Merge pull request #9779 from JuliaLang/anj/triangle

Redesign of triangular matrix types

Author: andreasnoack

NEWS.md

@@ -154,6 +154,8 @@ Library improvements
 
   * copy(DArray) will now make a copy of the DArray ([#9745])
 
+  * Split `Triangular` type into `UpperTriangular`, `LowerTriangular`, `UnitUpperTriagular` and `UnitLowerTriangular` ([#9779])
+
 Deprecated or removed
 ---------------------
 

base/exports.jl

@@ -62,6 +62,7 @@ export
     IOBuffer,
     IOStream,
     LocalProcess,
+    LowerTriangular,
     MathConst,
     Matrix,
     MergeSort,
@@ -108,9 +109,9 @@ export
     SymTridiagonal,
     Timer,
     TmStruct,
-    Triangular,
     Tridiagonal,
     UnitRange,
+    UpperTriangular,
     UTF16String,
     UTF32String,
     Val,

base/linalg.jl

@@ -30,7 +30,8 @@ export
     SVD,
     Hermitian,
     Symmetric,
-    Triangular,
+    LowerTriangular,
+    UpperTriangular,
     Diagonal,
     UniformScaling,
 

base/linalg/bidiag.jl

@@ -114,24 +114,23 @@ end
 /(A::Bidiagonal, B::Number) = Bidiagonal(A.dv/B, A.ev/B, A.isupper)
 ==(A::Bidiagonal, B::Bidiagonal) = (A.dv==B.dv) && (A.ev==B.ev) && (A.isupper==B.isupper)
 
-SpecialMatrix = Union(Diagonal, Bidiagonal, SymTridiagonal, Tridiagonal, Triangular)
+SpecialMatrix = Union(Diagonal, Bidiagonal, SymTridiagonal, Tridiagonal, TriangularUnion)
 *(A::SpecialMatrix, B::SpecialMatrix)=full(A)*full(B)
 
 #Generic multiplication
 for func in (:*, :Ac_mul_B, :A_mul_Bc, :/, :A_rdiv_Bc)
     @eval begin
         ($func){T}(A::Bidiagonal{T}, B::AbstractVector{T}) = ($func)(full(A), B)
-        #($func){T}(A::AbstractArray{T}, B::Triangular{T}) = ($func)(full(A), B)
     end
 end
 
 
 #Linear solvers
-A_ldiv_B!(A::Union(Bidiagonal, Triangular), b::AbstractVector) = naivesub!(A, b)
-At_ldiv_B!(A::Union(Bidiagonal, Triangular), b::AbstractVector) = naivesub!(transpose(A), b)
-Ac_ldiv_B!(A::Union(Bidiagonal, Triangular), b::AbstractVector) = naivesub!(ctranspose(A), b)
+A_ldiv_B!(A::Union(Bidiagonal, TriangularUnion), b::AbstractVector) = naivesub!(A, b)
+At_ldiv_B!(A::Union(Bidiagonal, TriangularUnion), b::AbstractVector) = naivesub!(transpose(A), b)
+Ac_ldiv_B!(A::Union(Bidiagonal, TriangularUnion), b::AbstractVector) = naivesub!(ctranspose(A), b)
 for func in (:A_ldiv_B!, :Ac_ldiv_B!, :At_ldiv_B!) @eval begin
-    function ($func)(A::Union(Bidiagonal, Triangular), B::AbstractMatrix)
+    function ($func)(A::Union(Bidiagonal, TriangularUnion), B::AbstractMatrix)
         tmp = similar(B[:,1])
         n = size(B, 1)
         for i = 1:size(B,2)
@@ -143,7 +142,7 @@ for func in (:A_ldiv_B!, :Ac_ldiv_B!, :At_ldiv_B!) @eval begin
     end
 end end
 for func in (:A_ldiv_Bt!, :Ac_ldiv_Bt!, :At_ldiv_Bt!) @eval begin
-    function ($func)(A::Union(Bidiagonal, Triangular), B::AbstractMatrix)
+    function ($func)(A::Union(Bidiagonal, TriangularUnion), B::AbstractMatrix)
         tmp = similar(B[:, 2])
         m, n = size(B)
         nm = n*m

base/linalg/cholesky.jl

@@ -18,12 +18,35 @@ immutable CholeskyPivoted{T,S<:AbstractMatrix} <: Factorization{T}
 end
 CholeskyPivoted{T}(UL::AbstractMatrix{T}, uplo::Char, piv::Vector{BlasInt}, rank::BlasInt, tol::Real, info::BlasInt) = CholeskyPivoted{T,typeof(UL)}(UL, uplo, piv, rank, tol, info)
 
-function chol!{T<:BlasFloat}(A::StridedMatrix{T}, uplo::Symbol=:U)
+function chol!{T<:BlasFloat}(A::StridedMatrix{T})
+    C, info = LAPACK.potrf!('U', A)
+    return @assertposdef UpperTriangular(C) info
+end
+function chol!{T<:BlasFloat}(A::StridedMatrix{T}, uplo::Symbol)
     C, info = LAPACK.potrf!(char_uplo(uplo), A)
-    return @assertposdef Triangular{eltype(C),typeof(C),uplo,false}(C) info
+    return @assertposdef uplo == :U ? UpperTriangular(C) : LowerTriangular(C) info
 end
 
-function chol!{T}(A::AbstractMatrix{T}, uplo::Symbol=:U)
+function chol!{T}(A::AbstractMatrix{T})
+    n = chksquare(A)
+    @inbounds begin
+        for k = 1:n
+            for i = 1:k - 1
+                A[k,k] -= A[i,k]'A[i,k]
+            end
+            A[k,k] = chol!(A[k,k], uplo)
+            AkkInv = inv(A[k,k])
+            for j = k + 1:n
+                for i = 1:k - 1
+                    A[k,j] -= A[i,k]'A[i,j]
+                end
+                A[k,j] = A[k,k]'\A[k,j]
+            end
+        end
+    end
+    return UpperTriangular(A)
+end
+function chol!{T}(A::AbstractMatrix{T}, uplo::Symbol)
     n = chksquare(A)
     @inbounds begin
         if uplo == :L
@@ -58,7 +81,7 @@ function chol!{T}(A::AbstractMatrix{T}, uplo::Symbol=:U)
             throw(ArgumentError("uplo must be either :U or :L but was $(uplo)"))
         end
     end
-    return Triangular(A, uplo, false)
+    return uplo == :U ? UpperTriangular(A) : LowerTriangular(A)
 end
 
 function cholfact!{T<:BlasFloat}(A::StridedMatrix{T}, uplo::Symbol=:U; pivot=false, tol=0.0)
@@ -118,14 +141,14 @@ size(C::Union(Cholesky, CholeskyPivoted)) = size(C.UL)
 size(C::Union(Cholesky, CholeskyPivoted), d::Integer) = size(C.UL,d)
 
 function getindex{T,S,UpLo}(C::Cholesky{T,S,UpLo}, d::Symbol)
-    d == :U && return Triangular(UpLo == d ? C.UL : C.UL',:U)
-    d == :L && return Triangular(UpLo == d ? C.UL : C.UL',:L)
-    d == :UL && return Triangular(C.UL, UpLo)
+    d == :U && return UpperTriangular(UpLo == d ? C.UL : C.UL')
+    d == :L && return LowerTriangular(UpLo == d ? C.UL : C.UL')
+    d == :UL && return UpLo == :U ? UpperTriangular(C.UL) : LowerTriangular(C.UL)
     throw(KeyError(d))
 end
 function getindex{T<:BlasFloat}(C::CholeskyPivoted{T}, d::Symbol)
-    d == :U && return Triangular(symbol(C.uplo) == d ? C.UL : C.UL', :U)
-    d == :L && return Triangular(symbol(C.uplo) == d ? C.UL : C.UL', :L)
+    d == :U && return UpperTriangular(symbol(C.uplo) == d ? C.UL : C.UL')
+    d == :L && return LowerTriangular(symbol(C.uplo) == d ? C.UL : C.UL')
     d == :p && return C.piv
     if d == :P
         n = size(C, 1)
@@ -142,8 +165,8 @@ show{T,S<:AbstractMatrix,UpLo}(io::IO, C::Cholesky{T,S,UpLo}) = (println("$(type
 
 A_ldiv_B!{T<:BlasFloat,S<:AbstractMatrix}(C::Cholesky{T,S,:U}, B::StridedVecOrMat{T}) = LAPACK.potrs!('U', C.UL, B)
 A_ldiv_B!{T<:BlasFloat,S<:AbstractMatrix}(C::Cholesky{T,S,:L}, B::StridedVecOrMat{T}) = LAPACK.potrs!('L', C.UL, B)
-A_ldiv_B!{T,S<:AbstractMatrix}(C::Cholesky{T,S,:L}, B::StridedVecOrMat) = Ac_ldiv_B!(Triangular(C.UL, :L, false), A_ldiv_B!(Triangular(C.UL, :L, false), B))
-A_ldiv_B!{T,S<:AbstractMatrix}(C::Cholesky{T,S,:U}, B::StridedVecOrMat) = A_ldiv_B!(Triangular(C.UL, :U, false), Ac_ldiv_B!(Triangular(C.UL, :U, false), B))
+A_ldiv_B!{T,S<:AbstractMatrix}(C::Cholesky{T,S,:L}, B::StridedVecOrMat) = Ac_ldiv_B!(LowerTriangular(C.UL), A_ldiv_B!(LowerTriangular(C.UL), B))
+A_ldiv_B!{T,S<:AbstractMatrix}(C::Cholesky{T,S,:U}, B::StridedVecOrMat) = A_ldiv_B!(UpperTriangular(C.UL), Ac_ldiv_B!(UpperTriangular(C.UL), B))
 
 function A_ldiv_B!{T<:BlasFloat}(C::CholeskyPivoted{T}, B::StridedVector{T})
     chkfullrank(C)
@@ -161,8 +184,8 @@ function A_ldiv_B!{T<:BlasFloat}(C::CholeskyPivoted{T}, B::StridedMatrix{T})
     end
     B
 end
-A_ldiv_B!(C::CholeskyPivoted, B::StridedVector) = C.uplo=='L' ? Ac_ldiv_B!(Triangular(C.UL, symbol(C.uplo), false), A_ldiv_B!(Triangular(C.UL, symbol(C.uplo), false), B[C.piv]))[invperm(C.piv)] : A_ldiv_B!(Triangular(C.UL, symbol(C.uplo), false), Ac_ldiv_B!(Triangular(C.UL, symbol(C.uplo), false), B[C.piv]))[invperm(C.piv)]
-A_ldiv_B!(C::CholeskyPivoted, B::StridedMatrix) = C.uplo=='L' ? Ac_ldiv_B!(Triangular(C.UL, symbol(C.uplo), false), A_ldiv_B!(Triangular(C.UL, symbol(C.uplo), false), B[C.piv,:]))[invperm(C.piv),:] : A_ldiv_B!(Triangular(C.UL, symbol(C.uplo), false), Ac_ldiv_B!(Triangular(C.UL, symbol(C.uplo), false), B[C.piv,:]))[invperm(C.piv),:]
+A_ldiv_B!(C::CholeskyPivoted, B::StridedVector) = C.uplo=='L' ? Ac_ldiv_B!(LowerTriangular(C.UL), A_ldiv_B!(LowerTriangular(C.UL), B[C.piv]))[invperm(C.piv)] : A_ldiv_B!(UpperTriangular(C.UL), Ac_ldiv_B!(UpperTriangular(C.UL), B[C.piv]))[invperm(C.piv)]
+A_ldiv_B!(C::CholeskyPivoted, B::StridedMatrix) = C.uplo=='L' ? Ac_ldiv_B!(LowerTriangular(C.UL), A_ldiv_B!(LowerTriangular(C.UL), B[C.piv,:]))[invperm(C.piv),:] : A_ldiv_B!(UpperTriangular(C.UL), Ac_ldiv_B!(UpperTriangular(C.UL), B[C.piv,:]))[invperm(C.piv),:]
 
 function det{T,S,UpLo}(C::Cholesky{T,S,UpLo})
     dd = one(T)

base/linalg/dense.jl

@@ -307,15 +307,15 @@ function sqrtm{T<:Real}(A::StridedMatrix{T})
     issym(A) && return sqrtm(Symmetric(A))
     n = chksquare(A)
     SchurF = schurfact(complex(A))
-    R = full(sqrtm(Triangular(SchurF[:T], :U, false)))
+    R = full(sqrtm(UpperTriangular(SchurF[:T])))
     retmat = SchurF[:vectors]*R*SchurF[:vectors]'
     all(imag(retmat) .== 0) ? real(retmat) : retmat
 end
 function sqrtm{T<:Complex}(A::StridedMatrix{T})
     ishermitian(A) && return sqrtm(Hermitian(A))
     n = chksquare(A)
     SchurF = schurfact(A)
-    R = full(sqrtm(Triangular(SchurF[:T], :U, false)))
+    R = full(sqrtm(UpperTriangular(SchurF[:T])))
     SchurF[:vectors]*R*SchurF[:vectors]'
 end
 sqrtm(a::Number) = (b = sqrt(complex(a)); imag(b) == 0 ? real(b) : b)
@@ -325,9 +325,9 @@ function inv{S}(A::StridedMatrix{S})
     T = typeof(one(S)/one(S))
     Ac = convert(AbstractMatrix{T}, A)
     if istriu(Ac)
-        Ai = inv(Triangular(A, :U, false))
+        Ai = inv(UpperTriangular(A))
     elseif istril(Ac)
-        Ai = inv(Triangular(A, :L, false))
+        Ai = inv(LowerTriangular(A))
     else
         Ai = inv(lufact(Ac))
     end
@@ -377,7 +377,7 @@ function factorize{T}(A::Matrix{T})
                 if utri1
                     return Bidiagonal(diag(A), diag(A, -1), false)
                 end
-                return Triangular(A, :L)
+                return LowerTriangular(A)
             end
             if utri
                 return Bidiagonal(diag(A), diag(A, 1), true)
@@ -392,7 +392,7 @@ function factorize{T}(A::Matrix{T})
             end
         end
         if utri
-            return Triangular(A, :U)
+            return UpperTriangular(A)
         end
         if herm
             try
@@ -413,9 +413,9 @@ function (\)(A::StridedMatrix, B::StridedVecOrMat)
     m, n = size(A)
     if m == n
         if istril(A)
-            return istriu(A) ? \(Diagonal(A),B) : \(Triangular(A, :L),B)
+            return istriu(A) ? \(Diagonal(A),B) : \(LowerTriangular(A),B)
         end
-        istriu(A) && return \(Triangular(A, :U),B)
+        istriu(A) && return \(UpperTriangular(A),B)
         return \(lufact(A),B)
     end
     return qrfact(A,pivot=eltype(A)<:BlasFloat)\B

base/linalg/diagonal.jl

@@ -8,7 +8,8 @@ Diagonal(A::Matrix) = Diagonal(diag(A))
 convert{T}(::Type{Diagonal{T}}, D::Diagonal{T}) = D
 convert{T}(::Type{Diagonal{T}}, D::Diagonal) = Diagonal{T}(convert(Vector{T}, D.diag))
 convert{T}(::Type{AbstractMatrix{T}}, D::Diagonal) = convert(Diagonal{T}, D)
-convert{T}(::Type{Triangular}, A::Diagonal{T}) = Triangular{T, Diagonal{T}, :U, false}(A)
+convert{T}(::Type{UpperTriangular}, A::Diagonal{T}) = UpperTriangular(A)
+convert{T}(::Type{LowerTriangular}, A::Diagonal{T}) = LowerTriangular(A)
 
 function similar{T}(D::Diagonal, ::Type{T}, d::(Int,Int))
     d[1] == d[2] || throw(ArgumentError("Diagonal matrix must be square"))

base/linalg/factorization.jl

@@ -274,7 +274,7 @@ function A_mul_Bc!{T}(A::AbstractMatrix{T},Q::QRPackedQ{T})
     end
     A
 end
-A_mul_Bc(A::Triangular, B::Union(QRCompactWYQ,QRPackedQ)) = A_mul_Bc(full(A), B)
+A_mul_Bc(A::TriangularUnion, B::Union(QRCompactWYQ,QRPackedQ)) = A_mul_Bc(full(A), B)
 function A_mul_Bc{TA,TB}(A::AbstractArray{TA}, B::Union(QRCompactWYQ{TB},QRPackedQ{TB}))
     TAB = promote_type(TA,TB)
     A_mul_Bc!(size(A,2)==size(B.factors,1) ? (TA == TAB ? copy(A) : convert(AbstractMatrix{TAB}, A)) :
@@ -284,8 +284,8 @@ function A_mul_Bc{TA,TB}(A::AbstractArray{TA}, B::Union(QRCompactWYQ{TB},QRPacke
               convert(AbstractMatrix{TAB}, B))
 end
 
-A_ldiv_B!{T<:BlasFloat}(A::QRCompactWY{T}, b::StridedVector{T}) = (A_ldiv_B!(Triangular(A[:R], :U), sub(Ac_mul_B!(A[:Q], b), 1:size(A, 2))); b)
-A_ldiv_B!{T<:BlasFloat}(A::QRCompactWY{T}, B::StridedMatrix{T}) = (A_ldiv_B!(Triangular(A[:R], :U), sub(Ac_mul_B!(A[:Q], B), 1:size(A, 2), 1:size(B, 2))); B)
+A_ldiv_B!{T<:BlasFloat}(A::QRCompactWY{T}, b::StridedVector{T}) = (A_ldiv_B!(UpperTriangular(A[:R]), sub(Ac_mul_B!(A[:Q], b), 1:size(A, 2))); b)
+A_ldiv_B!{T<:BlasFloat}(A::QRCompactWY{T}, B::StridedMatrix{T}) = (A_ldiv_B!(UpperTriangular(A[:R]), sub(Ac_mul_B!(A[:Q], B), 1:size(A, 2), 1:size(B, 2))); B)
 
 # Julia implementation similarly to xgelsy
 function A_ldiv_B!{T<:BlasFloat}(A::QRPivoted{T}, B::StridedMatrix{T}, rcond::Real)
@@ -313,7 +313,7 @@ function A_ldiv_B!{T<:BlasFloat}(A::QRPivoted{T}, B::StridedMatrix{T}, rcond::Re
         # if cond(r[1:rnk, 1:rnk])*rcond < 1 break end
     end
     C, τ = LAPACK.tzrzf!(A.factors[1:rnk,:])
-    A_ldiv_B!(Triangular(C[1:rnk,1:rnk],:U),sub(Ac_mul_B!(getq(A),sub(B, 1:mA, 1:nrhs)),1:rnk,1:nrhs))
+    A_ldiv_B!(UpperTriangular(C[1:rnk,1:rnk]),sub(Ac_mul_B!(getq(A),sub(B, 1:mA, 1:nrhs)),1:rnk,1:nrhs))
     B[rnk+1:end,:] = zero(T)
     LAPACK.ormrz!('L', iseltype(B, Complex) ? 'C' : 'T', C, τ, sub(B,1:nA,1:nrhs))
     B[1:nA,:] = sub(B, 1:nA, :)[invperm(A[:p]::Vector{BlasInt}),:]

base/linalg/generic.jl

@@ -1,5 +1,9 @@
 ## linalg.jl: Some generic Linear Algebra definitions
 
+# Fall back arithmetic
++(A::AbstractMatrix, B::AbstractMatrix) = full(A) + full(B)
+-(A::AbstractMatrix, B::AbstractMatrix) = full(A) - full(B)
+
 scale(X::AbstractArray, s::Number) = scale!(copy(X), s)
 scale(s::Number, X::AbstractArray) = scale!(copy(X), s)
 
@@ -424,7 +428,7 @@ function elementaryRightTrapezoid!(A::AbstractMatrix, row::Integer)
 end
 
 function det(A::AbstractMatrix)
-    (istriu(A) || istril(A)) && return det(Triangular(A, :U, false))
+    (istriu(A) || istril(A)) && return det(UpperTriangular(A))
     return det(lufact(A))
 end
 det(x::Number) = x

base/linalg/lu.jl

@@ -114,8 +114,8 @@ function getindex{T,S<:StridedMatrix}(A::LU{T,S}, d::Symbol)
 end
 
 A_ldiv_B!{T<:BlasFloat, S<:StridedMatrix}(A::LU{T, S}, B::StridedVecOrMat{T}) = @assertnonsingular LAPACK.getrs!('N', A.factors, A.ipiv, B) A.info
-A_ldiv_B!{T,S<:StridedMatrix}(A::LU{T,S}, b::StridedVector) = A_ldiv_B!(Triangular(A.factors, :U, false), A_ldiv_B!(Triangular(A.factors, :L, true), b[ipiv2perm(A.ipiv, length(b))]))
-A_ldiv_B!{T,S<:StridedMatrix}(A::LU{T,S}, B::StridedMatrix) = A_ldiv_B!(Triangular(A.factors, :U, false), A_ldiv_B!(Triangular(A.factors, :L, true), B[ipiv2perm(A.ipiv, size(B, 1)),:]))
+A_ldiv_B!{T,S<:StridedMatrix}(A::LU{T,S}, b::StridedVector) = A_ldiv_B!(UpperTriangular(A.factors), A_ldiv_B!(UnitLowerTriangular(A.factors), b[ipiv2perm(A.ipiv, length(b))]))
+A_ldiv_B!{T,S<:StridedMatrix}(A::LU{T,S}, B::StridedMatrix) = A_ldiv_B!(UpperTriangular(A.factors), A_ldiv_B!(UnitLowerTriangular(A.factors), B[ipiv2perm(A.ipiv, size(B, 1)),:]))
 At_ldiv_B{T<:BlasFloat,S<:StridedMatrix}(A::LU{T,S}, B::StridedVecOrMat{T}) = @assertnonsingular LAPACK.getrs!('T', A.factors, A.ipiv, copy(B)) A.info
 Ac_ldiv_B{T<:BlasComplex,S<:StridedMatrix}(A::LU{T,S}, B::StridedVecOrMat{T}) = @assertnonsingular LAPACK.getrs!('C', A.factors, A.ipiv, copy(B)) A.info
 At_ldiv_Bt{T<:BlasFloat,S<:StridedMatrix}(A::LU{T,S}, B::StridedVecOrMat{T}) = @assertnonsingular LAPACK.getrs!('T', A.factors, A.ipiv, transpose(B)) A.info

base/linalg/sparse.jl

@@ -126,7 +126,7 @@ end
 *{TvA,TiA}(X::BitArray{2}, A::SparseMatrixCSC{TvA,TiA}) = invoke(*, (AbstractMatrix, SparseMatrixCSC), X, A)
 # TODO: Tridiagonal * Sparse should be implemented more efficiently
 *{TX,TvA,TiA}(X::Tridiagonal{TX}, A::SparseMatrixCSC{TvA,TiA}) = invoke(*, (Tridiagonal, AbstractMatrix), X, A)
-*{TvA,TiA}(X::Triangular, A::SparseMatrixCSC{TvA,TiA}) = full(X)*A
+*{TvA,TiA}(X::TriangularUnion, A::SparseMatrixCSC{TvA,TiA}) = full(X)*A
 function *{TX,TvA,TiA}(X::AbstractMatrix{TX}, A::SparseMatrixCSC{TvA,TiA})
     mX, nX = size(X)
     nX == A.m || throw(DimensionMismatch())

base/linalg/special.jl

@@ -4,8 +4,12 @@
 convert{T}(::Type{Bidiagonal}, A::Diagonal{T})=Bidiagonal(A.diag, zeros(T, size(A.diag,1)-1), true)
 convert{T}(::Type{SymTridiagonal}, A::Diagonal{T})=SymTridiagonal(A.diag, zeros(T, size(A.diag,1)-1))
 convert{T}(::Type{Tridiagonal}, A::Diagonal{T})=Tridiagonal(zeros(T, size(A.diag,1)-1), A.diag, zeros(T, size(A.diag,1)-1))
-convert(::Type{Triangular}, A::Diagonal) = Triangular(full(A), :L)
-convert(::Type{Triangular}, A::Bidiagonal) = Triangular(full(A), A.isupper ? :U : :L)
+convert(::Type{UpperTriangular}, A::Diagonal) = UpperTriangular(full(A), :L)
+convert(::Type{UnitUpperTriangular}, A::Diagonal) = UnitUpperTriangular(full(A), :L)
+convert(::Type{LowerTriangular}, A::Diagonal) = LowerTriangular(full(A), :L)
+convert(::Type{UnitLowerTriangular}, A::Diagonal) = UnitLowerTriangular(full(A), :L)
+convert(::Type{LowerTriangular}, A::Bidiagonal) = !A.isupper ? LowerTriangular(full(A)) : throw(ArgumentError("Bidiagonal matrix must have lower off diagonal to be converted to LowerTriangular"))
+convert(::Type{UpperTriangular}, A::Bidiagonal) = A.isupper ? UpperTriangular(full(A)) : throw(ArgumentError("Bidiagonal matrix must have upper off diagonal to be converted to UpperTriangular"))
 convert(::Type{Matrix}, D::Diagonal) = diagm(D.diag)
 
 function convert(::Type{Diagonal}, A::Union(Bidiagonal, SymTridiagonal))
@@ -42,12 +46,12 @@ function convert(::Type{SymTridiagonal}, A::Tridiagonal)
     SymTridiagonal(A.d, A.dl)
 end
 
-function convert(::Type{Diagonal}, A::Triangular)
+function convert(::Type{Diagonal}, A::TriangularUnion)
     full(A) == diagm(diag(A)) || throw(ArgumentError("Matrix cannot be represented as Diagonal"))
     Diagonal(diag(A))
 end
 
-function convert(::Type{Bidiagonal}, A::Triangular)
+function convert(::Type{Bidiagonal}, A::TriangularUnion)
     fA = full(A)
     if fA == diagm(diag(A)) + diagm(diag(fA, 1), 1)
         return Bidiagonal(diag(A), diag(fA,1), true)
@@ -58,9 +62,9 @@ function convert(::Type{Bidiagonal}, A::Triangular)
     end
 end
 
-convert(::Type{SymTridiagonal}, A::Triangular) = convert(SymTridiagonal, convert(Tridiagonal, A))
+convert(::Type{SymTridiagonal}, A::TriangularUnion) = convert(SymTridiagonal, convert(Tridiagonal, A))
 
-function convert(::Type{Tridiagonal}, A::Triangular)
+function convert(::Type{Tridiagonal}, A::TriangularUnion)
     fA = full(A)
     if fA == diagm(diag(A)) + diagm(diag(fA, 1), 1) + diagm(diag(fA, -1), -1)
         return Tridiagonal(diag(fA, -1), diag(A), diag(fA,1))
@@ -82,7 +86,7 @@ macro commutative(myexpr)
 end
 
 for op in (:+, :-)
-    SpecialMatrices = [:Diagonal, :Bidiagonal, :Tridiagonal, :Triangular, :Matrix]
+    SpecialMatrices = [:Diagonal, :Bidiagonal, :Tridiagonal, :Matrix]
     for (idx, matrixtype1) in enumerate(SpecialMatrices) #matrixtype1 is the sparser matrix type
         for matrixtype2 in SpecialMatrices[idx+1:end] #matrixtype2 is the denser matrix type
             @eval begin #TODO quite a few of these conversions are NOT defined...
@@ -93,7 +97,7 @@ for op in (:+, :-)
     end
 
     for  matrixtype1 in (:SymTridiagonal,)                      #matrixtype1 is the sparser matrix type
-        for matrixtype2 in (:Tridiagonal, :Triangular, :Matrix) #matrixtype2 is the denser matrix type
+        for matrixtype2 in (:Tridiagonal, :Matrix) #matrixtype2 is the denser matrix type
             @eval begin
                 ($op)(A::($matrixtype1), B::($matrixtype2)) = ($op)(convert(($matrixtype2), A), B)
                 ($op)(A::($matrixtype2), B::($matrixtype1)) = ($op)(A, convert(($matrixtype2), B))
@@ -111,7 +115,7 @@ for op in (:+, :-)
     end
 end
 
-A_mul_Bc!(A::Triangular, B::QRCompactWYQ) = A_mul_Bc!(full!(A),B)
-A_mul_Bc!(A::Triangular, B::QRPackedQ) = A_mul_Bc!(full!(A),B)
-A_mul_Bc(A::Triangular, B::Union(QRCompactWYQ,QRPackedQ)) = A_mul_Bc(full(A), B)
+A_mul_Bc!(A::TriangularUnion, B::QRCompactWYQ) = A_mul_Bc!(full!(A),B)
+A_mul_Bc!(A::TriangularUnion, B::QRPackedQ) = A_mul_Bc!(full!(A),B)
+A_mul_Bc(A::TriangularUnion, B::Union(QRCompactWYQ,QRPackedQ)) = A_mul_Bc(full(A), B)