Overall error bound of lowest eigenvalue of real symm matrix

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Overall error bound of lowest eigenvalue of real symm matrix

Postby njca12 » Fri May 12, 2006 6:53 pm

Hi,

I'm using dsievx subroutine to calculate the lowest eigenvalue of a real symmetric matrix, and i want to determine if this matrix is definite positive. Time ago i used nag f02bff routine. An alternative procedure is given by using lapack dsievx subroutine, however with this subroutine not overall error bound are obtained. Is there any procedure to obtain this error?

The aim of work is calculate the lowest eigenvalue of a matrix and determine if this is definite positive.

Is there other alternative method to obtain these?

Thanks in advance
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Postby Julien Langou » Mon May 15, 2006 2:30 pm

Hello,

the recommended way to check if a matrix is symmetric positive definite is to first check if it's symmetric (...) and then run Cholesky on it. Run Cholesky means perform a Cholesky factorization of the matrix, and to do so you can use dpotrf. If the factorization fails (a_ii <= 0, INFO>0 ) then the matrix is not positive definite, if the factorization sucesses then the matrix is positive definite (INFO=0).

The implementation of Cholesky in LAPACK is the left-looking variant. This choice has been made on the assumption that the general case is that the factorization would not fail. Another variant ("bordered" variant of Cholesky) can be implemented, this implemetation will fail faster. So if the failure mode is the general expected case, that's the one you might want to use. (See Cholesky Decomposition for more information.)

So now, if your problem is to find the minimum eigenvalue (and thus consequently solving the symmetric-definiteness question). You'll go with dsyevx. We do not provide error bounds though, and I can not really tell what NAG routine f02bff was returning. You can certainly post-process the minimum eigenvalue and the associated eigenvector to compute some backward error ...


Julien
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