Allow rank computation for QRPivoted matrices

NEWS.md

@@ -87,6 +87,8 @@ Standard library changes
 
 #### LinearAlgebra
 
+* `rank` can now take a `QRPivoted` matrix to allow rank estimation via QR factorization ([#54283]).
+
 #### Logging
 
 #### Printf

stdlib/LinearAlgebra/src/qr.jl

@@ -535,6 +535,13 @@ function ldiv!(A::QRCompactWY{T}, B::AbstractMatrix{T}) where {T}
     return B
 end
 
+function rank(A::QRPivoted; atol::Real=0, rtol::Real=min(size(A)...) * eps(real(float(one(eltype(A.Q))))) * iszero(atol))
+    m = min(size(A)...)
+    m == 0 && return 0
+    tol = max(atol, rtol*abs(A.R[1,1]))
+    return something(findfirst(i -> abs(A.R[i,i]) <= tol, 1:m), m+1) - 1
+end
+
 # Julia implementation similar to xgelsy
 function ldiv!(A::QRPivoted{T,<:StridedMatrix}, B::AbstractMatrix{T}, rcond::Real) where {T<:BlasFloat}
     require_one_based_indexing(B)

stdlib/LinearAlgebra/test/qr.jl

@@ -528,4 +528,16 @@ end
     end
 end
 
+@testset "issue #53214" begin
+    # Test that the rank of a QRPivoted matrix is computed correctly
+    @test rank(qr([1.0 0.0; 0.0 1.0], ColumnNorm())) == 2
+    @test rank(qr([1.0 0.0; 0.0 0.9], ColumnNorm()), rtol=0.95) == 1
+    @test rank(qr([1.0 0.0; 0.0 0.9], ColumnNorm()), atol=0.95) == 1
+    @test rank(qr([1.0 0.0; 0.0 1.0], ColumnNorm()), rtol=1.01) == 0
+    @test rank(qr([1.0 0.0; 0.0 1.0], ColumnNorm()), atol=1.01) == 0
+
+    @test rank(qr([1.0 2.0; 2.0 4.0], ColumnNorm())) == 1
+    @test rank(qr([1.0 2.0 3.0; 4.0 5.0 6.0 ; 7.0 8.0 9.0], ColumnNorm())) == 2
+end
+
 end # module TestQR