))), Hilbert(4))<1e-12))
	assert(mreldif(pinv(pinv(Hilbert(5))), Hilbert(5))<1e-11))
	...

-- Bill
wgould@stata.com
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