Editing User:Antonio Pino:anotherwiki

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After this little holiday, I am back to work on gsoc. As Philip Nienhuis noted on the [http://octave.1599824.n4.nabble.com/Re-Query-for-project-on-improving-matrix-functions-td4669292.html#a4669297 mailing list] there is a fair amount of work done on Matrix Functions by Higham himself. Nonetheless, it was made to work with matlab so the first issue to tack is making Gnu Octave work with it, changing it where necessary. Starting from there, then new algorithms should be introduced to the system.
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==note==
First of all, I am still interested in going on with the project.

Concerning the algorithms, I have done a review of Golub and Van Loan's chapter on matrix functions (2013) and concluded that: for funm the Schur-Parlett algorithm [0] is the most appropriate, since it seems very reliable (i.e. numerically stable), for expm the new scaling and squaring approach (I hope it is the one Prof. Caliari referred to), while its analog will be used for logm. As for sqrtm, one of the latest results is [3], although a more general p-th root algorithm by Smith [4] or Higham/Lin [5] could do the job. Bear in mind that this response is barely a review of them. If you have a more recent source I would be grateful.

Note that today has been strange day for me: the last before a Little intermission of this term (Holy Week). I will need a little time for planning the new tentative time line, and do the new proposal; by the way, as google-melange has frozen here the initial one, I shall put it in the wiki.octave page or here as a comment.

Therefore, I will go for compatibility first and then for further improvements.

Thank Carnë Draug for the comments, I didn't know Octave already made use of the mctoolbox in the gallery function.

Antonio Pino


[0] P.I. Davies and N.J. Higham (2003). "A Schur-Parlett Algorithm for Computing Matrix Functions", SIAM. J. Matrix Anal. Applic. 25, 464-485. <Vhttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.6150&rep=rep1&type=pdf>

[1] A.H. Al-Mohy and N.J. Higham (2009). "A New Scaling and Squaring Algorithm for the Matrix Exponential," SIAM J. Matrix Anal. Applic. 31, 970-989 <http://eprints.ma.man.ac.uk/1217/01/covered/MIMS_ep2009_9.pdf>

[2] S.H. Cheng, N.J. Higham, C.S. Kenney, and A.J. Laub (2001). "Approximating the Logarithm of a Matrix to Specified Accuracy," SIA M J. Matrix Anal. Applic. 22, 1112- 1125. <http://www.maths.manchester.ac.uk/~higham/narep/narep353.ps.gz>

[3] A. Frommer and B. Hashemi (2009). "Verified Computation of Square Roots of a Matrix," Matrix Anal. Applic. 31, 1279-1302. <http://www.researchgate.net/publication/220656516_Verified_Computation_of_Square_Roots_of_a_Matrix>

[4] M. I. Smith (2003). "A Schur Algorithm For Computing Matrix Pth Roots", SIAM J. MATRIX ANAL. APPL. 24, 4, 971-989. <http://www.maths.manchester.ac.uk/~higham/narep/narep392.ps.gz>

[5] N. J. Higham and L. Lin (2011). "A Schur-Parlett Algorithm for Fractional Powers of a Matrix". Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester. <http://eprints.ma.man.ac.uk/1677/01/covered/MIMS_ep2010_91.pdf>
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'''Update:'''

Part of the work is already done by Prof. N.J. Higham and is available under a GPLv3+ license: [http://www.ma.man.ac.uk/~higham/mftoolbox/ The Matrix Function Toolbox][3] which is closely related to the book by the same author[4]. A [http://www.ma.man.ac.uk/~higham/mctoolbox toolbox for matrix computations][4] (The Matrix Computation Toolbox) is also provided by the same author, under the same license. Finally, a funm function is provided in the page of the NAMF project under GPLv3+. One might suggest that there is still room for improvement; because as Marco Caliari noted the toolboxes are from 2008. A review of the literature needs to be done in order to use more recent algorithms when writing the new functions.


'''TENTATIVE TIMELINE'''

'''weeks 1-2'''

The start should be soft for I am having the finals in this period. At this point the list of algorithms to be used must be completely defined; that is, a final review of the literature is to be done.

Work on the toolboxes starts here.

'''weeks 3-4'''

''Milestone 0'': the toolboxes are working and packaged.

'''weeks 4-7'''

funm 

''Milestone 1'': general purpose funm based on a Schur-Parlett algorithm)

'''weeks 8-9'''

expm and logm

'''weeks 10-11'''

sqrtm

p-th roots (p = integer) algorithm [?]

''Milestone 2'': logm, sqrtm, funm

'''week 12'''

implement trigonometric functions (I submitted  a snippet of cosm [http://octave.1599824.n4.nabble.com/GSoC-2015-improving-matrix-funtcions-td4669227.html to the mailing list], [http://agora.octave.org/snippet/tqUQ/ updated])

'''weeks 13-14'''

Pencils down. Run tests on the Matrix Functions and write/review their documentation.

<!-- references
these feel like I have just one source, and I am citing it over and over again...
--->
[1] N.J. Higham. A New sqrtm for MATLAB.  Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999.

[2] G.H. Golub and C.F. Van Loan. Matrix Computations, 4th Edition. The Johns Hopkins University Press, Baltimore, USA, 2013.

[3] N. J. Higham. The Matrix Function Toolbox. http://www.ma.man.ac.uk/~higham/mftoolbox

[4] N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, USA, 2008.

[5] N. J. Higham. The Matrix Computation Toolbox. http://www.ma.man.ac.uk/~higham/mctoolbox

[6]  M. I. Smith (2003). [http://www.maths.manchester.ac.uk/~higham/narep/narep392.ps.gz A Schur Algorithm For Computing Matrix Pth Roots], SIAM J. MATRIX ANAL. APPL. 24, 4, 971-989.