[Numpy-discussion] numarray.linear_algebra is slow (and a partial fix?)

[Tim Hochberg]

I discovered that some (all?) of the functions in 
numarray.linear_algebra are very slow when operating on small matrices. 
In particular, determinant and inverse are both more than 15 times 
slower than their NumPy counterparts when operating on 4x4 matrices. I 
assume that this is simply a result of numarray's higher overhead.

Normally the overhead of numarray is not much of a problem since when 
I'm operating on lots of small data chunks I can usually agregate them 
into larger chunks and operate on the big chunks. This is, of course, 
the standard way to get decent performance in either numarray or NumPy. 
However, because the functions in linear_algebra take only rank-2 (or 1 
in some cases) arrays, their is no way to aggregate the small operations 
and thus things run quite slow.

In order to address this I rewrote some of the functions in 
linear_algebra to allow an additional, optional, dimension on the input 
arrays. Rank-3 arrays are treated as being a set of matrices that are 
indexed along the first axis of A. Thus determinant(A) is essentially 
equivalent to array(map(determinant, A)) when A is rank-3. See the 
attached file for more detail.

By this trick and by some relentless tuning, I got the numarray 
functions to run at about the same speed as their NumPy counterparts 
when computing the determinants and inverses of 1000 4x4 matrices. 
That's a humungous speedup.

Is this approach worth pursuing for linear_algebra in general? I'll be 
using these myself since I need the speed, although I may back out some 
of the more aggresive  tuning so I don't get bit if numarray's internals 
change. I'll gladly donate this code to numarray if it's wanted, and I'm 
willing to help convert the rest, although it probaly wouldn't happen as 
fast as this stuff since I don't need it myself presently.

-tim

[Use this with caution at this point -- I just got finished with a 
tuning spree and there may well be some bugs]
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