Re: logm robustness

[Philip Nienhuis]

Tommy Guy <guyrt7 at ****> wrote:
> Yes - I agree that a citation is needed.  That is why I am questioning
> the comment in the logm and sqrtm methods.
> 
> Found a citation that mentions Schur methods as superior
> 
> http://eprints.ma.man.ac.uk/156/01/covered/MIMS_ep2006_13.pdf
> 
> This paper also has an answer to the question of what kind of matrices
> are unstable.
> 
> Richard T. Guy
> 
> 2010/4/20 Jordi Gutiérrez Hermoso <address@hidden>:
>> On 19 April 2010 20:26, Tommy Guy <address@hidden> wrote:
>>> logm is using eigenvalues, which was listed as lacking robustness.
>>
>> [citation needed]

I see that logm.m uses more or less the same basic strategy that funm.m uses
(octave-forge in the linear-algebra-<version> package).

There's an old discussion relating to this on bug-octave archives from 2001
about funm.m (that I contributed) which is very basic and suffers from
probably the same issues.
See:
  http://www.octave.org/old-list-archives/bug-octave/2001/64
  "numerical artifacts by defective input to matrix functions" and follow-up
posts.

I made funm.m and use(d) it only for my field of expertise where the
relevant matrices are always diagonizable - so for me It Just Works.
Nevertheless funm.m should be improved too in similar fashion as logm.m to
be more widely useful. 
I'm no mathematician however so I can't contribute much further. 

FWIW I once found a better alternative invoking Schur, but as jwe pointed
out, that alternative might have been pirated from the competition, see
  http://www.octave.org/old-list-archives/help-octave/2004/760

Another alternative may be to check how scilab does their collection of
matrix functions and borrow from them - scilab's license permitting. But
IIRC (it's many years ago that I had a look at their docs) scilab's matrix
functions are fairly basic too. 

Philip

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