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Add tril and triu for special matrices #12574

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Aug 12, 2015
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Add tril and triu for special matrices #12574

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bb98832
Added tril/triu methods & tests for bidiags
kshyatt Aug 11, 2015
cdfe7aa
Added triu/tril for tridiagonal matrices
kshyatt Aug 11, 2015
0434772
tril/triu for Upper/LowerTriangular types
kshyatt Aug 11, 2015
5b75f43
fixed triu/tril for diagonals
kshyatt Aug 11, 2015
8692f66
Added triu/tril for symmetric and Hermitian matrices
kshyatt Aug 11, 2015
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39 changes: 38 additions & 1 deletion base/linalg/bidiag.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -109,6 +109,44 @@ ctranspose(M::Bidiagonal) = Bidiagonal(conj(M.dv), conj(M.ev), !M.isupper)
istriu(M::Bidiagonal) = M.isupper || all(M.ev .== 0)
istril(M::Bidiagonal) = !M.isupper || all(M.ev .== 0)

function tril(M::Bidiagonal, k::Integer=0)
n = length(M.dv)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif M.isupper && k < 0
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Maybe this could be:

    elseif k > 0
          return M
    elseif M.isupper
          if k < 0
               return Bidiagonal(zeros(M.dv),zeros(M.ev),true)
          else
               return Bidiagonal(M.dv,zeros(M.ev),true)
          end
    elseif k < -1
       return Bidiagonal(zeros(M.dv),zeros(M.ev),false)
    elseif k == -1
       return Bidiagonal(zeros(M.dv),M.ev,false)
    else
       return M
    end

This avoids the rechecking of M.isupper and passing it to Bidiagonal.

return Bidiagonal(zeros(M.dv),zeros(M.ev),M.isupper)
elseif k < -1
return Bidiagonal(zeros(M.dv),zeros(M.ev),M.isupper)
elseif !M.isupper && k == 0
return M
elseif M.isupper && k == 0
return Bidiagonal(M.dv,zeros(M.ev),M.isupper)
elseif !M.isupper && k == -1
return Bidiagonal(zeros(M.dv),M.ev,M.isupper)
elseif k > 0
return M
end
end

function triu(M::Bidiagonal, k::Integer=0)
n = length(M.dv)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif !M.isupper && k > 0
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Same as above with tril

return Bidiagonal(zeros(M.dv),zeros(M.ev),M.isupper)
elseif k > 1
return Bidiagonal(zeros(M.dv),zeros(M.ev),M.isupper)
elseif M.isupper && k == 0
return M
elseif !M.isupper && k == 0
return Bidiagonal(M.dv,zeros(M.ev),M.isupper)
elseif M.isupper && k == 1
return Bidiagonal(zeros(M.dv),M.ev,M.isupper)
elseif k < 0
return M
end
end

function diag{T}(M::Bidiagonal{T}, n::Integer=0)
if n==0
return M.dv
Expand Down Expand Up @@ -249,4 +287,3 @@ function eigvecs{T}(M::Bidiagonal{T})
Q #Actually Triangular
end
eigfact(M::Bidiagonal) = Eigen(eigvals(M), eigvecs(M))

4 changes: 2 additions & 2 deletions base/linalg/diagonal.jl
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Expand Up @@ -55,8 +55,8 @@ isposdef(D::Diagonal) = all(D.diag .> 0)

factorize(D::Diagonal) = D

tril!(D::Diagonal,i::Integer) = i == 0 ? D : zeros(D)
triu!(D::Diagonal,i::Integer) = i == 0 ? D : zeros(D)
tril(D::Diagonal,i::Integer=0) = i == 0 ? D : zeros(D)
triu(D::Diagonal,i::Integer=0) = i == 0 ? D : zeros(D)

==(Da::Diagonal, Db::Diagonal) = Da.diag == Db.diag
-(A::Diagonal)=Diagonal(-A.diag)
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6 changes: 6 additions & 0 deletions base/linalg/symmetric.jl
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Expand Up @@ -56,6 +56,12 @@ end
ctranspose(A::Hermitian) = A
trace(A::Hermitian) = real(trace(A.data))

#tril/triu
tril(A::Hermitian,k::Integer=0) = tril(A.data,k)
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I don't think these are right. Aren't half the elements in A.data potentially garbage?

julia> A = rand(4,4)
4x4 Array{Float64,2}:
 0.992492  0.88037   0.40931   0.311544
 0.623913  0.93559   0.310044  0.598238
 0.393736  0.051534  0.103072  0.74386
 0.747731  0.684462  0.813315  0.0329609

julia> Symmetric(A)
tril(4x4 Base.LinAlg.Symmetric{Float64,Array{Float64,2}}:
 0.992492  0.88037   0.40931   0.311544
 0.88037   0.93559   0.310044  0.598238
 0.40931   0.310044  0.103072  0.74386
 0.311544  0.598238  0.74386   0.0329609

julia> tril(Symmetric(A))
4x4 Array{Float64,2}:
 0.992492  0.0       0.0       0.0
 0.623913  0.93559   0.0       0.0
 0.393736  0.051534  0.103072  0.0
 0.747731  0.684462  0.813315  0.0329609

julia> full(Symmetric(A))
4x4 Array{Float64,2}:
 0.992492  0.88037   0.40931   0.311544
 0.88037   0.93559   0.310044  0.598238
 0.40931   0.310044  0.103072  0.74386
 0.311544  0.598238  0.74386   0.0329609

julia> tril(full(Symmetric(A)))
4x4 Array{Float64,2}:
 0.992492  0.0       0.0       0.0
 0.88037   0.93559   0.0       0.0
 0.40931   0.310044  0.103072  0.0
 0.311544  0.598238  0.74386   0.0329609

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Wow. K, on it.

The lower triangle is garbage.

triu(A::Hermitian,k::Integer=0) = triu(A.data,k)
tril(A::Symmetric,k::Integer=0) = tril(A.data,k)
triu(A::Symmetric,k::Integer=0) = triu(A.data,k)

## Matvec
A_mul_B!{T<:BlasFloat,S<:StridedMatrix}(y::StridedVector{T}, A::Symmetric{T,S}, x::StridedVector{T}) = BLAS.symv!(A.uplo, one(T), A.data, x, zero(T), y)
A_mul_B!{T<:BlasComplex,S<:StridedMatrix}(y::StridedVector{T}, A::Hermitian{T,S}, x::StridedVector{T}) = BLAS.hemv!(A.uplo, one(T), A.data, x, zero(T), y)
Expand Down
60 changes: 60 additions & 0 deletions base/linalg/triangular.jl
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Expand Up @@ -146,6 +146,66 @@ istril(A::UnitLowerTriangular) = true
istriu(A::UpperTriangular) = true
istriu(A::UnitUpperTriangular) = true

function tril(A::UpperTriangular,k::Integer=0)
n = size(A,1)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k < 0
return UpperTriangular(zeros(A.data))
elseif k == 0
return UpperTriangular(diagm(diag(A)))
else
return UpperTriangular(triu(tril(A.data,k)))
end
end

triu(A::UpperTriangular,k::Integer=0) = UpperTriangular(triu(triu(A.data),k))

function tril(A::UnitUpperTriangular,k::Integer=0)
n = size(A,1)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k < 0
return UnitUpperTriangular(zeros(A.data))
elseif k == 0
return UnitUpperTriangular(eye(A))
else
return UnitUpperTriangular(triu(tril(A.data,k)))
end
end

triu(A::UnitUpperTriangular,k::Integer=0) = UnitUpperTriangular(triu(triu(A.data),k))

function triu(A::LowerTriangular,k::Integer=0)
n = size(A,1)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k > 0
return LowerTriangular(zeros(A.data))
elseif k == 0
return LowerTriangular(diagm(diag(A)))
else
return LowerTriangular(tril(triu(A.data,k)))
end
end

tril(A::LowerTriangular,k::Integer=0) = LowerTriangular(tril(tril(A.data),k))

function triu(A::UnitLowerTriangular,k::Integer=0)
n = size(A,1)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k > 0
return UnitLowerTriangular(zeros(A.data))
elseif k == 0
return UnitLowerTriangular(eye(A))
else
return UnitLowerTriangular(tril(triu(A.data,k)))
end
end

tril(A::UnitLowerTriangular,k::Integer=0) = UnitLowerTriangular(tril(tril(A.data),k))

transpose{T,S}(A::LowerTriangular{T,S}) = UpperTriangular{T, S}(transpose(A.data))
transpose{T,S}(A::UnitLowerTriangular{T,S}) = UnitUpperTriangular{T, S}(transpose(A.data))
transpose{T,S}(A::UpperTriangular{T,S}) = LowerTriangular{T, S}(transpose(A.data))
Expand Down
70 changes: 70 additions & 0 deletions base/linalg/tridiag.jl
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Expand Up @@ -144,6 +144,41 @@ eigmin(A::SymTridiagonal) = eigvals(A, 1:1)[1]
eigvecs(A::SymTridiagonal) = eigfact(A)[:vectors]
eigvecs{T<:BlasFloat,Eigenvalue<:Real}(A::SymTridiagonal{T}, eigvals::Vector{Eigenvalue}) = LAPACK.stein!(A.dv, A.ev, eigvals)

#tril and triu

istriu(M::SymTridiagonal) = all(M.ev .== 0)
istril(M::SymTridiagonal) = all(M.ev .== 0)

function tril(M::SymTridiagonal, k::Integer=0)
n = length(M.dv)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k < -1
return SymTridiagonal(zeros(M.dv),zeros(M.ev))
elseif k == -1
return Tridiagonal(M.ev,zeros(M.dv),zeros(M.ev))
elseif k == 0
return Tridiagonal(M.ev,M.dv,zeros(M.ev))
elseif k >= 1
return M
end
end

function triu(M::SymTridiagonal, k::Integer=0)
n = length(M.dv)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k > 1
return SymTridiagonal(zeros(M.dv),zeros(M.ev))
elseif k == 1
return Tridiagonal(zeros(M.ev),zeros(M.dv),M.ev)
elseif k == 0
return Tridiagonal(zeros(M.ev),M.dv,M.ev)
elseif k <= -1
return M
end
end

###################
# Generic methods #
###################
Expand Down Expand Up @@ -319,6 +354,41 @@ function getindex{T}(A::Tridiagonal{T}, i::Integer, j::Integer)
end
end

#tril and triu

istriu(M::Tridiagonal) = all(M.ev .== 0)
istril(M::Tridiagonal) = all(M.ev .== 0)

function tril(M::Tridiagonal, k::Integer=0)
n = length(M.d)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k < -1
return Tridiagonal(zeros(M.dl),zeros(M.d),zeros(M.du))
elseif k == -1
return Tridiagonal(M.dl,zeros(M.d),zeros(M.du))
elseif k == 0
return Tridiagonal(M.dl,M.d,zeros(M.du))
elseif k >= 1
return M
end
end

function triu(M::Tridiagonal, k::Integer=0)
n = length(M.d)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k > 1
return Tridiagonal(zeros(M.dl),zeros(M.d),zeros(M.du))
elseif k == 1
return Tridiagonal(zeros(M.dl),zeros(M.d),M.du)
elseif k == 0
return Tridiagonal(zeros(M.dl),M.d,M.du)
elseif k <= -1
return M
end
end

###################
# Generic methods #
###################
Expand Down
22 changes: 22 additions & 0 deletions test/linalg/bidiag.jl
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Expand Up @@ -42,6 +42,28 @@ for relty in (Float32, Float64, BigFloat), elty in (relty, Complex{relty})
@test func(func(T)) == T
end

debug && println("triu and tril")
@test tril(Bidiagonal(dv,ev,'U'),-1) == Bidiagonal(zeros(dv),zeros(ev),'U')
@test tril(Bidiagonal(dv,ev,'L'),-1) == Bidiagonal(zeros(dv),ev,'L')
@test tril(Bidiagonal(dv,ev,'U'),-2) == Bidiagonal(zeros(dv),zeros(ev),'U')
@test tril(Bidiagonal(dv,ev,'L'),-2) == Bidiagonal(zeros(dv),zeros(ev),'L')
@test tril(Bidiagonal(dv,ev,'U'),1) == Bidiagonal(dv,ev,'U')
@test tril(Bidiagonal(dv,ev,'L'),1) == Bidiagonal(dv,ev,'L')
@test tril(Bidiagonal(dv,ev,'U')) == Bidiagonal(dv,zeros(ev),'U')
@test tril(Bidiagonal(dv,ev,'L')) == Bidiagonal(dv,ev,'L')
@test_throws ArgumentError tril(Bidiagonal(dv,ev,'U'),n+1)

@test triu(Bidiagonal(dv,ev,'L'),1) == Bidiagonal(zeros(dv),zeros(ev),'L')
@test triu(Bidiagonal(dv,ev,'U'),1) == Bidiagonal(zeros(dv),ev,'U')
@test triu(Bidiagonal(dv,ev,'U'),2) == Bidiagonal(zeros(dv),zeros(ev),'U')
@test triu(Bidiagonal(dv,ev,'L'),2) == Bidiagonal(zeros(dv),zeros(ev),'L')
@test triu(Bidiagonal(dv,ev,'U'),-1) == Bidiagonal(dv,ev,'U')
@test triu(Bidiagonal(dv,ev,'L'),-1) == Bidiagonal(dv,ev,'L')
@test triu(Bidiagonal(dv,ev,'L')) == Bidiagonal(dv,zeros(ev),'L')
@test triu(Bidiagonal(dv,ev,'U')) == Bidiagonal(dv,ev,'U')
@test_throws ArgumentError triu(Bidiagonal(dv,ev,'U'),n+1)


debug && println("Linear solver")
Tfull = full(T)
condT = cond(map(Complex128,Tfull))
Expand Down
8 changes: 4 additions & 4 deletions test/linalg/diagonal.jl
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Expand Up @@ -84,10 +84,10 @@ for relty in (Float32, Float64, BigFloat), elty in (relty, Complex{relty})
@test_approx_eq D/D2 Diagonal(D.diag./D2.diag)

# test triu/tril
@test triu!(copy(D),1) == zeros(D)
@test triu!(copy(D),0) == D
@test tril!(copy(D),1) == zeros(D)
@test tril!(copy(D),0) == D
@test triu(D,1) == zeros(D)
@test triu(D,0) == D
@test tril(D,1) == zeros(D)
@test tril(D,0) == D

# factorize
@test factorize(D) == D
Expand Down
6 changes: 6 additions & 0 deletions test/linalg/symmetric.jl
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Expand Up @@ -59,6 +59,12 @@ let n=10
@test ctranspose(Symmetric(asym)) == Symmetric(conj(asym))
@test ctranspose(Hermitian(asym)) == asym

#tril/triu
@test triu(Symmetric(asym),1) == triu(asym,1)
@test tril(Symmetric(asym),1) == tril(asym,1)
@test triu(Hermitian(asym),1) == triu(asym,1)
@test tril(Hermitian(asym),1) == tril(asym,1)

eltya == BigFloat && continue # Revisit when implemented in julia
d, v = eig(asym)
@test_approx_eq asym*v[:,1] d[1]*v[:,1]
Expand Down
19 changes: 19 additions & 0 deletions test/linalg/triangular.jl
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Expand Up @@ -85,6 +85,25 @@ for elty1 in (Float32, Float64, Complex64, Complex128, BigFloat, Int)
@test !istril(A1)
end

#tril/triu
if uplo1 == :L
@test tril(A1) == A1
@test tril(A1,-1) == t1(tril(full(A1),-1))
@test tril(A1,1) == t1(tril(tril(full(A1),1)))
@test_throws ArgumentError tril(A1,n+1)
@test triu(A1) == t1(diagm(diag(A1)))
@test triu(A1,-1) == t1(tril(triu(A1.data,-1)))
@test_throws ArgumentError triu(A1,n+1)
else
@test triu(A1) == A1
@test triu(A1,1) == t1(triu(full(A1),1))
@test triu(A1,-1) == t1(triu(triu(full(A1),-1)))
@test_throws ArgumentError triu(A1,n+1)
@test tril(A1) == t1(diagm(diag(A1)))
@test tril(A1,1) == t1(triu(tril(A1.data,1)))
@test_throws ArgumentError tril(A1,n+1)
end

# factorize
@test factorize(A1) == A1

Expand Down
19 changes: 19 additions & 0 deletions test/linalg/tridiag.jl
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Expand Up @@ -115,6 +115,25 @@ for elty in (Float32, Float64, Complex64, Complex128, Int)

# issue #1490
@test_approx_eq_eps det(ones(elty, 3,3)) zero(elty) 3*eps(real(one(elty)))

#tril/triu
@test_throws ArgumentError tril(SymTridiagonal(d,dl),n+1)
@test_throws ArgumentError tril(Tridiagonal(dl,d,du),n+1)
@test tril(SymTridiagonal(d,dl)) == Tridiagonal(dl,d,zeros(dl))
@test tril(SymTridiagonal(d,dl),1) == Tridiagonal(dl,d,dl)
@test tril(SymTridiagonal(d,dl),-1) == Tridiagonal(dl,zeros(d),zeros(dl))
@test tril(Tridiagonal(dl,d,du)) == Tridiagonal(dl,d,zeros(du))
@test tril(Tridiagonal(dl,d,du),1) == Tridiagonal(dl,d,du)
@test tril(Tridiagonal(dl,d,du),-1) == Tridiagonal(dl,zeros(d),zeros(du))

@test_throws ArgumentError triu(SymTridiagonal(d,dl),n+1)
@test_throws ArgumentError triu(Tridiagonal(dl,d,du),n+1)
@test triu(SymTridiagonal(d,dl)) == Tridiagonal(zeros(dl),d,dl)
@test triu(SymTridiagonal(d,dl),-1) == Tridiagonal(dl,d,dl)
@test triu(SymTridiagonal(d,dl),1) == Tridiagonal(zeros(dl),zeros(d),dl)
@test triu(Tridiagonal(dl,d,du)) == Tridiagonal(zeros(dl),d,du)
@test triu(Tridiagonal(dl,d,du),-1) == Tridiagonal(dl,d,du)
@test triu(Tridiagonal(dl,d,du),1) == Tridiagonal(zeros(dl),zeros(d),du)
end
end

Expand Down