add fixes for \

examples/lapack.jl

@@ -0,0 +1,41 @@
+using SemiseparableMatrices
+
+# A benefit of our approach to QR is that it is in an LAPACK-compatible format. Thus for moderate n our
+# O(n) factorization can either be combined with our O(n) solver, or LAPack's optimised O(n^2) solver.
+# Here is a demonstration of using this. For small n the dense QR is only marginally slower:
+
+n = 1000
+l, m, r, p = 4, 5, 2, 3
+B = brandn(n,n,l,m)
+U,V = randn(n,r), randn(n,r)
+W,S = randn(n,p), randn(n,p)
+b = randn(n)
+
+A = BandedPlusSemiseparableMatrix(B,(U,V),(W,S))
+Ã = Matrix(A) # dense A
+@time F = qr(A); # BPS QR, 0.01s
+@time F̃ = qr(Ã); # Dense QRCompactWY, 0.04s
+F̄ = QR(Matrix(F.factors), F.τ); # convert to dense format
+
+@time F\b; # 0.02s
+@time F̃\b; # 0.001s
+@time F̄\b; # 0.004s
+
+# But if we make n much bigger dense QR becomes prohibitively slow:
+
+n = 10_000
+l, m, r, p = 4, 5, 2, 3
+B = brandn(n,n,l,m)
+U,V = randn(n,r), randn(n,r)
+W,S = randn(n,p), randn(n,p)
+b = randn(n)
+
+A = BandedPlusSemiseparableMatrix(B,(U,V),(W,S))
+Ã = Matrix(A) # dense A
+@time F = qr(A); # BPS QR, 0.11s
+@time F̃ = qr(Ã); # Dense QRCompactWY, 27s
+@time F̄ = QR(Matrix(F.factors), F.τ); # convert to dense format 2s, A 10x speedup!
+
+@time F.R\(F.Q'b); # 0.001s
+@time F̃\b; # 0.08s
+@time F̄\b; # 0.25s

src/BandedPlusSemiseparableMatrix.jl

@@ -50,7 +50,8 @@ function getindex(A::BandedPlusSemiseparableMatrix, k::Integer, j::Integer)
     end
 end
 
-function ldiv!(R::UpperTriangular{<:Any,<:BandedPlusSemiseparableMatrix}, b::StridedVector)
+function ArrayLayouts.ldiv!(R::UpperTriangular{<:Any,<:BandedPlusSemiseparableMatrix}, b::StridedVecOrMat)
+    @assert b isa AbstractVector
     F = parent(R)
     n, p = size(F.W)
     l, m = bandwidths(F.B)

src/semiseparableqr.jl

@@ -860,7 +860,8 @@ function getproperty(F::QR{<:Any,<:BandedPlusSemiseparableMatrix}, d::Symbol)
     end
 end
 
-function lmul!(adjQ::AdjointQ{<:Any,<:QRPackedQ{<:Any,<:BandedPlusSemiseparableMatrix}}, b::StridedVector)
+function ArrayLayouts.lmul!(adjQ::AdjointQ{<:Any,<:QRPackedQ{<:Any,<:BandedPlusSemiseparableMatrix}}, b::StridedVecOrMat)
+    @assert b isa AbstractVector
     Q = parent(adjQ)
     F = Q.factors
     τ = Q.τ
@@ -902,6 +903,7 @@ function lmul!(adjQ::AdjointQ{<:Any,<:QRPackedQ{<:Any,<:BandedPlusSemiseparableM
     end
     #b[n] += F.U[n,:]'*h
     b[n] += view(F.U, n, :)' * h
+    b
 end
 
 function Uᵀb_lookup_table(F, b)

test/test_qr.jl

@@ -51,5 +51,7 @@ using SemiseparableMatrices: BandedPlusSemiseparableQRPerturbedFactors
         x = triu(F) \ b
         ldiv!(R, b)
         @test b ≈ x
+
+        @test F \ b_true ≈ x
     end
 end