Pre-multiplying Matrices

[lukem12345]

Currently, given a decapode like: * -f-> * -g-> * -h-> *, where each arrow f,g,h multiplies by a matrix A,B,C, respectively, the generated function will perform three matrix-vector multiplications.

We would like to pre-multiply these matrices into D=CBA, so the generated function will perform only one matrix-vector multiplication.

A working example of this optimization should focus on editing the combinatorial data structure of the decapode. This will require a function that is the inverse of expand_operators.

[lukem12345]

We note that we can take this idea of contracting chains further by eliminating branches.

In other words, a binary tree of depth 2, (with update steps at the leaves) requires (assuming all functions on edges are unique) 6 matrices, 2 intermediary vectors, and performs 6 matrix-vector multiplications. This could be re-written as an equivalent 1-deep 4-ary tree which requires just 4 matrices, 0 intermediary vectors, and performs 4 matrix-vector multiplications. This result cannot be improved upon, unless edge pruning of duplicate operations is performed.

This speed-up and memory improvement of an entire decapode in this fashion is unlikely to occur, since Op2s are likely to break this downwards tree-like property.