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More general division by triangular matrices #837

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Oct 16, 2020
Merged

More general division by triangular matrices #837

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225 changes: 46 additions & 179 deletions src/triangular.jl
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Original file line number Diff line number Diff line change
@@ -1,11 +1,19 @@
@inline transpose(A::LinearAlgebra.LowerTriangular{<:Any,<:StaticMatrix}) =
LinearAlgebra.UpperTriangular(transpose(A.data))
@inline adjoint(A::LinearAlgebra.LowerTriangular{<:Any,<:StaticMatrix}) =
LinearAlgebra.UpperTriangular(adjoint(A.data))
@inline transpose(A::LinearAlgebra.UpperTriangular{<:Any,<:StaticMatrix}) =
LinearAlgebra.LowerTriangular(transpose(A.data))
@inline adjoint(A::LinearAlgebra.UpperTriangular{<:Any,<:StaticMatrix}) =
LinearAlgebra.LowerTriangular(adjoint(A.data))
@inline transpose(A::LowerTriangular{<:Any,<:StaticMatrix}) =
UpperTriangular(transpose(A.data))
@inline adjoint(A::LowerTriangular{<:Any,<:StaticMatrix}) =
UpperTriangular(adjoint(A.data))
@inline transpose(A::UnitLowerTriangular{<:Any,<:StaticMatrix}) =
UnitUpperTriangular(transpose(A.data))
@inline adjoint(A::UnitLowerTriangular{<:Any,<:StaticMatrix}) =
UnitUpperTriangular(adjoint(A.data))
@inline transpose(A::UpperTriangular{<:Any,<:StaticMatrix}) =
LowerTriangular(transpose(A.data))
@inline adjoint(A::UpperTriangular{<:Any,<:StaticMatrix}) =
LowerTriangular(adjoint(A.data))
@inline transpose(A::UnitUpperTriangular{<:Any,<:StaticMatrix}) =
UnitLowerTriangular(transpose(A.data))
@inline adjoint(A::UnitUpperTriangular{<:Any,<:StaticMatrix}) =
UnitLowerTriangular(adjoint(A.data))
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It's surprising that Base doesn't seem to include these rules.

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Is it worth opening an issue in Julia repository?

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Sure. I can't think of a reason these shouldn't be structure-preserving.

@inline Base.:*(A::Adjoint{<:Any,<:StaticVector}, B::LinearAlgebra.AbstractTriangular{<:Any,<:StaticMatrix}) =
adjoint(adjoint(B) * adjoint(A))
@inline Base.:*(A::Transpose{<:Any,<:StaticVector}, B::LinearAlgebra.AbstractTriangular{<:Any,<:StaticMatrix}) =
Expand All @@ -15,193 +23,52 @@
@inline Base.:*(A::LinearAlgebra.AbstractTriangular{<:Any,<:StaticMatrix}, B::Transpose{<:Any,<:StaticVector}) =
transpose(transpose(B) * transpose(A))

const StaticULT = Union{UpperTriangular{<:Any,<:StaticMatrix},LowerTriangular{<:Any,<:StaticMatrix}}
const StaticULT{TA} = Union{UpperTriangular{TA,<:StaticMatrix},LowerTriangular{TA,<:StaticMatrix},UnitUpperTriangular{TA,<:StaticMatrix},UnitLowerTriangular{TA,<:StaticMatrix}}

@inline Base.:\(A::StaticULT, B::StaticVecOrMat) = _A_ldiv_B(Size(A), Size(B), A, B)
@inline Base.:\(A::StaticULT, B::StaticVecOrMatLike) = _A_ldiv_B(Size(A), Size(B), A, B)
@inline Base.:/(A::StaticVecOrMatLike, B::StaticULT) = transpose(transpose(B) \ transpose(A))

@generated function _A_ldiv_B(::Size{sa}, ::Size{sb}, A::UpperTriangular{<:TA,<:StaticMatrix}, B::StaticVecOrMat{TB}) where {sa,sb,TA,TB}
@generated function _A_ldiv_B(::Size{sa}, ::Size{sb}, A::StaticULT{TA}, B::StaticVecOrMatLike{TB}) where {sa,sb,TA,TB}
m = sb[1]
n = length(sb) > 1 ? sb[2] : 1
if m != sa[1]
throw(DimensionMismatch("right hand side B needs first dimension of size $(sa[1]), has size $m"))
end

X = [Symbol("X_$(i)_$(j)") for i = 1:m, j = 1:n]
init = [:($(X[i,j]) = B[$(LinearIndices(sb)[i, j])]) for i = 1:m, j = 1:n]

code = Expr(:block)
for k = 1:n
for j = m:-1:1
if k == 1
push!(code.args, :(A.data[$(LinearIndices(sa)[j, j])] == zero(A.data[$(LinearIndices(sa)[j, j])]) && throw(LinearAlgebra.SingularException($j))))
isunitdiag = A <: Union{UnitUpperTriangular, UnitLowerTriangular}
isupper = A <: Union{UnitUpperTriangular, UpperTriangular}

j_range = isupper ? (m:-1:1) : (1:m)

init = gen_by_access(B, :B) do access_b
init_exprs = [:($(X[i,j]) = $(uplo_access(sb, :b, i, j, access_b))) for i = 1:m, j = 1:n]
code = Expr(:block, init_exprs...)
for k = 1:n
for j = j_range
if !isunitdiag && k == 1
push!(code.args, :(A.data[$(LinearIndices(sa)[j, j])] == zero(A.data[$(LinearIndices(sa)[j, j])]) && throw(LinearAlgebra.SingularException($j))))
end
if isunitdiag
push!(code.args, :($(X[j,k]) = oneunit(TA) \ $(X[j,k])))
else
push!(code.args, :($(X[j,k]) = A.data[$(LinearIndices(sa)[j, j])] \ $(X[j,k])))
end
i_range = isupper ? (j-1:-1:1) : (j+1:m)
for i = i_range
push!(code.args, :($(X[i,k]) -= A.data[$(LinearIndices(sa)[i, j])]*$(X[j,k])))
end
end
push!(code.args, :($(X[j,k]) = A.data[$(LinearIndices(sa)[j, j])] \ $(X[j,k])))
for i = j-1:-1:1
push!(code.args, :($(X[i,k]) -= A.data[$(LinearIndices(sa)[i, j])]*$(X[j,k])))
end
end
end

return quote
@_inline_meta
@inbounds $(Expr(:block, init...))
@inbounds $code
TAB = typeof((zero(TA)*zero(TB) + zero(TA)*zero(TB))/one(TA))
@inbounds return similar_type(B, TAB)(tuple($(X...)))
end
end

@generated function _A_ldiv_B(::Size{sa}, ::Size{sb}, A::LowerTriangular{<:TA,<:StaticMatrix}, B::StaticVecOrMat{TB}) where {sa,sb,TA,TB}
m = sb[1]
n = length(sb) > 1 ? sb[2] : 1
if m != sa[1]
throw(DimensionMismatch("right hand side B needs first dimension of size $(sa[1]), has size $m"))
end

X = [Symbol("X_$(i)_$(j)") for i = 1:m, j = 1:n]
init = [:($(X[i,j]) = B[$(LinearIndices(sb)[i, j])]) for i = 1:m, j = 1:n]

code = Expr(:block)
for k = 1:n
for j = 1:m
if k == 1
push!(code.args, :(A.data[$(LinearIndices(sa)[j, j])] == zero(A.data[$(LinearIndices(sa)[j, j])]) && throw(LinearAlgebra.SingularException($j))))
end
push!(code.args, :($(X[j,k]) = A.data[$(LinearIndices(sa)[j, j])] \ $(X[j,k])))
for i = j+1:m
push!(code.args, :($(X[i,k]) -= A.data[$(LinearIndices(sa)[i, j])]*$(X[j,k])))
end
end
end

return quote
@_inline_meta
@inbounds $(Expr(:block, init...))
@inbounds $code
TAB = typeof((zero(TA)*zero(TB) + zero(TA)*zero(TB))/one(TA))
@inbounds return similar_type(B, TAB)(tuple($(X...)))
end
end

@generated function _Ac_ldiv_B(::Size{sa}, ::Size{sb}, A::UpperTriangular{<:TA,<:StaticMatrix}, B::StaticVecOrMat{TB}) where {sa,sb,TA,TB}
m = sb[1]
n = length(sb) > 1 ? sb[2] : 1
if m != sa[1]
throw(DimensionMismatch("right hand side B needs first dimension of size $(sa[1]), has size $m"))
end

X = [Symbol("X_$(i)_$(j)") for i = 1:m, j = 1:n]

code = Expr(:block)
for k = 1:n
for j = 1:m
ex = :(B[$(LinearIndices(sb)[j, k])])
for i = 1:j-1
ex = :($ex - A.data[$(LinearIndices(sa)[i, j])]'*$(X[i,k]))
end
if k == 1
push!(code.args, :(A.data[$(LinearIndices(sa)[j, j])] == zero(A.data[$(LinearIndices(sa)[j, j])]) && throw(LinearAlgebra.SingularException($j))))
end
push!(code.args, :($(X[j,k]) = A.data[$(LinearIndices(sa)[j, j])]' \ $ex))
end
end

return quote
@_inline_meta
@inbounds $code
TAB = typeof((zero(TA)*zero(TB) + zero(TA)*zero(TB))/one(TA))
@inbounds return similar_type(B, TAB)(tuple($(X...)))
end
end

@generated function _At_ldiv_B(::Size{sa}, ::Size{sb}, A::UpperTriangular{<:TA,<:StaticMatrix}, B::StaticVecOrMat{TB}) where {sa,sb,TA,TB}
m = sb[1]
n = length(sb) > 1 ? sb[2] : 1
if m != sa[1]
throw(DimensionMismatch("right hand side B needs first dimension of size $(sa[1]), has size $m"))
end

X = [Symbol("X_$(i)_$(j)") for i = 1:m, j = 1:n]

code = Expr(:block)
for k = 1:n
for j = 1:m
ex = :(B[$(LinearIndices(sb)[j, k])])
for i = 1:j-1
ex = :($ex - A.data[$(LinearIndices(sa)[i, j])]*$(X[i,k]))
end
if k == 1
push!(code.args, :(A.data[$(LinearIndices(sa)[j, j])] == zero(A.data[$(LinearIndices(sa)[j, j])]) && throw(LinearAlgebra.SingularException($j))))
end
push!(code.args, :($(X[j,k]) = A.data[$(LinearIndices(sa)[j, j])] \ $ex))
end
end

return quote
@_inline_meta
@inbounds $code
TAB = typeof((zero(TA)*zero(TB) + zero(TA)*zero(TB))/one(TA))
@inbounds return similar_type(B, TAB)(tuple($(X...)))
end
end

@generated function _Ac_ldiv_B(::Size{sa}, ::Size{sb}, A::LowerTriangular{<:TA,<:StaticMatrix}, B::StaticVecOrMat{TB}) where {sa,sb,TA,TB}
m = sb[1]
n = length(sb) > 1 ? sb[2] : 1
if m != sa[1]
throw(DimensionMismatch("right hand side B needs first dimension of size $(sa[1]), has size $m"))
end

X = [Symbol("X_$(i)_$(j)") for i = 1:m, j = 1:n]

code = Expr(:block)
for k = 1:n
for j = m:-1:1
ex = :(B[$(LinearIndices(sb)[j, k])])
for i = m:-1:j+1
ex = :($ex - A.data[$(LinearIndices(sa)[i, j])]'*$(X[i,k]))
end
if k == 1
push!(code.args, :(A.data[$(LinearIndices(sa)[j, j])] == zero(A.data[$(LinearIndices(sa)[j, j])]) && throw(LinearAlgebra.SingularException($j))))
end
push!(code.args, :($(X[j,k]) = A.data[$(LinearIndices(sa)[j, j])]' \ $ex))
end
end

return quote
@_inline_meta
@inbounds $code
TAB = typeof((zero(TA)*zero(TB) + zero(TA)*zero(TB))/one(TA))
@inbounds return similar_type(B, TAB)(tuple($(X...)))
end
end

@generated function _At_ldiv_B(::Size{sa}, ::Size{sb}, A::LowerTriangular{<:TA,<:StaticMatrix}, B::StaticVecOrMat{TB}) where {sa,sb,TA,TB}
m = sb[1]
n = length(sb) > 1 ? sb[2] : 1
if m != sa[1]
throw(DimensionMismatch("right hand side B needs first dimension of size $(sa[1]), has size $m"))
end

X = [Symbol("X_$(i)_$(j)") for i = 1:m, j = 1:n]

code = Expr(:block)
for k = 1:n
for j = m:-1:1
ex = :(B[$(LinearIndices(sb)[j, k])])
for i = m:-1:j+1
ex = :($ex - A.data[$(LinearIndices(sa)[i, j])]*$(X[i,k]))
end
if k == 1
push!(code.args, :(A.data[$(LinearIndices(sa)[j, j])] == zero(A.data[$(LinearIndices(sa)[j, j])]) && throw(LinearAlgebra.SingularException($j))))
end
push!(code.args, :($(X[j,k]) = A.data[$(LinearIndices(sa)[j, j])] \ $ex))
end
return code
end

return quote
@_inline_meta
@inbounds $code
b = mul_parent(B)
Tb = TB
@inbounds $init
TAB = typeof((zero(TA)*zero(TB) + zero(TA)*zero(TB))/one(TA))
@inbounds return similar_type(B, TAB)(tuple($(X...)))
end
Expand Down
29 changes: 19 additions & 10 deletions test/triangular.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -102,27 +102,36 @@ end
@testset "Triangular-matrix division" begin
for n in (1, 2, 3, 4),
eltyA in (Float64, ComplexF64, Int),
(t, uplo) in ((UpperTriangular, :U), (LowerTriangular, :L)),
eltyB in (Float64, ComplexF64)
(t, uplo) in ((UpperTriangular, :U), (LowerTriangular, :L), (UnitUpperTriangular, :U)),
eltyB in (Float64, ComplexF64),
tb in (identity, LowerTriangular, Symmetric)

A = t(eltyA == Int ? rand(1:7, n, n) : convert(Matrix{eltyA}, (eltyA <: Complex ? complex.(randn(n, n), randn(n, n)) : randn(n, n)) |> t -> cholesky(t't).U |> t -> uplo == :U ? t : adjoint(t)))
B = convert(Matrix{eltyB}, eltyA <: Complex ? real(A)*ones(n, n) : A*ones(n, n))
B = tb(convert(Matrix{eltyB}, eltyA <: Complex ? real(A)*ones(n, n) : A*ones(n, n)))
SA = t(SMatrix{n,n}(A.data))
SB = SMatrix{n,n}(B)
SB = tb(SMatrix{n,n}(parent(B)))

@test (SA\SB[:,1])::SVector{n} ≈ A\B[:,1]
if tb === identity
@test (SA\SB[:,1])::SVector{n} ≈ A\B[:,1]
@test (transpose(SA)\SB[:,1])::SVector{n} ≈ transpose(A)\B[:,1]
@test (SA'\SB[:,1])::SVector{n} ≈ A'\B[:,1]
end
@test (SA\SB)::SMatrix{n,n} ≈ A\B
@test (transpose(SA)\SB[:,1])::SVector{n} ≈ transpose(A)\B[:,1]
@test (transpose(SA)\SB)::SMatrix{n,n} ≈ transpose(A)\B
@test (SA'\SB[:,1])::SVector{n} ≈ A'\B[:,1]
@test (SA'\SB)::SMatrix{n,n} ≈ A'\B

@test_throws DimensionMismatch SA\ones(SVector{n+2,eltyB})
@test_throws DimensionMismatch transpose(SA)\ones(SVector{n+2,eltyB})
@test_throws DimensionMismatch SA'\ones(SVector{n+2,eltyB})

@test_throws LinearAlgebra.SingularException t(zeros(SMatrix{n,n,eltyA}))\ones(SVector{n,eltyB})
@test_throws LinearAlgebra.SingularException t(transpose(zeros(SMatrix{n,n,eltyA})))\ones(SVector{n,eltyB})
@test_throws LinearAlgebra.SingularException t(zeros(SMatrix{n,n,eltyA}))'\ones(SVector{n,eltyB})
if t != UnitUpperTriangular
@test_throws LinearAlgebra.SingularException t(zeros(SMatrix{n,n,eltyA}))\ones(SVector{n,eltyB})
@test_throws LinearAlgebra.SingularException t(transpose(zeros(SMatrix{n,n,eltyA})))\ones(SVector{n,eltyB})
@test_throws LinearAlgebra.SingularException t(zeros(SMatrix{n,n,eltyA}))'\ones(SVector{n,eltyB})
end

@test (SB/SA)::SMatrix{n,n} ≈ B/A
@test (SB/transpose(SA))::SMatrix{n,n} ≈ B/transpose(A)
@test (SB/SA')::SMatrix{n,n} ≈ B/A'
end
end