# 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.