| Re: generalized eigenvectors [message #30232 is a reply to message #30223] |
Mon, 15 April 2002 14:54   |
Ralf Flicker
Messages: 19
Registered: October 2001
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Junior Member |
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Hey Randall - this is not really related to the original question,
but since you seem to know what you're on about I figured I should
ask you.
In working on large sparse arrays it has become crucial to make the
SVD more efficient than O(n^3), but that seems to be easier said
than done. Do you know of an efficient implementation of the Laczos
bidiagonalization with selective reorthogonalization? I have not
been able to accomplish it - with complete, explicit
reorthogonalization it actually gets even worse than O(n^3).
More to the point, has anyone _ever_ managed to bring down the SVD
significantly below O(n^3) for sparse arrays? Pointers and
suggestions welcome.
aloha
ralf
Randall Skelton wrote:
>
> I am not sure I understand what you mean by a "Generalized Eigenvalue
> problem".
>
> IDL does have routines for computing the Eigenvalues and Eigenvectors.
> From the IDL help:
>
> EIGENQL - Compute eigenvectors of a real, symmetric array, given the array.
> EIGENVEC - Compute eigenvectors of a real, nonsymmetric array, given
> the array and its eigenvalues.
> ELMHES - Reduce a real, nonsymmetric array to upper-Hessenberg form.
> HQR - Compute the eigenvalues of an upper-Hessenberg array.
> TRIQL - Compute eigenvalues and eigenvectors of a real, symmetric,
> tridiagonal array.
> TRIRED - Use Householder's method to reduce a real, symmetric array to
> tridiagonal form.
>
> Most of these methods are described in the Numerical recipies books.
> See http://www.ulib.org/webRoot/Books/Numerical_Recipes/
>
> In IDL the user must decide if the input matrix is symmetric or not then
> use the appropriate tools. The Matlab EIG function basically uses the
> same tools (to a first order), but automatically determines the "best"
> method based on your input matrix... I personally find Matlab's auto-magic
> approach to be more trouble than it is worth. Moreover, it promotes the
> idea that you don't really need to understand the numerical problem you
> are trying to solve...
>
> The general approach is:
>
> - If your matrix is real and symmetric, convert to the tridiagonal form
> (TRIRED) and then use the QR procedure (TRIQL) to iteratively find the
> eigenvalues/vectors from the tridiagonal array.
>
> - If your matrix is real and not symmetric, reduce to Hessenberg form
> using the Householder's transformation method (ELMHES) and then use the QR
> procedure (HQR) to get the eigenvalues/vectors of the upper Hessenberg
> matrix.
>
> - Unfortunately, there is no built in IDL code for the QZ (Golub and Van
> Loan, 1989) algorithm which can be modified to work for complex matricies.
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