Subject: Linear independence of a basis set

From: "David M. Schrader" <[email-masked]>
Date: Wed, 09 Mar 1994 14:06:31 -0600 (CST)

Netters:

I am looking for a numerical test for linear independence of a basis set.  I
know that the overlap matrix S is singular for a linearly dependent basis set,
and that det(S) = 1 for an orthonormal basis set.  Is det(S) a good criterion
for linear independence?  I am sure many people have thought and written about
this, and that there is a better criterion.

This comes up for me because I am trying to expand some first-order decay
kinetics data in a basis set which is the eigenfunctions of the Laplace
transform, convoluted by the instrumental resolution function.  This is a long
way from quantum chemistry, my usual research area, but the problem comes up
there too, of course, in the expansion of molecular orbitals in an basis of
atomic orbitals.

I will appreciate hearing from anyone who can help me.

Dave Schrader
Marquette University
Milwaukee, WI, USA
[email-masked]

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From: "David J. Schneider" <[email-masked]>
Date: Wed, 9 Mar 1994 16:28:35 -0500

David,

Direct computation of determinants should usually be avoided because
of numerical instability problems.  A much better test is a singular
value decomposition of the overlap (Gram) matrix.  Small singular
values indicate "near" linear dependence in a sense that can be made
mathematically precise.  For example, see "Matrix Computations" (G.
Golub and C. Van Loan, Johns Hopkins Univ. Press, 1983) or "Matrix
Analysis" (R. Horn and C. Johnson, Cambridge Univ. Press, 1985).  You
should be able to obtain free, high quality, general purpose SVD codes
from netlib.  Both C and F77 are available.  Good luck.

		Dave Schneider 
		626 Engineering and Theory Center
		Cornell University 
		Ithaca, NY  14853

		phone : (607) 254-4510
		fax: (607) 254-8888
		E-mail: [email-masked]

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   Date: Wed, 09 Mar 1994 14:06:31 -0600 (CST)
   From: "David M. Schrader" <[email-masked]>

   Netters:

   Dave Schrader
   Marquette University
   Milwaukee, WI, USA

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From: [email-masked] (Zdenko Tomasic)
Date: Wed, 9 Mar 94 16:10:19 CST

SVD(Singular Value Decomposition) is the way to go, see e.g. Golub,
Van Loan: Matrix Computations. There are LAPACK drivers in netlib for
it too.

Det(S) is totally unreliable, as it can be tiny even though there is
good linear independence. The above book is quite clear on that point.

ZT