Question

I have a matrix B that is square and dense, and a matrix A that is rectangular and sparse.

Is there a way to efficiently compute the product B^-1 * A?

So far, I use (in numpy)

tmp = B.inv()
return tmp * A

which, I believe, makes us of A’s sparsity. I was thinking about using the sparse method
numpy.sparse.linalg.spsolve, but this requires B, and not A, to be sparse.

Is there another way to speed things up?

Answer 1

Since the matrix to be inverted is dense, spsolve is not the tool you want. In addition, it is bad numerical practice to calculate the inverse of a matrix and multiply it by another – you are much better off using LU decomposition, which is supported by scipy.

Another point is that unless you are using the matrix class (I think that the ndarray class is better, this is something of a question of taste), you need to use dot instead of the multiplication operator. And if you want to efficiently multiply a sparse matrix by a dense matrix, you need to use the dot method of the sparse matrix. Unfortunately this only works if the first matrix is sparse, so you need to use the trick which Anycorn suggested of taking the transpose to swap the order of operations.

Here is a lazy implementation which doesn’t use the LU decomposition, but which should otherwise be efficient:

B_inv = scipy.linalg.inv(B)
C = (A.transpose().dot(B_inv.transpose())).transpose()

Doing it properly with the LU decomposition involves finding a way to efficiently multiply a triangular matrix by a sparse matrix, which currently eludes me.