From: Spencer Graves <spencer.graves_at_pdf.com>

Date: Fri 15 Jul 2005 - 03:59:43 EST

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> It has to do with what exactly you want to test. Knowing the determinant

*> does not tell you if the matrix is close to non-positive definite or not.
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*> For numerical work, a comparison of the smallest eigenvalue to the largest
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*> is usually the most useful indication of possible problems in
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*> computations. An alternative is to try a Choleski decomposition, which
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*> may be faster but is less informative.
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*> Given how fast eigenvalues can be computed by current algorithms (and note
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*> the comments in ?eigen) I would suggest not worrying about speed until you
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*> need to (probably never).
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Date: Fri 15 Jul 2005 - 03:59:43 EST

To reinforce Prof. Ripley's comment that, "Knowing the determinant does not tell you if the matrix is close to non-positive definite", note that the determinant of the negative of the identity matrix, (-diag(k)), is (-1)^k; if k is even, the determinant is positive. This silly example connects to real cases, as for example a 3x3 matrix of rank 1 with eigenvalues (1, -1e-20, -1e-21). The last two eigenvalues are buried in the noise relative to the largest (under standard double precision arithmatec).

spencer graves

Prof Brian Ripley wrote:

> On Wed, 13 Jul 2005, Makram Talih wrote:

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>>Dear R-users, >> >>Is there a preferred method for testing whether a real symmetric matrix is >>positive definite? [modulo machine rounding errors.] >> >>The obvious way of computing eigenvalues via "E <- eigen(A, symmetric=T, >>only.values=T)$values" and returning the result of "!any(E <= 0)" seems >>less efficient than going through the LU decomposition invoked in >>"determinant.matrix(A)" and checking the sign and (log) modulus of the >>determinant. >> >>I suppose this has to do with the underlying C routines. Any thoughts or >>anecdotes?

> It has to do with what exactly you want to test. Knowing the determinant

-- Spencer Graves, PhD Senior Development Engineer PDF Solutions, Inc. 333 West San Carlos Street Suite 700 San Jose, CA 95110, USA spencer.graves@pdf.com www.pdf.com <http://www.pdf.com> Tel: 408-938-4420 Fax: 408-280-7915 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.htmlReceived on Fri Jul 15 04:08:58 2005

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