Re: [GSoC2014] Looking for a mentor

[Philip Nienhuis]

mfasi wrote
> On 11/03/2014 22:40, Philip Nienhuis wrote:
>> mfasi wrote
>>> Hi all,
>>>
>>> I am a little bit sorry for spamming the maintainers list, but I was
>>> suggested to do so on IRC - and I am in good company. As the subject
>>> says,
>>> I am looking for a mentor to apply for the Google Summer of Code 2014.
>>> The
>>> topic I am interested in is the implementation of some matrix functions
>>> in
>>> Octave (project 
>> *
>>> Improve logm, sqrtm, funm
>> *
>>> ), as I already have a little experience on the domain. Further
>>> information on my 
>>> public application draft <http://wiki.octave.org/User:Mfasi>  
>>> . Thank you!
>> 
>> Have you found a mentor yet?
>> I regret I am too little proficient in linear algebra to help you out,
>> although I wrote the existing funm.m in the OF linear-algebra package.
>> While
>> that works OK for my aims (relatively small (n < 100), well conditioned,
>> positive semi-definite symmetric matrices) I'd love to see a more robust
>> funm.m implementation. But be warned: as Higham wrote in his email to our
>> maintainers list, writing a robust funm.m from scratch is no easy task.
>> 
>> Just some remarks:
>> 
>> As to logm(), AFAIK that one has already been optimized by "Tommy Guy"
>> some
>> years back, along the lines set out by Higham (IIRC Tommy Guy implemented
>> some of Higham's methods). I am not aware of things left out (maybe
>> complex?).
>> 
>> As you can see in the OF linear-algebra package, some matrix functions
>> (trig
>> and hyperbolic) have been implemented, by invoking expm(), in the thfm.m
>> function in that same package. funm.m delegates those trig and hyperbolic
>> matrix functions to thfm.m. 
>> It would be wise to keep that structure intact.
> <snip>
> Hi Philip,
> 
> I have not found a mentor yet, and the fact that you are the only one
> that has answered on that topic so far let me think you are the only one
> that could be interested. Moreover, the fact of being the author of the
> actual funm () makes you one of the most recommended people to mentor
> this project. In conclusion, I hope you will change your mind about it.
> :-D

Fist of all, my math skills do lag a bit behind on my programming skills,
and there's no reason to boast about the latter either ;-)

Seriously, the only thing I can help you with is in coding itself. As to
linear algebra we'd have to rely on the other specialists here.


> Talking about serious stuff, I think you are absolutely right about the
> behaviour of the current funm () function. The basic weakness, as far as
> I can see, is that it uses a special case of the definition by means of
> the Jordan decomposition, and I would expect it to be not accurate when
> the eigenvectors are very ill-conditioned and to have sometimes a
> non-predictable outcome when the Jordan blocks have size bigger than
> one, as you pointed out in the source comments. Maybe it will be a
> little slow for very big matrices, requiring the computation of the
> diagonal decomposition, but I am not sure that this can be completely
> eliminated. AFAIK, the best approach for general functions should be the
> Schur-Parlett recurrence, that basically computes the matrix function on
> the diagonal block of the Schur decomposition and then builds back the
> matrix using Sylvester equations. It is robust in general but there are
> some issues that have to be addressed to get a good implementations.

I expect that about any method with "Schur" in the name will be better than
the straightforward diagonalization that funm-as-it-stands invokes :-)

Have you already looked at Higham's stuff? (it's GPL-ed and available on his
web site).


> On the other hand, according to the documentation, the logm () function
> is implemented using some Padé approximants, so I guess it should work
> for complex in general, but you are probably right when dealing with
> branch points and branch cut. I will try to look at the logm () code to
> shed some light on the question.

TommyGuy has put quite some effort in it.

Anyway, good luck,

Philip



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