Generalized eigenvalue problem

Robert Israel — Mon, 04 Feb 2008 06:01:17 Z

I'm not a Matlab user, but it looks like this is supposed to solve the generalized eigenvalue problem

A x = lambda B x

using the Cholesky factorization of B. Except for the Cholesky part, Maple's analogue would be

> LinearAlgebra[Eigenvectors](A,B);

generalized positive-definite symmetric eigenvector problem

acer — Mon, 04 Feb 2008 08:29:51 Z

One can see which external compiled function Maple's LinearAlgebra:-Eigenvectors uses, for a generalized eigenvector problem A.x=lambda.B.x where A is symmetric floating-point and B is symmetric positive-definite floating-point.

> A := Matrix(1,shape=symmetric,datatype=float):
> B := Matrix(1,shape=symmetric,datatype\
> =float,attributes=[positive_definite]):
> infolevel[LinearAlgebra]:=1:
> LinearAlgebra:-Eigenvectors(A,B):
Eigenvectors:   "calling external function"
Eigenvectors:   "NAG"   hw_f02fdf

A web search for f02fdf gives this link which documents that function. It says, of the parameter array B, "the upper or lower triangle of B (as specified by UPLO) is overwritten by the triangular factor U or L from the Cholesky factorization of B".

This suggests that, with the shape and attributes on Maple Matrices A and B as in the example above, LinearAlgebra:-Eigenvectors will use a method involving the Cholesky facorization of B. It goes on to say, in the Further Comments section, "F02FDF calls routines from LAPACK in Chapter F08."

For some years now, Matlab has been using LAPACK. See this link from 2000 for an early note on that. It appears from a mirror of the release notes of Matlab 6.0 that the eig function was enhanced to solve exactly the same positive-definite symmetric generalized eigenproblem with the syntax eig(A,B,'chol').

I wouldn't be surprised if these two products schemes were not (at least when originally introduced) very similar implementations of an alternative to the usual QZ/QR algorithm. Notice however a difference in behaviour of the two systems. In Maple it is the shape and attributes of the Matrices whch allow the routine to select the algorithm. The algorithm cannot otherwise be forced. In Matlab the data is pretty much without qualities, and no clever method deduction can be done I think. But the method can be forced by an option to the routine.

ps. Is "without qualities" an eigen-pun, in German? Is the word qualities better translated here as Qualitaten or as Eigenschaften?

acer

Thx

Axel Vogt — Mon, 04 Feb 2008 13:11:02 Z

Thanks for both the answers, it helps to find the docu
http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/eig.html

PS: eigen-pun is a nice word ... but that has to wait until ash wednesday :-)

Yes, seems to be an adequate translation, mostly I use dict.tu-chemnitz.de/ as dictionary

MaplePrimes - answers and comments on Question, Matlab <---> Maple equivalent commands

Generalized eigenvalue problem

generalized positive-definite symmetric eigenvector problem

Thx