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Topic: Computation of the matrix exponential
Replies: 6   Last Post: Feb 10, 2013 5:00 AM

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Chip Eastham

Posts: 2,341
Registered: 12/13/04
Re: Computation of the matrix exponential
Posted: May 25, 2010 1:59 AM
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On May 24, 8:37 pm, luca <luca.frammol...@gmail.com> wrote:
> Hi,
>
> i have the following problem: given a 3x3 real matrix, compute exp(A).
>
> I need a really fast way to do this. I have searched a bit with
> google, but it seems to me that
> computing the matrix exponential is not so simple, at least if your
> matrix does not have a special
> structure (for example A=diagonal matrix).
>
> I have found a simple method that use the diagonalization of A. If A
> has 3 distinct eigenvalues, than compute
> A=PDP^-1, where P is the matrix of the eigenvectors, D is a diagonal
> matrix (whose diagonal elements are
> the eigenvalues of A). Than, exp(A) = P exp(D) P^-1. Since P^-1 is
> fast enough and exp(D) is simple
> to compute, this should be a fast method.
>
> But, the problem is: i am not sure that the matrix A will always have
> 3 distinct eigenvalues...what happens
> if this does not happen? Can i use that formula even if 2 (or all
> three) eigenvalues are equal?
>
> Are there any other ways to compute exp(A) in a fast way?
>
> Thank you,
> Luca


You'll probably find the "updated" discussion of matrix
exponentiation by Cleve Moler and Charles Van Loan to be
helpful:

[Nineteen Dubious Ways to Compute the Exponential of a
Matrix, Twenty-Five Years Later]
http://www.cs.cornell.edu/cv/ResearchPDF/19ways+.pdf

Your statement that you need to compute the matrix
exponential "really fast" suggests that you intend
to do this repeatedly. A bit more information about
the reason for doing this might lead to a more focused
evaluation of algorithms.

regards, chip



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