improve cat design

This is type-piracy, but we cannot change that (until
JuliaLang/julia#2326), so at least do not make these method
intersections unnecessary slow and complicated for everyone who does not
care about SparseArrays and does not load it, and unreliable for
everyone who does load it.

src/linalg.jl

@@ -1323,6 +1323,13 @@ const _Triangular_SparseKronArrays = UpperOrLowerTriangular{<:Any,<:_SparseKronA
 const _Annotated_SparseKronArrays = Union{_Triangular_SparseKronArrays, _Symmetric_SparseKronArrays, _Hermitian_SparseKronArrays}
 const _SparseKronGroup = Union{_SparseKronArrays, _Annotated_SparseKronArrays}
 
+const _SpecialArrays = Union{Diagonal, Bidiagonal, Tridiagonal, SymTridiagonal}
+const _Symmetric_DenseArrays{T,A<:Matrix} = Symmetric{T,A}
+const _Hermitian_DenseArrays{T,A<:Matrix} = Hermitian{T,A}
+const _Triangular_DenseArrays{T,A<:Matrix} = UpperOrLowerTriangular{<:Any,A} # AbstractTriangular{T,A}
+const _Annotated_DenseArrays = Union{_SpecialArrays, _Triangular_DenseArrays, _Symmetric_DenseArrays, _Hermitian_DenseArrays}
+const _DenseConcatGroup = Union{Number, Vector, Adjoint{<:Any,<:Vector}, Transpose{<:Any,<:Vector}, Matrix, _Annotated_DenseArrays}
+
 @inline function kron!(C::SparseMatrixCSC, A::AbstractSparseMatrixCSC, B::AbstractSparseMatrixCSC)
     mA, nA = size(A); mB, nB = size(B)
     mC, nC = mA*mB, nA*nB

src/sparsevector.jl

@@ -4,7 +4,6 @@
 
 import Base: sort!, findall, copy!
 import LinearAlgebra: promote_to_array_type, promote_to_arrays_
-using LinearAlgebra: _SpecialArrays, _DenseConcatGroup
 
 ### The SparseVector
 
@@ -1175,24 +1174,10 @@ function _absspvec_vcat(X::AbstractSparseVector{Tv,Ti}...) where {Tv,Ti}
     SparseVector(len, rnzind, rnzval)
 end
 
-hcat(Xin::Union{Vector, AbstractSparseVector}...) = hcat(map(sparse, Xin)...)
-vcat(Xin::Union{Vector, AbstractSparseVector}...) = vcat(map(sparse, Xin)...)
-
 ### Concatenation of un/annotated sparse/special/dense vectors/matrices
-
-const _SparseArrays = Union{AbstractSparseVector,
-                            AbstractSparseMatrixCSC,
-                            Adjoint{<:Any,<:AbstractSparseVector},
-                            Transpose{<:Any,<:AbstractSparseVector}}
-const _SparseConcatArrays = Union{_SpecialArrays, _SparseArrays}
-
-const _Symmetric_SparseConcatArrays = Symmetric{<:Any,<:_SparseConcatArrays}
-const _Hermitian_SparseConcatArrays = Hermitian{<:Any,<:_SparseConcatArrays}
-const _Triangular_SparseConcatArrays = UpperOrLowerTriangular{<:Any,<:_SparseConcatArrays}
-const _Annotated_SparseConcatArrays = Union{_Triangular_SparseConcatArrays, _Symmetric_SparseConcatArrays, _Hermitian_SparseConcatArrays}
-# It's important that _SparseConcatGroup is a larger union than _DenseConcatGroup to make
-# sparse cat-methods less specific and to kick in only if there is some sparse array present
-const _SparseConcatGroup = Union{_DenseConcatGroup, _SparseConcatArrays, _Annotated_SparseConcatArrays}
+# by type-pirating and subverting the Base.cat design by making these a subtype of the normal methods for it
+# and re-defining all of it here. See https://github.com/JuliaLang/julia/issues/2326
+# for what would have been a more principled way of doing this.
 
 # Concatenations involving un/annotated sparse/special matrices/vectors should yield sparse arrays
 
@@ -1204,23 +1189,55 @@ _sparse(A) = _makesparse(A)
 _makesparse(x::Number) = x
 _makesparse(x::AbstractVector) = convert(SparseVector, issparse(x) ? x : sparse(x))::SparseVector
 _makesparse(x::AbstractMatrix) = convert(SparseMatrixCSC, issparse(x) ? x : sparse(x))::SparseMatrixCSC
+anysparse() = false
+anysparse(X) = X isa AbstractArray && issparse(X)
+anysparse(X, Xs...) = anysparse(X) || anysparse(Xs...)
+
+function hcat(X::Union{Vector, AbstractSparseVector}...)
+    if anysparse(X...)
+        X = map(sparse, X)
+    end
+    return cat(X...; dims=Val(2))
+end
+function vcat(X::Union{Vector, AbstractSparseVector}...)
+    if anysparse(X...)
+        X = map(sparse, X)
+    end
+    return cat(X...; dims=Val(1))
+end
+
+# type-pirate the Base.cat design by making this a subtype of the existing method for it
+# in future versions of Julia (v1.10+), in which https://github.com/JuliaLang/julia/issues/2326 is not fixed yet, the <:Number constraint could be relaxed
+# but see also https://github.com/JuliaSparse/SparseArrays.jl/issues/71
+const _SparseConcatGroup = Union{AbstractVecOrMat{<:Number},Number}
 
 # `@constprop :aggressive` allows `dims` to be propagated as constant improving return type inference
-Base.@constprop :aggressive function Base._cat(dims, Xin::_SparseConcatGroup...)
-    X = (_sparse(first(Xin)), map(_makesparse, Base.tail(Xin))...)
-    T = promote_eltype(Xin...)
+Base.@constprop :aggressive function Base._cat(dims, X::_SparseConcatGroup...)
+    T = promote_eltype(X...)
+    if anysparse(X...)
+        X = (_sparse(first(X)), map(_makesparse, Base.tail(X))...)
+    end
     return Base._cat_t(dims, T, X...)
 end
-function hcat(Xin::_SparseConcatGroup...)
-    X = (_sparse(first(Xin)), map(_makesparse, Base.tail(Xin))...)
+function hcat(X::_SparseConcatGroup...)
+    if anysparse(X...)
+        X = (_sparse(first(X)), map(_makesparse, Base.tail(X))...)
+    end
     return cat(X..., dims=Val(2))
 end
-function vcat(Xin::_SparseConcatGroup...)
-    X = (_sparse(first(Xin)), map(_makesparse, Base.tail(Xin))...)
+function vcat(X::_SparseConcatGroup...)
+    if anysparse(X...)
+        X = (_sparse(first(X)), map(_makesparse, Base.tail(X))...)
+    end
     return cat(X..., dims=Val(1))
 end
-hvcat(rows::Tuple{Vararg{Int}}, X::_SparseConcatGroup...) =
-    vcat(_hvcat_rows(rows, X...)...)
+function hvcat(rows::Tuple{Vararg{Int}}, X::_SparseConcatGroup...)
+    if anysparse(X...)
+        vcat(_hvcat_rows(rows, X...)...)
+    else
+        typed_hvcat(promote_eltypeof(X...), rows, X...)
+    end
+end
 function _hvcat_rows((row1, rows...)::Tuple{Vararg{Int}}, X::_SparseConcatGroup...)
     if row1 ≤ 0
         throw(ArgumentError("length of block row must be positive, got $row1"))
@@ -1237,7 +1254,7 @@ end
 _hvcat_rows(::Tuple{}, X::_SparseConcatGroup...) = ()
 
 # make sure UniformScaling objects are converted to sparse matrices for concatenation
-promote_to_array_type(A::Tuple{Vararg{Union{_SparseConcatGroup,UniformScaling}}}) = SparseMatrixCSC
+promote_to_array_type(A::Tuple{Vararg{Union{_SparseConcatGroup,UniformScaling}}}) = anysparse(A...) ? SparseMatrixCSC : Matrix
 promote_to_arrays_(n::Int, ::Type{SparseMatrixCSC}, J::UniformScaling) = sparse(J, n, n)
 
 """