User:Antonio Pino:anotherwiki
a starter[edit]
...just experimenting with subpages, more info on [wikipedia]
[...]
After this little holiday (Holy Week), I am back to work on gsoc. As Philip Nienhuis noted on the 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.
a note[edit]
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> Leave a comment
updated task outline[edit]
...moved the definitive version to my wiki, User:Antonio_Pino