“Fix” #53451 -- allow zero-row QR factorization bypassing LAPACK

stdlib/LinearAlgebra/src/lapack.jl

@@ -470,7 +470,7 @@ for (gebrd, gelqf, geqlf, geqrf, geqp3, geqrt, geqrt3, gerqf, getrf, elty, relty
             end
             lda = max(1, stride(A,2))
             work = Vector{$elty}(undef, nb*n)
-            if n > 0
+            if minmn > 0
                 info = Ref{BlasInt}()
                 ccall((@blasfunc($geqrt), libblastrampoline), Cvoid,
                     (Ref{BlasInt}, Ref{BlasInt}, Ref{BlasInt}, Ptr{$elty},
@@ -496,7 +496,7 @@ for (gebrd, gelqf, geqlf, geqrf, geqp3, geqrt, geqrt3, gerqf, getrf, elty, relty
             if p != n || q != n
                 throw(DimensionMismatch("block reflector T has dimensions ($p,$q), but should have dimensions ($n,$n)"))
             end
-            if n > 0
+            if n > 0    # this implies `m > 0` because of `m >= n`
                 info = Ref{BlasInt}()
                 ccall((@blasfunc($geqrt3), libblastrampoline), Cvoid,
                     (Ref{BlasInt}, Ref{BlasInt}, Ptr{$elty}, Ref{BlasInt},

stdlib/LinearAlgebra/test/qr.jl

@@ -504,4 +504,28 @@ end
     @test x ≈ xf
 end
 
+@testset "issue #53451" begin
+    # in the issue it was noted that QR factorizations of zero-column matrices
+    # were possible, but zero row-matrices errored, because LAPACK does not
+    # accept these empty matrices. now, the `geqrt!` call should be forwarded only
+    # if both matrix dimensions are positive.
+
+    for dimA in (0, 1, 2, 4)
+        for F in (Float32, Float64, ComplexF32, ComplexF64, BigFloat)
+            # this should have worked before, Q is square, and R is 0 × 0:
+            A_zero_cols = rand(F, dimA, 0)
+            qr_zero_cols = qr(A_zero_cols)
+            @test size(qr_zero_cols.Q) == (dimA, dimA)
+            @test size(qr_zero_cols.R) == (0, 0)
+            @test qr_zero_cols.Q == LinearAlgebra.I(dimA)
+
+            # this should work now, Q is 0 × 0, and R has `dimA` columns:
+            A_zero_rows = rand(F, 0, dimA)
+            qr_zero_rows = qr(A_zero_rows)
+            @test size(qr_zero_rows.Q) == (0, 0)
+            @test size(qr_zero_rows.R) == (0, dimA)
+        end
+    end
+end
+
 end # module TestQR