On Tue, Apr 29, 2008 at 1:22 PM, Timothy Hochberg <[EMAIL PROTECTED]> wrote:
> > Let me throw out a couple of more thoughts: > > First, there seems to be disagreement about what a row_vector and > column_vector are (and even if they are sensible concepts, but let's leave > that aside for moment). One school of thought is that they are > one-dimensional objects that have some orientation (hence row/column). They > correspond, more or less, to covariant and contravariant tensors, although I > can never recall which is which. The second view, which I suspect is > influenced by MatLab and its ilk, is that they are 2-dimensional 1xN and > Nx1 arrays. It's my view that the pseudo tensor approach is more powerful, > but it does require some metainformation be added to the array. This > metadata can either take the form of making the different objects different > classes, which leads to the matrix/row/column formulation, or adding some > sort of tag to the array object (proposal #5, which so far lacks any > detail). > > Second, most of the stuff that we have been discussing so far is primarily > about notational convenience. However, there is matrix related stuff that is > at best poorly supported now, namely operations on stacks of arrays (or > vectors). As a concrete example, I at times need to work with stacks of > small matrices. If I do the operations one by one, the overhead is > prohibitive, however, most of that overhead can be avoided. For example, I > rewrote some of the linalg routines to work on stacks of matrices and > inverse is seven times faster for a 100x10x10 array (a stack of 100 10x10 > matrices) when operating on a stack than when operating on the matrices one > at a time. This is a result of sharing the setup overhead, the C routines > that called are the same in either case. > Some operations on stacks of small matrices are easy to get, for instance, +,-,*,/, and matrix multiply. The last is the interesting one. If A and B are stacks of matrices with the same number of dimensions with the matrices stored in the last two indices, then sum(A[...,:,:,newaxis]*B[...,newaxis,:,:], axis=-2) is the matrix-wise multiplication of the two stacks. If B is replaced by a stack of 1D vectors, x, it is even simpler: sum(A[...,:,:]*x[...,newaxis,:], axis=-1) This doesn't go through BLAS, but for large stacks of small matrices it might be even faster than BLAS because BLAS is kinda slow for small matrices. Chuck
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