Make cholesky handle AbstractMatrix (JuliaLang#44076)

Co-authored-by: Sheehan Olver <s.olver@imperial.ac.uk>

stdlib/LinearAlgebra/src/cholesky.jl

@@ -179,7 +179,9 @@ Base.iterate(C::CholeskyPivoted, ::Val{:done}) = nothing
 
 # make a copy that allow inplace Cholesky factorization
 @inline choltype(A) = promote_type(typeof(sqrt(oneunit(eltype(A)))), Float32)
-@inline cholcopy(A) = copy_oftype(A, choltype(A))
+@inline cholcopy(A::StridedMatrix) = copy_oftype(A, choltype(A))
+@inline cholcopy(A::RealHermSymComplexHerm) = copy_oftype(A, choltype(A))
+@inline cholcopy(A::AbstractMatrix) = copy_similar(A, choltype(A))
 
 # _chol!. Internal methods for calling unpivoted Cholesky
 ## BLAS/LAPACK element types
@@ -269,9 +271,9 @@ function cholesky!(A::RealHermSymComplexHerm, ::NoPivot = NoPivot(); check::Bool
     return Cholesky(C.data, A.uplo, info)
 end
 
-### for StridedMatrices, check that matrix is symmetric/Hermitian
+### for AbstractMatrix, check that matrix is symmetric/Hermitian
 """
-    cholesky!(A::StridedMatrix, NoPivot(); check = true) -> Cholesky
+    cholesky!(A::AbstractMatrix, NoPivot(); check = true) -> Cholesky
 
 The same as [`cholesky`](@ref), but saves space by overwriting the input `A`,
 instead of creating a copy. An [`InexactError`](@ref) exception is thrown if
@@ -291,7 +293,7 @@ Stacktrace:
 [...]
 ```
 """
-function cholesky!(A::StridedMatrix, ::NoPivot = NoPivot(); check::Bool = true)
+function cholesky!(A::AbstractMatrix, ::NoPivot = NoPivot(); check::Bool = true)
     checksquare(A)
     if !ishermitian(A) # return with info = -1 if not Hermitian
         check && checkpositivedefinite(-1)
@@ -320,16 +322,16 @@ cholesky!(A::RealHermSymComplexHerm{<:Real}, ::RowMaximum; tol = 0.0, check::Boo
     throw(ArgumentError("generic pivoted Cholesky factorization is not implemented yet"))
 @deprecate cholesky!(A::RealHermSymComplexHerm{<:Real}, ::Val{true}; kwargs...) cholesky!(A, RowMaximum(); kwargs...) false
 
-### for StridedMatrices, check that matrix is symmetric/Hermitian
+### for AbstractMatrix, check that matrix is symmetric/Hermitian
 """
-    cholesky!(A::StridedMatrix, RowMaximum(); tol = 0.0, check = true) -> CholeskyPivoted
+    cholesky!(A::AbstractMatrix, RowMaximum(); tol = 0.0, check = true) -> CholeskyPivoted
 
 The same as [`cholesky`](@ref), but saves space by overwriting the input `A`,
 instead of creating a copy. An [`InexactError`](@ref) exception is thrown if the
 factorization produces a number not representable by the element type of `A`,
 e.g. for integer types.
 """
-function cholesky!(A::StridedMatrix, ::RowMaximum; tol = 0.0, check::Bool = true)
+function cholesky!(A::AbstractMatrix, ::RowMaximum; tol = 0.0, check::Bool = true)
     checksquare(A)
     if !ishermitian(A)
         C = CholeskyPivoted(A, 'U', Vector{BlasInt}(),convert(BlasInt, 1),
@@ -350,7 +352,7 @@ end
 
 Compute the Cholesky factorization of a dense symmetric positive definite matrix `A`
 and return a [`Cholesky`](@ref) factorization. The matrix `A` can either be a [`Symmetric`](@ref) or [`Hermitian`](@ref)
-[`StridedMatrix`](@ref) or a *perfectly* symmetric or Hermitian `StridedMatrix`.
+[`AbstractMatrix`](@ref) or a *perfectly* symmetric or Hermitian `AbstractMatrix`.
 
 The triangular Cholesky factor can be obtained from the factorization `F` via `F.L` and `F.U`,
 where `A ≈ F.U' * F.U ≈ F.L * F.L'`.
@@ -397,11 +399,11 @@ julia> C.L * C.U == A
 true
 ```
 """
-cholesky(A::Union{StridedMatrix,RealHermSymComplexHerm{<:Real,<:StridedMatrix}},
-    ::NoPivot=NoPivot(); check::Bool = true) = cholesky!(cholcopy(A); check = check)
+cholesky(A::AbstractMatrix, ::NoPivot=NoPivot(); check::Bool = true) =
+    cholesky!(cholcopy(A); check)
 @deprecate cholesky(A::Union{StridedMatrix,RealHermSymComplexHerm{<:Real,<:StridedMatrix}}, ::Val{false}; check::Bool = true) cholesky(A, NoPivot(); check) false
 
-function cholesky(A::Union{StridedMatrix{Float16},RealHermSymComplexHerm{Float16,<:StridedMatrix}}, ::NoPivot=NoPivot(); check::Bool = true)
+function cholesky(A::AbstractMatrix{Float16}, ::NoPivot=NoPivot(); check::Bool = true)
     X = cholesky!(cholcopy(A); check = check)
     return Cholesky{Float16}(X)
 end
@@ -413,7 +415,7 @@ end
 
 Compute the pivoted Cholesky factorization of a dense symmetric positive semi-definite matrix `A`
 and return a [`CholeskyPivoted`](@ref) factorization. The matrix `A` can either be a [`Symmetric`](@ref)
-or [`Hermitian`](@ref) [`StridedMatrix`](@ref) or a *perfectly* symmetric or Hermitian `StridedMatrix`.
+or [`Hermitian`](@ref) [`AbstractMatrix`](@ref) or a *perfectly* symmetric or Hermitian `AbstractMatrix`.
 
 The triangular Cholesky factor can be obtained from the factorization `F` via `F.L` and `F.U`,
 and the permutation via `F.p`, where `A[F.p, F.p] ≈ Ur' * Ur ≈ Lr * Lr'` with `Ur = F.U[1:F.rank, :]`
@@ -463,11 +465,15 @@ julia> l == C.L && u == C.U
 true
 ```
 """
-cholesky(A::Union{StridedMatrix,RealHermSymComplexHerm{<:Real,<:StridedMatrix}},
-    ::RowMaximum; tol = 0.0, check::Bool = true) =
-    cholesky!(cholcopy(A), RowMaximum(); tol = tol, check = check)
+cholesky(A::AbstractMatrix, ::RowMaximum; tol = 0.0, check::Bool = true) =
+    cholesky!(cholcopy(A), RowMaximum(); tol, check)
 @deprecate cholesky(A::Union{StridedMatrix,RealHermSymComplexHerm{<:Real,<:StridedMatrix}}, ::Val{true}; tol = 0.0, check::Bool = true) cholesky(A, RowMaximum(); tol, check) false
 
+function cholesky(A::AbstractMatrix{Float16}, ::RowMaximum; tol = 0.0, check::Bool = true)
+    X = cholesky!(cholcopy(A), RowMaximum(); tol, check)
+    return CholeskyPivoted{Float16}(X)
+end
+
 ## Number
 function cholesky(x::Number, uplo::Symbol=:U)
     C, info = _chol!(x, uplo)
@@ -524,7 +530,7 @@ end
 Base.propertynames(F::Cholesky, private::Bool=false) =
     (:U, :L, :UL, (private ? fieldnames(typeof(F)) : ())...)
 
-function getproperty(C::CholeskyPivoted{T}, d::Symbol) where T<:BlasFloat
+function getproperty(C::CholeskyPivoted{T}, d::Symbol) where {T}
     Cfactors = getfield(C, :factors)
     Cuplo    = getfield(C, :uplo)
     if d === :U
@@ -595,7 +601,7 @@ function ldiv!(C::CholeskyPivoted{T}, B::StridedMatrix{T}) where T<:BlasFloat
     B
 end
 
-function ldiv!(C::CholeskyPivoted, B::StridedVector)
+function ldiv!(C::CholeskyPivoted, B::AbstractVector)
     if C.uplo == 'L'
         ldiv!(adjoint(LowerTriangular(C.factors)),
             ldiv!(LowerTriangular(C.factors), permute!(B, C.piv)))
@@ -606,7 +612,7 @@ function ldiv!(C::CholeskyPivoted, B::StridedVector)
     invpermute!(B, C.piv)
 end
 
-function ldiv!(C::CholeskyPivoted, B::StridedMatrix)
+function ldiv!(C::CholeskyPivoted, B::AbstractMatrix)
     n = size(C, 1)
     for i in 1:size(B, 2)
         permute!(view(B, 1:n, i), C.piv)
@@ -624,15 +630,15 @@ function ldiv!(C::CholeskyPivoted, B::StridedMatrix)
     B
 end
 
-function rdiv!(B::StridedMatrix, C::Cholesky{<:Any,<:AbstractMatrix})
+function rdiv!(B::AbstractMatrix, C::Cholesky{<:Any,<:AbstractMatrix})
     if C.uplo == 'L'
         return rdiv!(rdiv!(B, adjoint(LowerTriangular(C.factors))), LowerTriangular(C.factors))
     else
         return rdiv!(rdiv!(B, UpperTriangular(C.factors)), adjoint(UpperTriangular(C.factors)))
     end
 end
 
-function LinearAlgebra.rdiv!(B::StridedMatrix, C::CholeskyPivoted)
+function LinearAlgebra.rdiv!(B::AbstractMatrix, C::CholeskyPivoted)
     n = size(C, 2)
     for i in 1:size(B, 1)
         permute!(view(B, i, 1:n), C.piv)

stdlib/LinearAlgebra/src/diagonal.jl

@@ -753,11 +753,11 @@ function cholesky!(A::Diagonal, ::NoPivot = NoPivot(); check::Bool = true)
     Cholesky(A, 'U', convert(BlasInt, info))
 end
 @deprecate cholesky!(A::Diagonal, ::Val{false}; check::Bool = true) cholesky!(A::Diagonal, NoPivot(); check) false
-
-cholesky(A::Diagonal, ::NoPivot = NoPivot(); check::Bool = true) =
-    cholesky!(cholcopy(A), NoPivot(); check = check)
 @deprecate cholesky(A::Diagonal, ::Val{false}; check::Bool = true) cholesky(A::Diagonal, NoPivot(); check) false
 
+@inline cholcopy(A::Diagonal) = copy_oftype(A, choltype(A))
+@inline cholcopy(A::RealHermSymComplexHerm{<:Real,<:Diagonal}) = copy_oftype(A, choltype(A))
+
 function getproperty(C::Cholesky{<:Any,<:Diagonal}, d::Symbol)
     Cfactors = getfield(C, :factors)
     if d in (:U, :L, :UL)

stdlib/LinearAlgebra/src/special.jl

@@ -376,3 +376,10 @@ Base._cat(dims, xs::_TypedDenseConcatGroup{T}...) where {T} = Base.cat_t(T, xs..
 vcat(A::_TypedDenseConcatGroup{T}...) where {T} = Base.typed_vcat(T, A...)
 hcat(A::_TypedDenseConcatGroup{T}...) where {T} = Base.typed_hcat(T, A...)
 hvcat(rows::Tuple{Vararg{Int}}, xs::_TypedDenseConcatGroup{T}...) where {T} = Base.typed_hvcat(T, rows, xs...)
+
+# factorizations
+function cholesky(S::RealHermSymComplexHerm{<:Real,<:SymTridiagonal}, ::NoPivot = NoPivot(); check::Bool = true)
+    T = choltype(eltype(S))
+    B = Bidiagonal{T}(diag(S, 0), diag(S, S.uplo == 'U' ? 1 : -1), sym_uplo(S.uplo))
+    cholesky!(Hermitian(B, sym_uplo(S.uplo)), NoPivot(); check = check)
+end

stdlib/LinearAlgebra/src/tridiag.jl

@@ -854,3 +854,12 @@ function dot(x::AbstractVector, A::Tridiagonal, y::AbstractVector)
     r += dot(adjoint(du[nx-1])*x₀ + adjoint(d[nx])*x₊, y[nx])
     return r
 end
+
+function cholesky(S::SymTridiagonal, ::NoPivot = NoPivot(); check::Bool = true)
+    if !ishermitian(S)
+        check && checkpositivedefinite(-1)
+        return Cholesky(S, 'U', convert(BlasInt, -1))
+    end
+    T = choltype(eltype(S))
+    cholesky!(Hermitian(Bidiagonal{T}(diag(S, 0), diag(S, 1), :U)), NoPivot(); check = check)
+end

stdlib/LinearAlgebra/test/cholesky.jl

@@ -124,8 +124,15 @@ end
         end
 
         # test cholesky of 2x2 Strang matrix
-        S = Matrix{eltya}(SymTridiagonal([2, 2], [-1]))
+        S = SymTridiagonal{eltya}([2, 2], [-1])
+        for uplo in (:U, :L)
+            @test Matrix(@inferred cholesky(Hermitian(S, uplo))) ≈ S
+            if eltya <: Real
+                @test Matrix(@inferred cholesky(Symmetric(S, uplo))) ≈ S
+            end
+        end
         @test Matrix(cholesky(S).U) ≈ [2 -1; 0 sqrt(eltya(3))] / sqrt(eltya(2))
+        @test Matrix(cholesky(S)) ≈ S
 
         # test extraction of factor and re-creating original matrix
         if eltya <: Real
@@ -371,6 +378,10 @@ end
     @test D ≈ CD.L * CD.U
     @test CD.info == 0
 
+    F = cholesky(Hermitian(I(3)))
+    @test F isa Cholesky{Float64,<:Diagonal}
+    @test Matrix(F) ≈ I(3)
+
     # real, failing
     @test_throws PosDefException cholesky(Diagonal([1.0, -2.0]))
     Dnpd = cholesky(Diagonal([1.0, -2.0]); check = false)
@@ -502,6 +513,15 @@ end
     @test B.U ≈ B32.U
     @test B.L ≈ B32.L
     @test B.UL ≈ B32.UL
+    @test Matrix(B) ≈ A
+    B = cholesky(A, RowMaximum())
+    B32 = cholesky(Float32.(A), RowMaximum())
+    @test B isa CholeskyPivoted{Float16,Matrix{Float16}}
+    @test B.U isa UpperTriangular{Float16, Matrix{Float16}}
+    @test B.L isa LowerTriangular{Float16, Matrix{Float16}}
+    @test B.U ≈ B32.U
+    @test B.L ≈ B32.L
+    @test Matrix(B) ≈ A
 end
 
 @testset "det and logdet" begin