AxisAlgorithms

AxisAlgorithms is a collection of filtering and linear algebra algorithms for multidimensional arrays. For algorithms that would typically apply along the columns of a matrix, you can instead pick an arbitrary axis (dimension).

Note that all functions come in two variants, a ! version that uses pre-allocated output (where the output is the first argument) and a version that allocates the output. Below, the ! versions will be described.

Tridiagonal and Woodbury inversion

If F is an LU-factorization of a tridiagonal matrix, or a Woodbury matrix created from such a factorization, then A_ldiv_B_md!(dest, F, src, axis) will solve the equation F\b for 1-dimensional slices along dimension axis. Unlike many linear algebra algorithms, this one is safe to use as a mutating algorithm with dest=src. The tridiagonal case does not create temporaries, and it has excellent cache behavior.

Matrix multiplication

Multiply a matrix M to all 1-dimensional slices along a particular dimension. Here you have two algorithms to choose from:

• A_mul_B_perm!(dest, M, src, axis) uses permutedims and standard BLAS-accelerated routines; it allocates temporary storage.
• A_mul_B_md!(dest, M, src, axis) is a non-allocating naive routine. This also has optimized implementations for sparse M and 2x2 matrices.

In general it is very difficult to get efficient cache behavior for multidimensional multiplication, and often using A_mul_B_perm! is the best strategy. However, there are cases where A_mul_B_md! is faster. It's a good idea to time both and see which works better for your case.