(rank, z) = aca!(M, U, V, tol=0.0; [kwargs...]): given an object M that implements ACAFact.{row!, col!} (see also the docstring for ACAFact.aca), compute an ACA factorization M ≈ U*V' by in-place modifying U and V. If tol>0.0, this population may terminate early, and the unused columns of U and V are NOT zeroed out, so you should be sure to look at the rank output and zero out the (rank+1):end columns of U and V. The rank output gives you the number of columns used (which may be less than size(U,2) if tol > 0), and thez` output is an internal quantity that you only need to keep if you intend to resume the factorization later.

This function is designed to be non-allocating and resumable. In particular, you can allocate bigger U and V than you need to achieve a factorization at tolerance tol1, and then with careful use of returned values resume the computation to factorize at a higher precision tol2. See the README and tests for examples.

WARNING: this function does not check compatibility of array dimensions. So if your U and V are not the right size, you might waste time waiting for that error to occur or simply get incorrect answers.

aca(M, rank::Int64; [tol=0.0, sz=size(M)]): takes any object representing a matrix that implements ACAFact.{row!, col!} and produces matrices U and V such that aM ≈ U*V'.

• rank: the maximum allowed rank of the approximation

• tol: the tolerance for the stopping criterion of || M - U*V' || < tol

• sz: a Tuple{Int64, Int64} providing the size of M (which may not have a method for size(x::typeof{M}))

The object M must implement ACAFact.{row!, col!}, which both look like

ACAFact.row!(buf, M, j)

and populate buf with the row (or column) of index j. Feel free to write your own methods for your special object.