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cutensor.jl
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cutensor.jl
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using CUDA.CUTENSOR
using LinearAlgebra
@test has_cutensor()
@testset "CuTensor type basics" begin
N = 2
dmax = 2^div(18,N)
dims = rand(2:dmax, N)
p = randperm(N)
indsA = collect(('a':'z')[1:N])
dimsA = dims
A = rand(Float64, dimsA...)
dA = CuArray(A)
p = randperm(N)
indsA = collect(('a':'z')[1:N])
ctA = CuTensor(dA, indsA)
@test length(ctA) == length(A)
@test size(ctA) == size(A)
@test size(ctA, 1) == size(A, 1)
@test ndims(ctA) == ndims(A)
@test strides(ctA) == strides(A)
@test eltype(ctA) == eltype(A)
end
@testset "Elementwise binary" begin
eltypes = ((Float16, Float16),
#(Float16, Float32),
(Float32, Float32),
#(Float32, Float64),
(Float64, Float64),
#(ComplexF16, ComplexF16), (ComplexF16, ComplexF32),
(ComplexF32, ComplexF32), #(ComplexF32, ComplexF64),
(ComplexF64, ComplexF64))
@testset for N=2:5
@testset for (eltyA, eltyC) in eltypes
# setup
eltyD = eltyC
dmax = 2^div(18,N)
dims = rand(2:dmax, N)
p = randperm(N)
indsA = collect(('a':'z')[1:N])
indsC = indsA[p]
dimsA = dims
dimsC = dims[p]
A = rand(eltyA, dimsA...)
dA = CuArray(A)
C = rand(eltyC, dimsC...)
dC = CuArray(C)
# simple case
opA = CUTENSOR.CUTENSOR_OP_IDENTITY
opC = CUTENSOR.CUTENSOR_OP_IDENTITY
dD = similar(dC, eltyD)
opAC = CUTENSOR.CUTENSOR_OP_ADD
dD = CUTENSOR.elementwiseBinary!(1, dA, indsA, opA, 1, dC, indsC, opC,
dD, indsC, opAC)
D = collect(dD)
@test D ≈ permutedims(A, p) .+ C
# using integers as indices
dD = CUTENSOR.elementwiseBinary!(1, dA, 1:N, opA, 1, dC, p, opC, dD, p, opAC)
D = collect(dD)
@test D ≈ permutedims(A, p) .+ C
# multiplication as binary operator
opAC = CUTENSOR.CUTENSOR_OP_MUL
dD = CUTENSOR.elementwiseBinary!(1, dA, indsA, opA, 1, dC, indsC, opC,
dD, indsC, opAC)
D = collect(dD)
@test D ≈ permutedims(A, p) .* C
# with non-trivial coefficients and conjugation
opA = eltyA <: Complex ? CUTENSOR.CUTENSOR_OP_CONJ :
CUTENSOR.CUTENSOR_OP_IDENTITY
opC = CUTENSOR.CUTENSOR_OP_IDENTITY
opAC = CUTENSOR.CUTENSOR_OP_ADD
α = rand(eltyD)
γ = rand(eltyD)
dD = CUTENSOR.elementwiseBinary!(α, dA, indsA, opA, γ, dC, indsC, opC,
dD, indsC, opAC)
D = collect(dD)
@test D ≈ α .* conj.(permutedims(A, p)) .+ γ .* C
# test in-place, and more complicated unary and binary operations
opA = eltyA <: Complex ? CUTENSOR.CUTENSOR_OP_IDENTITY :
CUTENSOR.CUTENSOR_OP_SQRT
# because we use rand, entries of A will be positive when elty is real
opC = eltyC <: Complex ? CUTENSOR.CUTENSOR_OP_CONJ :
CUTENSOR.CUTENSOR_OP_IDENTITY
opAC = eltyD <: Complex ? CUTENSOR.CUTENSOR_OP_ADD :
CUTENSOR.CUTENSOR_OP_MAX
α = rand(eltyD)
γ = rand(eltyD)
dD = CUTENSOR.elementwiseBinary!(α, dA, indsA, opA, γ, dC, indsC, opC,
dC, indsC, opAC)
D = collect(dC)
if eltyD <: Complex
if eltyA <: Complex
@test D ≈ α .* permutedims(A, p) .+ γ .* conj.(C)
else
@test D ≈ α .* sqrt.(convert.(eltyD, permutedims(A, p))) .+
γ .* conj.(C)
end
else
@test D ≈ max.(α .* sqrt.(convert.(eltyD, permutedims(A, p))), γ .* C)
end # # using host memory
# using CuTensor type
dA = CuArray(A)
dC = CuArray(C)
ctA = CuTensor(dA, indsA)
ctC = CuTensor(dC, indsC)
ctD = ctA + ctC
hD = collect(ctD.data)
@test hD ≈ permutedims(A, p) .+ C
ctD = ctA - ctC
hD = collect(ctD.data)
@test hD ≈ permutedims(A, p) .- C
α = rand(eltyD)
ctD = LinearAlgebra.axpy!(α, ctA, ctC)
hD = collect(ctD.data)
@test hD ≈ α.*permutedims(A, p) .+ C
γ = rand(eltyD)
ctD = LinearAlgebra.axpby!(α, ctA, γ, ctC)
hD = collect(ctD.data)
@test hD ≈ α.*permutedims(A, p) .+ γ.*C
end
end
end
@testset "Permutations" begin
eltypes = ((Float16, Float16),
#(Float16, Float32),
(Float32, Float32),
#(Float32, Float64),
(Float64, Float64),
#(ComplexF16, ComplexF16),
#(ComplexF16, ComplexF32),
(ComplexF32, ComplexF32),
#(ComplexF32, ComplexF64),
(ComplexF64, ComplexF64))
@testset for N=2:5
@testset for (eltyA, eltyC) in eltypes
# setup
dmax = 2^div(18,N)
dims = rand(2:dmax, N)
p = randperm(N)
indsA = collect(('a':'z')[1:N])
indsC = indsA[p]
dimsA = dims
dimsC = dims[p]
A = rand(eltyA, dimsA...)
dA = CuArray(A)
dC = similar(dA, eltyC, dimsC...)
# simple case
dC = CUTENSOR.permutation!(one(eltyA), dA, indsA, dC, indsC)
C = collect(dC)
@test C == permutedims(A, p) # exact equality
# with scalar
α = rand(eltyA)
dC = CUTENSOR.permutation!(α, dA, indsA, dC, indsC)
C = collect(dC)
@test C ≈ α * permutedims(A, p) # approximate, floating point rounding
end
end
end
@testset "Elementwise trinary" begin
eltypes = ((Float16, Float16, Float16),
#(Float16, Float32, Float32),
# (Float32, Float16, Float32),
(Float32, Float32, Float32),
# (Float32, Float32, Float64),
# (Float32, Float64, Float64),
# (Float64, Float32, Float64),
(Float64, Float64, Float64),
(ComplexF32, ComplexF32, ComplexF32),
(ComplexF64, ComplexF64, ComplexF64))
@testset for N=2:5
@testset for (eltyA, eltyB, eltyC) in eltypes
# setup
eltyD = eltyC
dmax = 2^div(18,N)
dims = rand(2:dmax, N)
pA = randperm(N)
ipA = invperm(pA)
pB = randperm(N)
ipB = invperm(pB)
indsC = collect(('a':'z')[1:N])
dimsC = dims
indsA = indsC[ipA]
dimsA = dims[ipA]
indsB = indsC[ipB]
dimsB = dims[ipB]
A = rand(eltyA, dimsA...)
dA = CuArray(A)
B = rand(eltyB, dimsB...)
dB = CuArray(B)
C = rand(eltyC, dimsC...)
dC = CuArray(C)
dD = similar(dC)
# simple case
opA = CUTENSOR.CUTENSOR_OP_IDENTITY
opB = CUTENSOR.CUTENSOR_OP_IDENTITY
opC = CUTENSOR.CUTENSOR_OP_IDENTITY
opAB = CUTENSOR.CUTENSOR_OP_ADD
opABC = CUTENSOR.CUTENSOR_OP_ADD
dD = CUTENSOR.elementwiseTrinary!(1, dA, indsA, opA, 1, dB, indsB, opB,
1, dC, indsC, opC, dD, indsC, opAB, opABC)
D = collect(dD)
@test D ≈ permutedims(A, pA) .+ permutedims(B, pB) .+ C
# using integers as indices
dD = CUTENSOR.elementwiseTrinary!(1, dA, ipA, opA, 1, dB, ipB, opB,
1, dC, 1:N, opC, dD, 1:N, opAB, opABC)
D = collect(dD)
@test D ≈ permutedims(A, pA) .+ permutedims(B, pB) .+ C
# multiplication as binary operator
opAB = CUTENSOR.CUTENSOR_OP_MUL
opABC = CUTENSOR.CUTENSOR_OP_ADD
dD = CUTENSOR.elementwiseTrinary!(1, dA, indsA, opA, 1, dB, indsB, opB,
1, dC, indsC, opC, dD, indsC, opAB, opABC)
D = collect(dD)
@test D ≈ (convert.(eltyD, permutedims(A, pA)) .* convert.(eltyD, permutedims(B, pB))) .+ C
opAB = CUTENSOR.CUTENSOR_OP_ADD
opABC = CUTENSOR.CUTENSOR_OP_MUL
dD = CUTENSOR.elementwiseTrinary!(1, dA, indsA, opA, 1, dB, indsB, opB,
1, dC, indsC, opC, dD, indsC, opAB, opABC)
D = collect(dD)
@test D ≈ (convert.(eltyD, permutedims(A, pA)) .+ convert.(eltyD, permutedims(B, pB))) .* C
opAB = CUTENSOR.CUTENSOR_OP_MUL
opABC = CUTENSOR.CUTENSOR_OP_MUL
dD = CUTENSOR.elementwiseTrinary!(1, dA, indsA, opA, 1, dB, indsB, opB,
1, dC, indsC, opC, dD, indsC, opAB, opABC)
D = collect(dD)
@test D ≈ convert.(eltyD, permutedims(A, pA)) .*
convert.(eltyD, permutedims(B, pB)) .* C
# with non-trivial coefficients and conjugation
α = rand(eltyD)
β = rand(eltyD)
γ = rand(eltyD)
opA = eltyA <: Complex ? CUTENSOR.CUTENSOR_OP_CONJ :
CUTENSOR.CUTENSOR_OP_IDENTITY
opAB = CUTENSOR.CUTENSOR_OP_ADD
opABC = CUTENSOR.CUTENSOR_OP_ADD
dD = CUTENSOR.elementwiseTrinary!(α, dA, indsA, opA, β, dB, indsB, opB,
γ, dC, indsC, opC, dD, indsC, opAB, opABC)
D = collect(dD)
@test D ≈ α .* conj.(permutedims(A, pA)) .+ β .* permutedims(B, pB) .+ γ .* C
opB = eltyB <: Complex ? CUTENSOR.CUTENSOR_OP_CONJ :
CUTENSOR.CUTENSOR_OP_IDENTITY
opAB = CUTENSOR.CUTENSOR_OP_ADD
opABC = CUTENSOR.CUTENSOR_OP_ADD
dD = CUTENSOR.elementwiseTrinary!(α, dA, indsA, opA, β, dB, indsB, opB,
γ, dC, indsC, opC, dD, indsC, opAB, opABC)
D = collect(dD)
@test D ≈ α .* conj.(permutedims(A, pA)) .+
β .* conj.(permutedims(B, pB)) .+ γ .* C
opA = CUTENSOR.CUTENSOR_OP_IDENTITY
opAB = CUTENSOR.CUTENSOR_OP_MUL
opABC = CUTENSOR.CUTENSOR_OP_ADD
dD = CUTENSOR.elementwiseTrinary!(α, dA, indsA, opA, β, dB, indsB, opB,
γ, dC, indsC, opC, dD, indsC, opAB, opABC)
D = collect(dD)
@test D ≈ α .* permutedims(A, pA) .* β .* conj.(permutedims(B, pB)) .+ γ .* C
# test in-place, and more complicated unary and binary operations
opA = eltyA <: Complex ? CUTENSOR.CUTENSOR_OP_IDENTITY :
CUTENSOR.CUTENSOR_OP_SQRT
opB = eltyB <: Complex ? CUTENSOR.CUTENSOR_OP_IDENTITY :
CUTENSOR.CUTENSOR_OP_SQRT
# because we use rand, entries of A will be positive when elty is real
opC = eltyC <: Complex ? CUTENSOR.CUTENSOR_OP_CONJ :
CUTENSOR.CUTENSOR_OP_IDENTITY
opAB = eltyD <: Complex ? CUTENSOR.CUTENSOR_OP_MUL :
CUTENSOR.CUTENSOR_OP_MIN
opABC = eltyD <: Complex ? CUTENSOR.CUTENSOR_OP_ADD :
CUTENSOR.CUTENSOR_OP_MAX
α = rand(eltyD)
β = rand(eltyD)
γ = rand(eltyD)
dD = CUTENSOR.elementwiseTrinary!(α, dA, indsA, opA, β, dB, indsB, opB,
γ, dC, indsC, opC, dC, indsC, opAB, opABC)
D = collect(dD)
if eltyD <: Complex
if eltyA <: Complex && eltyB <: Complex
@test D ≈ α .* permutedims(A, pA) .* β .* permutedims(B, pB) .+
γ .* conj.(C)
elseif eltyB <: Complex
@test D ≈ α .* sqrt.(convert.(eltyD, permutedims(A, pA))) .*
β .* permutedims(B, pB) .+ γ .* conj.(C)
elseif eltyB <: Complex
@test D ≈ α .* permutedims(A, pA) .*
β .* sqrt.(convert.(eltyD, permutedims(B, pB))) .+
γ .* conj.(C)
else
@test D ≈ α .* sqrt.(convert.(eltyD, permutedims(A, pA))) .*
β .* sqrt.(convert.(eltyD, permutedims(B, pB))) .+
γ .* conj.(C)
end
else
@test D ≈ max.(min.(α .* sqrt.(convert.(eltyD, permutedims(A, pA))),
β .* sqrt.(convert.(eltyD, permutedims(B, pB)))),
γ .* C)
end
end
end
end
@testset "Reduction" begin
eltypes = (#(Float16, Float16), #(Float16, Float32),
(Float32, Float32), #(Float32, Float64),
(Float64, Float64),
#(ComplexF16, ComplexF16), (ComplexF16, ComplexF32),
(ComplexF32, ComplexF32), #(ComplexF32, ComplexF64),
(ComplexF64, ComplexF64))
@testset for NA=2:5, NC = 1:NA-1
@testset for (eltyA, eltyC) in eltypes
# setup
eltyD = eltyC
dmax = 2^div(18,NA)
dims = rand(2:dmax, NA)
p = randperm(NA)
indsA = collect(('a':'z')[1:NA])
indsC = indsA[p][1:NC]
dimsA = dims
dimsC = dims[p][1:NC]
A = rand(eltyA, (dimsA...,))
dA = CuArray(A)
C = rand(eltyC, (dimsC...,))
dC = CuArray(C)
# setup
opA = CUTENSOR.CUTENSOR_OP_IDENTITY
opC = CUTENSOR.CUTENSOR_OP_IDENTITY
opReduce = CUTENSOR.CUTENSOR_OP_ADD
# simple case
dC = CUTENSOR.reduction!(1, dA, indsA, opA, 0, dC, indsC, opC, opReduce)
C = collect(dC)
@test reshape(C, (dimsC..., ones(Int,NA-NC)...)) ≈
sum(permutedims(A, p); dims = ((NC+1:NA)...,))
# using integers as indices
dC = CUTENSOR.reduction!(1, dA, collect(1:NA), opA, 0, dC, p[1:NC], opC, opReduce)
C = collect(dC)
@test reshape(C, (dimsC..., ones(Int,NA-NC)...)) ≈
sum(permutedims(A, p); dims = ((NC+1:NA)...,))
# multiplication as reduction operator
opReduce = CUTENSOR.CUTENSOR_OP_MUL
dC = CUTENSOR.reduction!(1, dA, indsA, opA, 0, dC, indsC, opC, opReduce)
C = collect(dC)
@test reshape(C, (dimsC..., ones(Int,NA-NC)...)) ≈
prod(permutedims(A, p); dims = ((NC+1:NA)...,)) atol=eps(Float16) rtol=Base.rtoldefault(Float16)
# NOTE: this test often yields values close to 0 that do not compare approximately
# with non-trivial coefficients and conjugation
opA = eltyA <: Complex ? CUTENSOR.CUTENSOR_OP_CONJ :
CUTENSOR.CUTENSOR_OP_IDENTITY
opC = CUTENSOR.CUTENSOR_OP_IDENTITY
opReduce = CUTENSOR.CUTENSOR_OP_ADD
C = rand(eltyC, (dimsC...,))
dC = CuArray(C)
α = rand(eltyC)
γ = rand(eltyC)
dC = CUTENSOR.reduction!(α, dA, indsA, opA, γ, dC, indsC, opC, opReduce)
@test reshape(collect(dC), (dimsC..., ones(Int,NA-NC)...)) ≈
α .* conj.(sum(permutedims(A, p); dims = ((NC+1:NA)...,))) .+ γ .* C
end
end
end
@testset "Contraction" begin
eltypes = ( (Float32, Float32, Float32, Float32),
#(Float32, Float32, Float32, Float16),
(ComplexF32, ComplexF32, ComplexF32, ComplexF32),
(Float64, Float64, Float64, Float64),
(Float64, Float64, Float64, Float32),
(ComplexF64, ComplexF64, ComplexF64, ComplexF64),
(ComplexF64, ComplexF64, ComplexF64, ComplexF32)
)
@testset for NoA=1:3, NoB=1:3, Nc=1:3
@testset for (eltyA, eltyB, eltyC, eltyCompute) in eltypes
# setup
dmax = 2^div(18, max(NoA+Nc, NoB+Nc, NoA+NoB))
dimsoA = rand(2:dmax, NoA)
loA = prod(dimsoA)
dimsoB = rand(2:dmax, NoB)
loB = prod(dimsoB)
dimsc = rand(2:dmax, Nc)
lc = prod(dimsc)
allinds = collect('a':'z')
indsoA = allinds[1:NoA]
indsoB = allinds[NoA .+ (1:NoB)]
indsc = allinds[NoA .+ NoB .+ (1:Nc)]
pA = randperm(NoA + Nc)
ipA = invperm(pA)
pB = randperm(Nc + NoB)
ipB = invperm(pB)
pC = randperm(NoA + NoB)
ipC = invperm(pC)
compute_rtol = (real(eltyCompute) == Float16 || real(eltyC) == Float16) ? 1e-2 : (real(eltyCompute) == Float32 ? 1e-4 : 1e-6)
dimsA = [dimsoA; dimsc][pA]
indsA = [indsoA; indsc][pA]
dimsB = [dimsc; dimsoB][pB]
indsB = [indsc; indsoB][pB]
dimsC = [dimsoA; dimsoB][pC]
indsC = [indsoA; indsoB][pC]
A = rand(eltyA, (dimsA...,))
mA = reshape(permutedims(A, ipA), (loA, lc))
dA = CuArray(A)
B = rand(eltyB, (dimsB...,))
dB = CuArray(B)
mB = reshape(permutedims(B, ipB), (lc, loB))
C = zeros(eltyC, (dimsC...,))
dC = CuArray(C)
# simple case
opA = CUTENSOR.CUTENSOR_OP_IDENTITY
opB = CUTENSOR.CUTENSOR_OP_IDENTITY
opC = CUTENSOR.CUTENSOR_OP_IDENTITY
opOut = CUTENSOR.CUTENSOR_OP_IDENTITY
dC = CUTENSOR.contraction!(1, dA, indsA, opA, dB, indsB, opB, 0, dC, indsC, opC, opOut, compute_type=eltyCompute)
C = collect(dC)
mC = reshape(permutedims(C, ipC), (loA, loB))
@test mC ≈ mA * mB rtol=compute_rtol
# simple case with plan storage
opA = CUTENSOR.CUTENSOR_OP_IDENTITY
opB = CUTENSOR.CUTENSOR_OP_IDENTITY
opC = CUTENSOR.CUTENSOR_OP_IDENTITY
opOut = CUTENSOR.CUTENSOR_OP_IDENTITY
plan = CUTENSOR.plan_contraction(dA, indsA, opA, dB, indsB, opB, dC, indsC, opC, opOut)
dC = CUTENSOR.contraction!(1, dA, indsA, opA, dB, indsB, opB, 0, dC, indsC, opC, opOut, plan=plan)
C = collect(dC)
mC = reshape(permutedims(C, ipC), (loA, loB))
@test mC ≈ mA * mB
# simple case with plan storage and compute type
opA = CUTENSOR.CUTENSOR_OP_IDENTITY
opB = CUTENSOR.CUTENSOR_OP_IDENTITY
opC = CUTENSOR.CUTENSOR_OP_IDENTITY
opOut = CUTENSOR.CUTENSOR_OP_IDENTITY
plan = CUTENSOR.plan_contraction(dA, indsA, opA, dB, indsB, opB, dC, indsC, opC, opOut, compute_type=eltyCompute)
dC = CUTENSOR.contraction!(1, dA, indsA, opA, dB, indsB, opB,
0, dC, indsC, opC, opOut, plan=plan, compute_type=eltyCompute)
C = collect(dC)
mC = reshape(permutedims(C, ipC), (loA, loB))
@test mC ≈ mA * mB rtol=compute_rtol
# with non-trivial α
α = rand(eltyCompute)
dC = CUTENSOR.contraction!(α, dA, indsA, opA, dB, indsB, opB, zero(eltyCompute), dC, indsC, opC, opOut, compute_type=eltyCompute)
C = collect(dC)
mC = reshape(permutedims(C, ipC), (loA, loB))
@test mC ≈ α * mA * mB rtol=compute_rtol
# with non-trivial β
C = rand(eltyC, (dimsC...,))
dC = CuArray(C)
α = rand(eltyCompute)
β = rand(eltyCompute)
copyto!(dC, C)
dD = CUTENSOR.contraction!(α, dA, indsA, opA, dB, indsB, opB, β, dC, indsC, opC, opOut, compute_type=eltyCompute)
D = collect(dD)
mC = reshape(permutedims(C, ipC), (loA, loB))
mD = reshape(permutedims(D, ipC), (loA, loB))
@test mD ≈ α * mA * mB + β * mC rtol=compute_rtol
# with CuTensor objects
if eltyCompute != Float32 && eltyC != Float16
ctA = CuTensor(dA, indsA)
ctB = CuTensor(dB, indsB)
ctC = CuTensor(dC, indsC)
ctC = LinearAlgebra.mul!(ctC, ctA, ctB)
C2, C2inds = collect(ctC)
mC = reshape(permutedims(C2, ipC), (loA, loB))
@test mC ≈ mA * mB
ctC = ctA * ctB
C2, C2inds = collect(ctC)
pC2 = convert.(Int, indexin(convert.(Char, C2inds), [indsoA; indsoB]))
mC = reshape(permutedims(C2, invperm(pC2)), (loA, loB))
@test mC ≈ mA * mB
end
# with conjugation flag for complex arguments
if !((NoA, NoB, Nc) in ((1,1,3), (1,2,3), (3,1,2)))
# not supported for these specific cases for unknown reason
if eltyA <: Complex
opA = CUTENSOR.CUTENSOR_OP_CONJ
opB = CUTENSOR.CUTENSOR_OP_IDENTITY
opOut = CUTENSOR.CUTENSOR_OP_IDENTITY
dC = CUTENSOR.contraction!(complex(1.0, 0.0), dA, indsA, opA, dB, indsB, opB,
0, dC, indsC, opC, opOut, compute_type=eltyCompute)
C = collect(dC)
mC = reshape(permutedims(C, ipC), (loA, loB))
@test mC ≈ conj(mA) * mB rtol=compute_rtol
end
if eltyB <: Complex
opA = CUTENSOR.CUTENSOR_OP_IDENTITY
opB = CUTENSOR.CUTENSOR_OP_CONJ
opOut = CUTENSOR.CUTENSOR_OP_IDENTITY
dC = CUTENSOR.contraction!(complex(1.0, 0.0), dA, indsA, opA, dB, indsB, opB,
complex(0.0, 0.0), dC, indsC, opC, opOut, compute_type=eltyCompute)
C = collect(dC)
mC = reshape(permutedims(C, ipC), (loA, loB))
@test mC ≈ mA*conj(mB) rtol=compute_rtol
end
if eltyA <: Complex && eltyB <: Complex
opA = CUTENSOR.CUTENSOR_OP_CONJ
opB = CUTENSOR.CUTENSOR_OP_CONJ
opOut = CUTENSOR.CUTENSOR_OP_IDENTITY
dC = CUTENSOR.contraction!(one(eltyCompute), dA, indsA, opA, dB, indsB, opB,
zero(eltyCompute), dC, indsC, opC, opOut, compute_type=eltyCompute)
C = collect(dC)
mC = reshape(permutedims(C, ipC), (loA, loB))
@test mC ≈ conj(mA)*conj(mB) rtol=compute_rtol
end
end
end
end
end