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RFC:Add generic Cholesky decomposition and make Cholesky parametric on matrix type #7236

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Aug 14, 2014
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RFC:Add generic Cholesky decomposition and make Cholesky parametric on matrix type #7236

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1 change: 1 addition & 0 deletions base/linalg.jl
View file
Original file line number Diff line number Diff line change
@@ -198,6 +198,7 @@ include("linalg/dense.jl")
include("linalg/tridiag.jl")
include("linalg/triangular.jl")
include("linalg/factorization.jl")
include("linalg/cholesky.jl")
include("linalg/lu.jl")

include("linalg/bunchkaufman.jl")
181 changes: 181 additions & 0 deletions base/linalg/cholesky.jl
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Original file line number Diff line number Diff line change
@@ -0,0 +1,181 @@
##########################
# Cholesky Factorization #
##########################
immutable Cholesky{T,S<:AbstractMatrix{T},UpLo} <: Factorization{T}
UL::S
end
immutable CholeskyPivoted{T} <: Factorization{T}
UL::Matrix{T}
uplo::Char
piv::Vector{BlasInt}
rank::BlasInt
tol::Real
info::BlasInt
end

function chol!{T<:BlasFloat}(A::StridedMatrix{T}, uplo::Symbol=:U)
C, info = LAPACK.potrf!(string(uplo)[1], A)
return @assertposdef Triangular(C, uplo, false) info
end

function chol!{T}(A::AbstractMatrix{T}, uplo::Symbol=:U)
n = chksquare(A)
@inbounds begin
if uplo == :L
for k = 1:n
for i = 1:k - 1
A[k,k] -= A[k,i]*A[k,i]'
end
A[k,k] = chol!(A[k,k], uplo)
AkkInv = inv(A[k,k]')
for j = 1:k
for i = k + 1:n
j == 1 && (A[i,k] = A[i,k]*AkkInv)
j < k && (A[i,k] -= A[i,j]*A[k,j]'*AkkInv)
end
end
end
elseif uplo == :U
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
else
throw(ArgumentError("uplo must be either :U or :L but was $(uplo)"))
end
end
return Triangular(A, uplo, false)
end

function cholfact!{T<:BlasFloat}(A::StridedMatrix{T}, uplo::Symbol=:U; pivot=false, tol=0.0)
uplochar = string(uplo)[1]
if pivot
A, piv, rank, info = LAPACK.pstrf!(uplochar, A, tol)
return CholeskyPivoted{T}(A, uplochar, piv, rank, tol, info)
end
return Cholesky{T,typeof(A),uplo}(chol!(A, uplo).data)
end
cholfact!(A::AbstractMatrix, uplo::Symbol=:U) = Cholesky{eltype(A),typeof(A),uplo}(chol!(A, uplo).data)

cholfact{T<:BlasFloat}(A::StridedMatrix{T}, uplo::Symbol=:U; pivot=false, tol=0.0) = cholfact!(copy(A), uplo, pivot=pivot, tol=tol)
function cholfact{T}(A::StridedMatrix{T}, uplo::Symbol=:U; pivot=false, tol=0.0)
S = promote_type(typeof(chol(one(T))),Float32)
S <: BlasFloat && return cholfact!(convert(AbstractMatrix{S}, A), uplo, pivot = pivot, tol = tol)
pivot && throw(ArgumentError("pivot only supported for Float32, Float64, Complex{Float32} and Complex{Float64}"))
S != T && return cholfact!(convert(AbstractMatrix{S}, A), uplo)
return cholfact!(copy(A), uplo)
end
function cholfact(x::Number, uplo::Symbol=:U)
xf = fill(chol!(x), 1, 1)
Cholesky{:U, eltype(xf), typeof(xf)}(xf)
end

chol(A::AbstractMatrix, uplo::Symbol=:U) = Triangular(chol!(copy(A), uplo), uplo, false)
function chol!(x::Number, uplo::Symbol=:U)
rx = real(x)
rx == abs(x) || throw(DomainError())
rxr = sqrt(rx)
convert(promote_type(typeof(x), typeof(rxr)), rxr)
end
chol(x::Number, uplo::Symbol=:U) = chol!(x, uplo)

function convert{Tnew,Told,S,UpLo}(::Type{Cholesky{Tnew}},C::Cholesky{Told,S,UpLo})
Cnew = convert(AbstractMatrix{Tnew}, C.UL)
Cholesky{Tnew, typeof(Cnew), UpLo}(Cnew)
end
function convert{T,S,UpLo}(::Type{Cholesky{T,S,UpLo}},C::Cholesky)
Cnew = convert(AbstractMatrix{T}, C.UL)
Cholesky{T, typeof(Cnew), UpLo}(Cnew)
end
convert{T}(::Type{Factorization{T}}, C::Cholesky) = convert(Cholesky{T}, C)
convert{T}(::Type{CholeskyPivoted{T}},C::CholeskyPivoted) = CholeskyPivoted(convert(AbstractMatrix{T},C.UL),C.uplo,C.piv,C.rank,C.tol,C.info)
convert{T}(::Type{Factorization{T}}, C::CholeskyPivoted) = convert(CholeskyPivoted{T}, C)

full{T,S}(C::Cholesky{T,S,:U}) = C[:U]'C[:U]
full{T,S}(C::Cholesky{T,S,:L}) = C[:L]*C[:L]'

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)
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 == :p && return C.piv
if d == :P
n = size(C, 1)
P = zeros(T, n, n)
for i=1:n
P[C.piv[i],i] = one(T)
end
return P
end
throw(KeyError(d))
end

show{T,S<:AbstractMatrix,UpLo}(io::IO, C::Cholesky{T,S,UpLo}) = (println("$(typeof(C)) with factor:");show(io,C[UpLo]))

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))

function A_ldiv_B!{T<:BlasFloat}(C::CholeskyPivoted{T}, B::StridedVector{T})
chkfullrank(C)
ipermute!(LAPACK.potrs!(C.uplo, C.UL, permute!(B, C.piv)), C.piv)
end
function A_ldiv_B!{T<:BlasFloat}(C::CholeskyPivoted{T}, B::StridedMatrix{T})
chkfullrank(C)
n = size(C, 1)
for i=1:size(B, 2)
permute!(sub(B, 1:n, i), C.piv)
end
LAPACK.potrs!(C.uplo, C.UL, B)
for i=1:size(B, 2)
ipermute!(sub(B, 1:n, i), C.piv)
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),:]

function det{T,S,UpLo}(C::Cholesky{T,S,UpLo})
dd = one(T)
for i in 1:size(C.UL,1) dd *= abs2(C.UL[i,i]) end
dd
end

det{T}(C::CholeskyPivoted{T}) = C.rank<size(C.UL,1) ? real(zero(T)) : prod(abs2(diag(C.UL)))

function logdet{T,S,UpLo}(C::Cholesky{T,S,UpLo})
dd = zero(T)
for i in 1:size(C.UL,1) dd += log(C.UL[i,i]) end
dd + dd # instead of 2.0dd which can change the type
end

inv{T<:BlasFloat,S<:AbstractMatrix}(C::Cholesky{T,S,:U}) = copytri!(LAPACK.potri!('U', copy(C.UL)), 'U', true)
inv{T<:BlasFloat,S<:AbstractMatrix}(C::Cholesky{T,S,:L}) = copytri!(LAPACK.potri!('L', copy(C.UL)), 'L', true)

function inv(C::CholeskyPivoted)
chkfullrank(C)
ipiv = invperm(C.piv)
copytri!(LAPACK.potri!(C.uplo, copy(C.UL)), C.uplo, true)[ipiv, ipiv]
end

chkfullrank(C::CholeskyPivoted) = C.rank<size(C.UL, 1) && throw(RankDeficientException(C.info))

rank(C::CholeskyPivoted) = C.rank
8 changes: 4 additions & 4 deletions base/linalg/dense.jl
View file
Original file line number Diff line number Diff line change
@@ -60,13 +60,13 @@ vecnorm1{T<:BlasReal}(x::Union(Array{T},StridedVector{T})) =
vecnorm2{T<:BlasFloat}(x::Union(Array{T},StridedVector{T})) =
length(x) < NRM2_CUTOFF ? generic_vecnorm2(x) : BLAS.nrm2(x)

function triu!{T}(M::Matrix{T}, k::Integer)
function triu!(M::AbstractMatrix, k::Integer)
m, n = size(M)
idx = 1
for j = 0:n-1
ii = min(max(0, j+1-k), m)
for i = (idx+ii):(idx+m-1)
M[i] = zero(T)
M[i] = zero(M[i])
end
idx += m
end
@@ -75,13 +75,13 @@ end

triu(M::Matrix, k::Integer) = triu!(copy(M), k)

function tril!{T}(M::Matrix{T}, k::Integer)
function tril!(M::AbstractMatrix, k::Integer)
m, n = size(M)
idx = 1
for j = 0:n-1
ii = min(max(0, j-k), m)
for i = idx:(idx+ii-1)
M[i] = zero(T)
M[i] = zero(M[i])
end
idx += m
end
120 changes: 0 additions & 120 deletions base/linalg/factorization.jl
View file
Original file line number Diff line number Diff line change
@@ -10,126 +10,6 @@ macro assertnonsingular(A, info)
:(($info)==0 ? $A : throw(SingularException($info)))
end

##########################
# Cholesky Factorization #
##########################
immutable Cholesky{T} <: Factorization{T}
UL::Matrix{T}
uplo::Char
end
immutable CholeskyPivoted{T} <: Factorization{T}
UL::Matrix{T}
uplo::Char
piv::Vector{BlasInt}
rank::BlasInt
tol::Real
info::BlasInt
end

function cholfact!{T<:BlasFloat}(A::StridedMatrix{T}, uplo::Symbol=:U; pivot=false, tol=0.0)
uplochar = string(uplo)[1]
if pivot
A, piv, rank, info = LAPACK.pstrf!(uplochar, A, tol)
return CholeskyPivoted{T}(A, uplochar, piv, rank, tol, info)
else
C, info = LAPACK.potrf!(uplochar, A)
return @assertposdef Cholesky(C, uplochar) info
end
end
cholfact{T<:BlasFloat}(A::StridedMatrix{T}, uplo::Symbol=:U; pivot=false, tol=0.0) = cholfact!(copy(A), uplo, pivot=pivot, tol=tol)
cholfact{T}(A::StridedMatrix{T}, uplo::Symbol=:U; pivot=false, tol=0.0) = (S = promote_type(typeof(sqrt(one(T))),Float32); S != T ? cholfact!(convert(AbstractMatrix{S},A), uplo, pivot=pivot, tol=tol) : cholfact!(copy(A), uplo, pivot=pivot, tol=tol)) # When julia Cholesky has been implemented, the promotion should be changed.
cholfact(x::Number) = @assertposdef Cholesky(fill(sqrt(x), 1, 1), :U) !(imag(x) == 0 && real(x) > 0)

chol(A::Union(Number, AbstractMatrix), uplo::Symbol) = cholfact(A, uplo)[uplo]
chol(A::Union(Number, AbstractMatrix)) = triu!(cholfact(A, :U).UL)

convert{T}(::Type{Cholesky{T}},C::Cholesky) = Cholesky(convert(AbstractMatrix{T},C.UL),C.uplo)
convert{T}(::Type{Factorization{T}}, C::Cholesky) = convert(Cholesky{T}, C)
convert{T}(::Type{CholeskyPivoted{T}},C::CholeskyPivoted) = CholeskyPivoted(convert(AbstractMatrix{T},C.UL),C.uplo,C.piv,C.rank,C.tol,C.info)
convert{T}(::Type{Factorization{T}}, C::CholeskyPivoted) = convert(CholeskyPivoted{T}, C)

function full{T<:BlasFloat}(C::Cholesky{T})
if C.uplo == 'U'
BLAS.trmm!('R', C.uplo, 'N', 'N', one(T), C.UL, tril!(C.UL'))
else
BLAS.trmm!('L', C.uplo, 'N', 'N', one(T), C.UL, triu!(C.UL'))
end
end

size(C::Union(Cholesky, CholeskyPivoted)) = size(C.UL)
size(C::Union(Cholesky, CholeskyPivoted), d::Integer) = size(C.UL,d)

function getindex(C::Cholesky, d::Symbol)
d == :U && return Triangular(triu!(symbol(C.uplo) == d ? C.UL : C.UL'),:U)
d == :L && return Triangular(tril!(symbol(C.uplo) == d ? C.UL : C.UL'),:L)
d == :UL && return Triangular(C.UL, symbol(C.uplo))
throw(KeyError(d))
end
function getindex{T<:BlasFloat}(C::CholeskyPivoted{T}, d::Symbol)
d == :U && return triu!(symbol(C.uplo) == d ? C.UL : C.UL')
d == :L && return tril!(symbol(C.uplo) == d ? C.UL : C.UL')
d == :p && return C.piv
if d == :P
n = size(C, 1)
P = zeros(T, n, n)
for i=1:n
P[C.piv[i],i] = one(T)
end
return P
end
throw(KeyError(d))
end

show(io::IO, C::Cholesky) = (println(io,"$(typeof(C)) with factor:");show(io,C[symbol(C.uplo)]))

A_ldiv_B!{T<:BlasFloat}(C::Cholesky{T}, B::StridedVecOrMat{T}) = LAPACK.potrs!(C.uplo, C.UL, B)
A_ldiv_B!(C::Cholesky, B::StridedVecOrMat) = C.uplo=='L' ? Ac_ldiv_B!(Triangular(C.UL,C.uplo,'N'), A_ldiv_B!(Triangular(C.UL,C.uplo,'N'), B)) : A_ldiv_B!(Triangular(C.UL,C.uplo,'N'), Ac_ldiv_B!(Triangular(C.UL,C.uplo,'N'), B))

function A_ldiv_B!{T<:BlasFloat}(C::CholeskyPivoted{T}, B::StridedVector{T})
chkfullrank(C)
ipermute!(LAPACK.potrs!(C.uplo, C.UL, permute!(B, C.piv)), C.piv)
end
function A_ldiv_B!{T<:BlasFloat}(C::CholeskyPivoted{T}, B::StridedMatrix{T})
chkfullrank(C)
n = size(C, 1)
for i=1:size(B, 2)
permute!(sub(B, 1:n, i), C.piv)
end
LAPACK.potrs!(C.uplo, C.UL, B)
for i=1:size(B, 2)
ipermute!(sub(B, 1:n, i), C.piv)
end
B
end
A_ldiv_B!(C::CholeskyPivoted, B::StridedVector) = C.uplo=='L' ? Ac_ldiv_B!(Triangular(C.UL,C.uplo,'N'), A_ldiv_B!(Triangular(C.UL,C.uplo,'N'), B[C.piv]))[invperm(C.piv)] : A_ldiv_B!(Triangular(C.UL,C.uplo,'N'), Ac_ldiv_B!(Triangular(C.UL,C.uplo,'N'), B[C.piv]))[invperm(C.piv)]
A_ldiv_B!(C::CholeskyPivoted, B::StridedMatrix) = C.uplo=='L' ? Ac_ldiv_B!(Triangular(C.UL,C.uplo,'N'), A_ldiv_B!(Triangular(C.UL,C.uplo,'N'), B[C.piv,:]))[invperm(C.piv),:] : A_ldiv_B!(Triangular(C.UL,C.uplo,'N'), Ac_ldiv_B!(Triangular(C.UL,C.uplo,'N'), B[C.piv,:]))[invperm(C.piv),:]

function det{T}(C::Cholesky{T})
dd = one(T)
for i in 1:size(C.UL,1) dd *= abs2(C.UL[i,i]) end
dd
end

det{T}(C::CholeskyPivoted{T}) = C.rank<size(C.UL,1) ? real(zero(T)) : prod(abs2(diag(C.UL)))

function logdet{T}(C::Cholesky{T})
dd = zero(T)
for i in 1:size(C.UL,1) dd += log(C.UL[i,i]) end
dd + dd # instead of 2.0dd which can change the type
end

inv(C::Cholesky) = copytri!(LAPACK.potri!(C.uplo, copy(C.UL)), C.uplo, true)

function inv(C::CholeskyPivoted)
chkfullrank(C)
ipiv = invperm(C.piv)
copytri!(LAPACK.potri!(C.uplo, copy(C.UL)), C.uplo, true)[ipiv, ipiv]
end

chkfullrank(C::CholeskyPivoted) = C.rank<size(C.UL, 1) && throw(RankDeficientException(C.info))

rank(C::CholeskyPivoted) = C.rank

####################
# QR Factorization #
####################
10 changes: 5 additions & 5 deletions base/linalg/triangular.jl
View file
Original file line number Diff line number Diff line change
@@ -133,11 +133,11 @@ function similar{T,S,UpLo,IsUnit,Tnew}(A::Triangular{T,S,UpLo,IsUnit}, ::Type{Tn
return Triangular{Tnew, typeof(A), UpLo, IsUnit}(A)
end

getindex{T,S}(A::Triangular{T,S,:L,true}, i::Integer, j::Integer) = i == j ? one(T) : (i > j ? A.data[i,j] : zero(T))
getindex{T,S}(A::Triangular{T,S,:L,false}, i::Integer, j::Integer) = i >= j ? A.data[i,j] : zero(T)
getindex{T,S}(A::Triangular{T,S,:U,true}, i::Integer, j::Integer) = i == j ? one(T) : (i < j ? A.data[i,j] : zero(T))
getindex{T,S}(A::Triangular{T,S,:U,false}, i::Integer, j::Integer) = i <= j ? A.data[i,j] : zero(T)
getindex{T,S,UpLo,IsUnit}(A::Triangular{T,S,UpLo,IsUnit}, i::Integer) = ((m, n) = divrem(i - 1, size(A,1)); A[m + 1, n + 1])
getindex{T,S}(A::Triangular{T,S,:L,true}, i::Integer, j::Integer) = i == j ? one(T) : (i > j ? A.data[i,j] : zero(A.data[i,j]))
getindex{T,S}(A::Triangular{T,S,:L,false}, i::Integer, j::Integer) = i >= j ? A.data[i,j] : zero(A.data[i,j])
getindex{T,S}(A::Triangular{T,S,:U,true}, i::Integer, j::Integer) = i == j ? one(T) : (i < j ? A.data[i,j] : zero(A.data[i,j]))
getindex{T,S}(A::Triangular{T,S,:U,false}, i::Integer, j::Integer) = i <= j ? A.data[i,j] : zero(A.data[i,j])
getindex(A::Triangular, i::Integer) = ((m, n) = divrem(i - 1, size(A,1)); A[m + 1, n + 1])

istril{T,S,UpLo,IsUnit}(A::Triangular{T,S,UpLo,IsUnit}) = UpLo == :L
istriu{T,S,UpLo,IsUnit}(A::Triangular{T,S,UpLo,IsUnit}) = UpLo == :U
55 changes: 27 additions & 28 deletions test/linalg1.jl
View file
Original file line number Diff line number Diff line change
@@ -37,38 +37,37 @@ for eltya in (Float32, Float64, Complex64, Complex128, BigFloat, Int)
debug && println("\ntype of a: ", eltya, " type of b: ", eltyb, "\n")

debug && println("(Automatic) upper Cholesky factor")
if eltya != BigFloat && eltyb != BigFloat # Note! Need to implement cholesky decomposition in julia
capd = factorize(apd)
r = capd[:U]
κ = cond(apd) #condition number

#Test error bound on reconstruction of matrix: LAWNS 14, Lemma 2.1
E = abs(apd - r'*r)
for i=1:n, j=1:n
@test E[i,j] <= (n+1)ε/(1-(n+1)ε)*real(sqrt(apd[i,i]*apd[j,j]))
end
E = abs(apd - full(capd))
for i=1:n, j=1:n
@test E[i,j] <= (n+1)ε/(1-(n+1)ε)*real(sqrt(apd[i,i]*apd[j,j]))
end

capd = factorize(apd)
r = capd[:U]
κ = cond(apd, 1) #condition number

#Test error bound on reconstruction of matrix: LAWNS 14, Lemma 2.1
E = abs(apd - r'*r)
for i=1:n, j=1:n
@test E[i,j] <= (n+1)ε/(1-(n+1)ε)*real(sqrt(apd[i,i]*apd[j,j]))
end
E = abs(apd - full(capd))
for i=1:n, j=1:n
@test E[i,j] <= (n+1)ε/(1-(n+1)ε)*real(sqrt(apd[i,i]*apd[j,j]))
end

#Test error bound on linear solver: LAWNS 14, Theorem 2.1
#This is a surprisingly loose bound...
x = capd\b
@test norm(x-apd\b)/norm(x) <= (3n^2 + n + n^3*ε)*ε/(1-(n+1)*ε)*κ
@test norm(apd*x-b)/norm(b) <= (3n^2 + n + n^3*ε)*ε/(1-(n+1)*ε)*κ
#Test error bound on linear solver: LAWNS 14, Theorem 2.1
#This is a surprisingly loose bound...
x = capd\b
@test norm(x-apd\b,1)/norm(x,1) <= (3n^2 + n + n^3*ε)*ε/(1-(n+1)*ε)*κ
@test norm(apd*x-b,1)/norm(b,1) <= (3n^2 + n + n^3*ε)*ε/(1-(n+1)*ε)*κ

@test_approx_eq apd * inv(capd) eye(n)
@test norm(a*(capd\(a'*b)) - b)/norm(b) <= ε*κ*n # Ad hoc, revisit
@test abs((det(capd) - det(apd))/det(capd)) <= ε*κ*n # Ad hoc, but statistically verified, revisit
@test_approx_eq logdet(capd) log(det(capd)) # logdet is less likely to overflow
@test_approx_eq apd * inv(capd) eye(n)
@test norm(a*(capd\(a'*b)) - b,1)/norm(b,1) <= ε*κ*n # Ad hoc, revisit
@test abs((det(capd) - det(apd))/det(capd)) <= ε*κ*n # Ad hoc, but statistically verified, revisit
@test_approx_eq logdet(capd) log(det(capd)) # logdet is less likely to overflow

debug && println("lower Cholesky factor")
lapd = cholfact(apd, :L)
@test_approx_eq full(lapd) apd
l = lapd[:L]
@test_approx_eq l*l' apd
end
lapd = cholfact(apd, :L)
@test_approx_eq full(lapd) apd
l = lapd[:L]
@test_approx_eq l*l' apd

debug && println("pivoted Choleksy decomposition")
if eltya != BigFloat && eltyb != BigFloat # Note! Need to implement pivoted cholesky decomposition in julia
13 changes: 13 additions & 0 deletions test/linalg4.jl
View file
Original file line number Diff line number Diff line change
@@ -331,6 +331,19 @@ for newtype in [Diagonal, Bidiagonal, SymTridiagonal, Triangular, Matrix]
@test full(convert(newtype, A)) == full(A)
end

# Test generic cholfact!
for elty in (Float32, Float64, Complex{Float32}, Complex{Float64})
if elty <: Complex
A = complex(randn(5,5), randn(5,5))
else
A = randn(5,5)
end
A = convert(Matrix{elty}, A'A)
for ul in (:U, :L)
@test_approx_eq full(cholfact(A, ul)[ul]) full(invoke(Base.LinAlg.chol!, (AbstractMatrix,Symbol),copy(A), ul))
end
end

# Issue #7886
x, r = LAPACK.gelsy!([0 1; 0 2; 0 3.], [2, 4, 6.])
@test_approx_eq x [0,2]