rank(::QRPivoted) method?

[stevengj]

Since pivoted QR is a rank-revealing calculation, it seems like we should have a LinearAlgebra.rank method that works directly on it. Something like:

function rank(F::QRPivoted; atol::Real=0, rtol::Real=min(size(F)...)*eps(real(float(one(eltype(F)))))*iszero(atol))
    m = min(size(F)...)
    m == 0 && return 0
    tol = max(atol, rtol*abs(F.R[1,1]))
    return something(findfirst(i -> abs(F.R[i,i]) <= tol, 1:m), m+1) - 1
end

should do it, no?

Of course, this won't necessarily return the same thing as the SVD-based rank (and is less reliable than SVD — it shouldn't be the default method for ::AbstractMatrix), but inconsistency is the fate of all numerical rank computations. In typical cases it should be close, one hopes, and if you have the QR you should be able to explicitly ask what numerical rank it reveals.

[jefftrojan]

Hi, I am trying to make an attempt on this, I have a question tho... Could you explain the motivation behind introducing a rank calculation based on the pivoted QR decomposition?

[stevengj]

It's way faster than the SVD (used by default for rank(A)), although it is somewhat less reliable. Moreover, if you already have the QR factorization, you might want to re-use it for the rank, and you should be allowed to query it for this information if you want.

See, for example, this discussion.

[aravindh-krishnamoorthy]

Hi, I am trying to make an attempt on this, I have a question tho... Could you explain the motivation behind introducing a rank calculation based on the pivoted QR decomposition?

This $2021$ blog post should also give you insight into the rank-revealing property of some matrix factorizations: What Is a Rank-Revealing Factorization?

[stevengj]

Similarly, it might be nice to have a nullspace(::QRPivoted) method and a cond(::QRPivoted) method (via cond(R)).