Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!uunet!mstan!amull From: amull@Morgan.COM (Andrew P. Mullhaupt) Newsgroups: comp.lang.apl Subject: Re: Syntax for extended matrix operations Summary: It can be quite important Message-ID: <590@jfk.Morgan.COM> Date: 8 Dec 89 21:35:01 GMT References: <539@s5.Morgan.COM> Organization: Morgan Stanley & Co. NY, NY Lines: 33 In article , raulmill@usc.edu (Raul Deluth Rockwell) writes: > > Seems to me that factorization tends to yield multiple results. This > could be dealt with by 'inventing' a new type, but how much would be >saved by this approach? If you want to escape from the incredibly painful limitation of APL to dense matrices, quite a bit. Also; there are different factorizations to fit different needs. You may find occasion to use Cholesky's method and at other times need for the SVD. That there is a pretty big factor. > > Comparing factorization with Guass-Jordan reduction, how many > operations are saved over the course of an algorithm? (Assume some > sort of ideal efficiency in the implementation of factorization). I > suppose I ought to do some of this myself, but I don't know what kind > of algorithms you are trying to optimize. Gauss-Jordan is not normally acceptable as such for a wide-spectrum quad-divide primitive. APL2, for example specifies that Householder QR (Hanson-Lawson method) is used. You actually lose operations to Gauss-Jordan on each call, and that's one of the motivations to save the factors. Matrix factorizations address a wide spectrum of numerical linear algebra problems, among them we are interested in the QR, SVD and Shur decompositions, and symmetric, polydiagonal, and sparse implementations. Good algorithms for these are well known, but not easily translated into time and space efficient APL. What might be the biggest saving is not having to worry about the same stuff each time you have to get some of the information from a column pivoted QR, for example. Later, Andrew Mullhaupt