How could I define my own division operation with QR decomposition?

(Feel free to skip this part and look at the minimal example below)

After a long time, I had to continue again with some old code, that stillyslalom already posted here, and because is a different question, I created a new post.
So, in the end of the post we have the code to compute eigenvectors, given the eigenvalues:

using LinearAlgebra, GenericLinearAlgebra
import LinearAlgebra.eigen

function eigen(A::Array{T,2}) where T <: Union{BigFloat,Complex{BigFloat}}
    ni, nj = size(A)
    Λ = eigvals(A)
    if norm(imag(Λ)) < eps(BigFloat)
        Λ = real(Λ)
    V = rand(eltype(Λ), nj, nj)
    bkp = similar(V, nj)
    for j = 1:nj
        Δ = BigFloat(Inf)
        bk = view(V,:,j)
        f = qr(A - Λ[j]*I)
        i = 0
        while Δ > eps(BigFloat) && i < 10
            bkp .= f\bk
            ck = norm(bkp)
            bkp ./= ck
            Δ = norm(bkp - bk)
            bk .= bkp
            i += 1
    return Λ, V

The eigenvector part seems to be a matrix multiplication operation. Then, for fun, I was debugging the code in hope to implement a GPU version of it.

My struggle is with the line bkp .= f\bk.

A minimal example with Float64 type:

using LinearAlgebra
A = rand(4,4)
f = qr(A)
b = rand(4)
C = f\b

f has type of LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}, and stores matrices Q and R.

My questions is: can I convert the division operation into standard matrix multiplication between Q,R and b?
(I don’t mind about high tunned algorithm, the GPU will do muliplication operations faster anyway)

Extra information for reference:

  • In the end. Looking at the internals of qr.jl file, the division operation becomes a Lapack call LAPACK.gemqrt!('L','T',A.factors,A.T,B).

  • A simplest code that I would like to have in GPU is:

using ArrayFire
Agpu = AFArray(A)
fgpu = qr(Agpu)
bgpu = AFArray(b)
cgpu = fgpu\bgpu

If I run this code, I got a message telling me that adjoint is not defined to AFArray type. So, I defined it as:
ArrayFire.adjoint(A::AFArray) = A'

but I got a StackOverFlow message :cry: