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Re: matrix functions
From: |
Bård Skaflestad |
Subject: |
Re: matrix functions |
Date: |
Tue, 21 Dec 2010 16:14:36 +0100 |
On Tue, 2010-12-21 at 15:02 +0100, Jordi Gutiérrez Hermoso wrote:
> On 21 December 2010 04:05, Bård Skaflestad <address@hidden> wrote:
>
> > I was about to suggest you use the 'funm' function, but now I see that
> > 'funm' is only available (by default) in That Other Software. On the
> > other hand, the 'linear-algebra' add-on package from OctaveForge does
> > contain a 'funm' implementation.
>
> Huh, that's awkward, because the docstring for expm mentions funm,
> which isn't included in core. I see the funm implementation is quite
> naïve and relies on the matrix being diagonalisable.
Naïve indeed. The implementation uses exactly the algorithm alluded to
by the OP and, as you say, implements even this algorithm poorly. I
almost feel like withdrawing my original reference to this particular
function...
> Even worse, it
> calls inv instead of using the backslash operator. Is this the best
> that can be done in general? Surely if expm works for defective
> matrices, something can be done in general if we assume analytic
> functions, which funm is assuming anyways?
There are at least two classical references in this area
* Davies, P. I. and N. J. Higham, "A Schur-Parlett algorithm for
computing matrix functions," SIAM J. Matrix Anal. Appl., Vol.
25, Number 2, pp. 464-485, 2003.
* Golub, G. H. and C. F. Van Loan, Matrix Computation, Third
Edition, Johns Hopkins University Press, 1996, p. 384.
And, of course, I would be remiss if I failed to mention Moler and Van
Loan's "Nineteen Dubious Ways to Compute the Exponential of a Matrix"
and "Nineteen Dubious Ways to Compute the Exponential of a Matrix,
Twenty-Five Years Later", although these papers are mostly about the
exponential.
That said, implementing some of these more advanced algorithms is not
entirely trivial. A bit of work and analysis is needed...
Regards,
--
Bård Skaflestad <address@hidden>
SINTEF ICT, Applied Mathematics
- matrix functions, CdeMills, 2010/12/21
- Re: matrix functions, Bård Skaflestad, 2010/12/21
- Re: matrix functions, CdeMills, 2010/12/21
- Re: matrix functions, Jordi Gutiérrez Hermoso, 2010/12/21
- Re: matrix functions,
Bård Skaflestad <=
- Re: matrix functions, Bård Skaflestad, 2010/12/21
- Re: matrix functions, Philip Nienhuis, 2010/12/21
- Re: matrix functions, Jordi Gutiérrez Hermoso, 2010/12/22
- Re: matrix functions, Philip Nienhuis, 2010/12/22
- Re: matrix functions, Miroslaw Kwasniak, 2010/12/28
- Re: matrix functions, Jordi Gutiérrez Hermoso, 2010/12/28