Solving a linear system without modifying and allocating arrays

General Usage Performance
1

Hi, I have a setup where I need to repeatedly solve a linear system like A \ B = C every timestep of a system. A is a matrix, B is a vector and both changing each timestep and more importantly A is reused later each timestep for other functions like a matrix multiplication e.g. D = A * C.

What I assume is a naive function might look like:

using LinearAlgebra
using BenchmarkTools

A = rand(Float32,10,10)
A_copy = similar(A)
B = rand(Float32,10)
C = similar(B)
D = similar(B)

function lu_div_mul!(A::Array{Float32,2},A_copy::Array{Float32,2},B::Array{Float32,1},C::Array{Float32,1},D::Array{Float32,1})
    copy!(A_copy,A)
    LU = lu!(A_copy)
    ldiv!(C,LU,B)
    mul!(D,A,C)
end

@btime lu_div_mul!($A,$A_copy,$B,$C,$D)
  843.077 ns (2 allocations: 144 bytes)

where the two allocations come from the lu factorisation. I would like to avoid these allocations and ideally the unneeded copy of A as in reality the dimensions of A can be rather large. Is there an existing function or method, something like lu!(A_lu,A), where the factorisation A_lu can be pre-allocated and reused, avoiding the matrix copy?

2

You can probably do that with LinearSolve.jl but it would be nice to have it in LinearAlgebra

3

Yes, this feature was recently added: define lu!(F::LU,A) for general matrices by longemen3000 · Pull Request #1131 · JuliaLang/LinearAlgebra.jl · GitHub

I’m not sure how long it will be until this appears in a Julia release, but the code from that PR is pretty short so you could easily copy it.

4

That’s excellent! Based on the docstring looks like this will come with Julia v1.12, until then I shall just copy the code as it is as you say, pretty short.

Thank you!