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mollusk
2018-02-10 10:27:19 -07:00
parent 94220957d7
commit 019f8e3064
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julia-0.6.2/share/julia/base/linalg/matmul.jl Normal file
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# This file is a part of Julia. License is MIT: https://julialang.org/license
# matmul.jl: Everything to do with dense matrix multiplication
matprod(x, y) = x*y + x*y
# multiply by diagonal matrix as vector
function scale!(C::AbstractMatrix, A::AbstractMatrix, b::AbstractVector)
m, n = size(A)
if size(A) != size(C)
throw(DimensionMismatch("size of A, $(size(A)), does not match size of C, $(size(C))"))
end
if n != length(b)
throw(DimensionMismatch("second dimension of A, $n, does not match length of b, $(length(b))"))
end
@inbounds for j = 1:n
bj = b[j]
for i = 1:m
C[i,j] = A[i,j]*bj
end
end
C
end
function scale!(C::AbstractMatrix, b::AbstractVector, A::AbstractMatrix)
m, n = size(A)
if size(A) != size(C)
throw(DimensionMismatch("size of A, $(size(A)), does not match size of C, $(size(C))"))
end
if m != length(b)
throw(DimensionMismatch("first dimension of A, $m, does not match length of b, $(length(b))"))
end
@inbounds for j = 1:n, i = 1:m
C[i,j] = A[i,j]*b[i]
end
C
end
# Dot products
vecdot(x::Union{DenseArray{T},StridedVector{T}}, y::Union{DenseArray{T},StridedVector{T}}) where {T<:BlasReal} = BLAS.dot(x, y)
vecdot(x::Union{DenseArray{T},StridedVector{T}}, y::Union{DenseArray{T},StridedVector{T}}) where {T<:BlasComplex} = BLAS.dotc(x, y)
function dot(x::Vector{T}, rx::Union{UnitRange{TI},Range{TI}}, y::Vector{T}, ry::Union{UnitRange{TI},Range{TI}}) where {T<:BlasReal,TI<:Integer}
if length(rx) != length(ry)
throw(DimensionMismatch("length of rx, $(length(rx)), does not equal length of ry, $(length(ry))"))
end
if minimum(rx) < 1 || maximum(rx) > length(x)
throw(BoundsError(x, rx))
end
if minimum(ry) < 1 || maximum(ry) > length(y)
throw(BoundsError(y, ry))
end
BLAS.dot(length(rx), pointer(x)+(first(rx)-1)*sizeof(T), step(rx), pointer(y)+(first(ry)-1)*sizeof(T), step(ry))
end
function dot(x::Vector{T}, rx::Union{UnitRange{TI},Range{TI}}, y::Vector{T}, ry::Union{UnitRange{TI},Range{TI}}) where {T<:BlasComplex,TI<:Integer}
if length(rx) != length(ry)
throw(DimensionMismatch("length of rx, $(length(rx)), does not equal length of ry, $(length(ry))"))
end
if minimum(rx) < 1 || maximum(rx) > length(x)
throw(BoundsError(x, rx))
end
if minimum(ry) < 1 || maximum(ry) > length(y)
throw(BoundsError(y, ry))
end
BLAS.dotc(length(rx), pointer(x)+(first(rx)-1)*sizeof(T), step(rx), pointer(y)+(first(ry)-1)*sizeof(T), step(ry))
end
At_mul_B(x::StridedVector{T}, y::StridedVector{T}) where {T<:BlasComplex} = BLAS.dotu(x, y)
# Matrix-vector multiplication
function (*)(A::StridedMatrix{T}, x::StridedVector{S}) where {T<:BlasFloat,S}
TS = promote_op(matprod, T, S)
A_mul_B!(similar(x, TS, size(A,1)), A, convert(AbstractVector{TS}, x))
end
function (*)(A::AbstractMatrix{T}, x::AbstractVector{S}) where {T,S}
TS = promote_op(matprod, T, S)
A_mul_B!(similar(x,TS,size(A,1)),A,x)
end
# these will throw a DimensionMismatch unless B has 1 row (or 1 col for transposed case):
A_mul_Bt(a::AbstractVector, B::AbstractMatrix) = A_mul_Bt(reshape(a,length(a),1),B)
A_mul_Bt(A::AbstractMatrix, b::AbstractVector) = A_mul_Bt(A,reshape(b,length(b),1))
A_mul_Bc(a::AbstractVector, B::AbstractMatrix) = A_mul_Bc(reshape(a,length(a),1),B)
A_mul_Bc(A::AbstractMatrix, b::AbstractVector) = A_mul_Bc(A,reshape(b,length(b),1))
(*)(a::AbstractVector, B::AbstractMatrix) = reshape(a,length(a),1)*B
A_mul_B!(y::StridedVector{T}, A::StridedVecOrMat{T}, x::StridedVector{T}) where {T<:BlasFloat} = gemv!(y, 'N', A, x)
for elty in (Float32,Float64)
@eval begin
function A_mul_B!(y::StridedVector{Complex{$elty}}, A::StridedVecOrMat{Complex{$elty}}, x::StridedVector{$elty})
Afl = reinterpret($elty,A,(2size(A,1),size(A,2)))
yfl = reinterpret($elty,y)
gemv!(yfl,'N',Afl,x)
return y
end
end
end
A_mul_B!(y::AbstractVector, A::AbstractVecOrMat, x::AbstractVector) = generic_matvecmul!(y, 'N', A, x)
function At_mul_B(A::StridedMatrix{T}, x::StridedVector{S}) where {T<:BlasFloat,S}
TS = promote_op(matprod, T, S)
At_mul_B!(similar(x,TS,size(A,2)), A, convert(AbstractVector{TS}, x))
end
function At_mul_B(A::AbstractMatrix{T}, x::AbstractVector{S}) where {T,S}
TS = promote_op(matprod, T, S)
At_mul_B!(similar(x,TS,size(A,2)), A, x)
end
At_mul_B!(y::StridedVector{T}, A::StridedVecOrMat{T}, x::StridedVector{T}) where {T<:BlasFloat} = gemv!(y, 'T', A, x)
At_mul_B!(y::AbstractVector, A::AbstractVecOrMat, x::AbstractVector) = generic_matvecmul!(y, 'T', A, x)
function Ac_mul_B(A::StridedMatrix{T}, x::StridedVector{S}) where {T<:BlasFloat,S}
TS = promote_op(matprod, T, S)
Ac_mul_B!(similar(x,TS,size(A,2)),A,convert(AbstractVector{TS},x))
end
function Ac_mul_B(A::AbstractMatrix{T}, x::AbstractVector{S}) where {T,S}
TS = promote_op(matprod, T, S)
Ac_mul_B!(similar(x,TS,size(A,2)), A, x)
end
Ac_mul_B!(y::StridedVector{T}, A::StridedVecOrMat{T}, x::StridedVector{T}) where {T<:BlasReal} = At_mul_B!(y, A, x)
Ac_mul_B!(y::StridedVector{T}, A::StridedVecOrMat{T}, x::StridedVector{T}) where {T<:BlasComplex} = gemv!(y, 'C', A, x)
Ac_mul_B!(y::AbstractVector, A::AbstractVecOrMat, x::AbstractVector) = generic_matvecmul!(y, 'C', A, x)
# Matrix-matrix multiplication
"""
```
*(A::AbstractMatrix, B::AbstractMatrix)
```
Matrix multiplication.
# Example
```jldoctest
julia> [1 1; 0 1] * [1 0; 1 1]
2×2 Array{Int64,2}:
2 1
1 1
```
"""
function (*)(A::AbstractMatrix{T}, B::AbstractMatrix{S}) where {T,S}
TS = promote_op(matprod, T, S)
A_mul_B!(similar(B, TS, (size(A,1), size(B,2))), A, B)
end
A_mul_B!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{T}) where {T<:BlasFloat} = gemm_wrapper!(C, 'N', 'N', A, B)
for elty in (Float32,Float64)
@eval begin
function A_mul_B!(C::StridedMatrix{Complex{$elty}}, A::StridedVecOrMat{Complex{$elty}}, B::StridedVecOrMat{$elty})
Afl = reinterpret($elty, A, (2size(A,1), size(A,2)))
Cfl = reinterpret($elty, C, (2size(C,1), size(C,2)))
gemm_wrapper!(Cfl, 'N', 'N', Afl, B)
return C
end
end
end
"""
A_mul_B!(Y, A, B) -> Y
Calculates the matrix-matrix or matrix-vector product ``A⋅B`` and stores the result in `Y`,
overwriting the existing value of `Y`. Note that `Y` must not be aliased with either `A` or
`B`.
# Example
```jldoctest
julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; Y = similar(B); A_mul_B!(Y, A, B);
julia> Y
2×2 Array{Float64,2}:
3.0 3.0
7.0 7.0
```
"""
A_mul_B!(C::AbstractMatrix, A::AbstractVecOrMat, B::AbstractVecOrMat) = generic_matmatmul!(C, 'N', 'N', A, B)
function At_mul_B(A::AbstractMatrix{T}, B::AbstractMatrix{S}) where {T,S}
TS = promote_op(matprod, T, S)
At_mul_B!(similar(B, TS, (size(A,2), size(B,2))), A, B)
end
At_mul_B!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{T}) where {T<:BlasFloat} = A===B ? syrk_wrapper!(C, 'T', A) : gemm_wrapper!(C, 'T', 'N', A, B)
At_mul_B!(C::AbstractMatrix, A::AbstractVecOrMat, B::AbstractVecOrMat) = generic_matmatmul!(C, 'T', 'N', A, B)
function A_mul_Bt(A::AbstractMatrix{T}, B::AbstractMatrix{S}) where {T,S}
TS = promote_op(matprod, T, S)
A_mul_Bt!(similar(B, TS, (size(A,1), size(B,1))), A, B)
end
A_mul_Bt!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{T}) where {T<:BlasFloat} = A===B ? syrk_wrapper!(C, 'N', A) : gemm_wrapper!(C, 'N', 'T', A, B)
for elty in (Float32,Float64)
@eval begin
function A_mul_Bt!(C::StridedMatrix{Complex{$elty}}, A::StridedVecOrMat{Complex{$elty}}, B::StridedVecOrMat{$elty})
Afl = reinterpret($elty, A, (2size(A,1), size(A,2)))
Cfl = reinterpret($elty, C, (2size(C,1), size(C,2)))
gemm_wrapper!(Cfl, 'N', 'T', Afl, B)
return C
end
end
end
A_mul_Bt!(C::AbstractVecOrMat, A::AbstractVecOrMat, B::AbstractVecOrMat) = generic_matmatmul!(C, 'N', 'T', A, B)
function At_mul_Bt(A::AbstractMatrix{T}, B::AbstractVecOrMat{S}) where {T,S}
TS = promote_op(matprod, T, S)
At_mul_Bt!(similar(B, TS, (size(A,2), size(B,1))), A, B)
end
At_mul_Bt!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{T}) where {T<:BlasFloat} = gemm_wrapper!(C, 'T', 'T', A, B)
At_mul_Bt!(C::AbstractMatrix, A::AbstractVecOrMat, B::AbstractVecOrMat) = generic_matmatmul!(C, 'T', 'T', A, B)
Ac_mul_B(A::StridedMatrix{T}, B::StridedMatrix{T}) where {T<:BlasReal} = At_mul_B(A, B)
Ac_mul_B!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{T}) where {T<:BlasReal} = At_mul_B!(C, A, B)
function Ac_mul_B(A::AbstractMatrix{T}, B::AbstractMatrix{S}) where {T,S}
TS = promote_op(matprod, T, S)
Ac_mul_B!(similar(B, TS, (size(A,2), size(B,2))), A, B)
end
Ac_mul_B!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{T}) where {T<:BlasComplex} = A===B ? herk_wrapper!(C,'C',A) : gemm_wrapper!(C,'C', 'N', A, B)
Ac_mul_B!(C::AbstractMatrix, A::AbstractVecOrMat, B::AbstractVecOrMat) = generic_matmatmul!(C, 'C', 'N', A, B)
A_mul_Bc(A::StridedMatrix{<:BlasFloat}, B::StridedMatrix{<:BlasReal}) = A_mul_Bt(A, B)
A_mul_Bc!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{<:BlasReal}) where {T<:BlasFloat} = A_mul_Bt!(C, A, B)
function A_mul_Bc(A::AbstractMatrix{T}, B::AbstractMatrix{S}) where {T,S}
TS = promote_op(matprod, T, S)
A_mul_Bc!(similar(B,TS,(size(A,1),size(B,1))),A,B)
end
A_mul_Bc!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{T}) where {T<:BlasComplex} = A===B ? herk_wrapper!(C, 'N', A) : gemm_wrapper!(C, 'N', 'C', A, B)
A_mul_Bc!(C::AbstractMatrix, A::AbstractVecOrMat, B::AbstractVecOrMat) = generic_matmatmul!(C, 'N', 'C', A, B)
Ac_mul_Bc(A::AbstractMatrix{T}, B::AbstractMatrix{S}) where {T,S} =
Ac_mul_Bc!(similar(B, promote_op(matprod, T, S), (size(A,2), size(B,1))), A, B)
Ac_mul_Bc!(C::StridedMatrix{T}, A::StridedVecOrMat{T}, B::StridedVecOrMat{T}) where {T<:BlasFloat} = gemm_wrapper!(C, 'C', 'C', A, B)
Ac_mul_Bc!(C::AbstractMatrix, A::AbstractVecOrMat, B::AbstractVecOrMat) = generic_matmatmul!(C, 'C', 'C', A, B)
Ac_mul_Bt!(C::AbstractMatrix, A::AbstractVecOrMat, B::AbstractVecOrMat) = generic_matmatmul!(C, 'C', 'T', A, B)
# Supporting functions for matrix multiplication
function copytri!(A::AbstractMatrix, uplo::Char, conjugate::Bool=false)
n = checksquare(A)
if uplo == 'U'
for i = 1:(n-1), j = (i+1):n
A[j,i] = conjugate ? conj(A[i,j]) : A[i,j]
end
elseif uplo == 'L'
for i = 1:(n-1), j = (i+1):n
A[i,j] = conjugate ? conj(A[j,i]) : A[j,i]
end
else
throw(ArgumentError("uplo argument must be 'U' (upper) or 'L' (lower), got $uplo"))
end
A
end
function gemv!(y::StridedVector{T}, tA::Char, A::StridedVecOrMat{T}, x::StridedVector{T}) where T<:BlasFloat
mA, nA = lapack_size(tA, A)
if nA != length(x)
throw(DimensionMismatch("second dimension of A, $nA, does not match length of x, $(length(x))"))
end
if mA != length(y)
throw(DimensionMismatch("first dimension of A, $mA, does not match length of y, $(length(y))"))
end
if mA == 0
return y
end
if nA == 0
return fill!(y,0)
end
stride(A, 1) == 1 && stride(A, 2) >= size(A, 1) && return BLAS.gemv!(tA, one(T), A, x, zero(T), y)
return generic_matvecmul!(y, tA, A, x)
end
function syrk_wrapper!(C::StridedMatrix{T}, tA::Char, A::StridedVecOrMat{T}) where T<:BlasFloat
nC = checksquare(C)
if tA == 'T'
(nA, mA) = size(A,1), size(A,2)
tAt = 'N'
else
(mA, nA) = size(A,1), size(A,2)
tAt = 'T'
end
if nC != mA
throw(DimensionMismatch("output matrix has size: $(nC), but should have size $(mA)"))
end
if mA == 0 || nA == 0
return fill!(C,0)
end
if mA == 2 && nA == 2
return matmul2x2!(C,tA,tAt,A,A)
end
if mA == 3 && nA == 3
return matmul3x3!(C,tA,tAt,A,A)
end
if stride(A, 1) == stride(C, 1) == 1 && stride(A, 2) >= size(A, 1) && stride(C, 2) >= size(C, 1)
return copytri!(BLAS.syrk!('U', tA, one(T), A, zero(T), C), 'U')
end
return generic_matmatmul!(C, tA, tAt, A, A)
end
function herk_wrapper!(C::Union{StridedMatrix{T}, StridedMatrix{Complex{T}}}, tA::Char, A::Union{StridedVecOrMat{T}, StridedVecOrMat{Complex{T}}}) where T<:BlasReal
nC = checksquare(C)
if tA == 'C'
(nA, mA) = size(A,1), size(A,2)
tAt = 'N'
else
(mA, nA) = size(A,1), size(A,2)
tAt = 'C'
end
if nC != mA
throw(DimensionMismatch("output matrix has size: $(nC), but should have size $(mA)"))
end
if mA == 0 || nA == 0
return fill!(C,0)
end
if mA == 2 && nA == 2
return matmul2x2!(C,tA,tAt,A,A)
end
if mA == 3 && nA == 3
return matmul3x3!(C,tA,tAt,A,A)
end
# Result array does not need to be initialized as long as beta==0
# C = Matrix{T}(mA, mA)
if stride(A, 1) == stride(C, 1) == 1 && stride(A, 2) >= size(A, 1) && stride(C, 2) >= size(C, 1)
return copytri!(BLAS.herk!('U', tA, one(T), A, zero(T), C), 'U', true)
end
return generic_matmatmul!(C,tA, tAt, A, A)
end
function gemm_wrapper(tA::Char, tB::Char,
A::StridedVecOrMat{T},
B::StridedVecOrMat{T}) where T<:BlasFloat
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
C = similar(B, T, mA, nB)
gemm_wrapper!(C, tA, tB, A, B)
end
function gemm_wrapper!(C::StridedVecOrMat{T}, tA::Char, tB::Char,
A::StridedVecOrMat{T},
B::StridedVecOrMat{T}) where T<:BlasFloat
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
if nA != mB
throw(DimensionMismatch("A has dimensions ($mA,$nA) but B has dimensions ($mB,$nB)"))
end
if C === A || B === C
throw(ArgumentError("output matrix must not be aliased with input matrix"))
end
if mA == 0 || nA == 0 || nB == 0
if size(C) != (mA, nB)
throw(DimensionMismatch("C has dimensions $(size(C)), should have ($mA,$nB)"))
end
return fill!(C,0)
end
if mA == 2 && nA == 2 && nB == 2
return matmul2x2!(C,tA,tB,A,B)
end
if mA == 3 && nA == 3 && nB == 3
return matmul3x3!(C,tA,tB,A,B)
end
if stride(A, 1) == stride(B, 1) == stride(C, 1) == 1 && stride(A, 2) >= size(A, 1) && stride(B, 2) >= size(B, 1) && stride(C, 2) >= size(C, 1)
return BLAS.gemm!(tA, tB, one(T), A, B, zero(T), C)
end
generic_matmatmul!(C, tA, tB, A, B)
end
# blas.jl defines matmul for floats; other integer and mixed precision
# cases are handled here
lapack_size(t::Char, M::AbstractVecOrMat) = (size(M, t=='N' ? 1:2), size(M, t=='N' ? 2:1))
function copy!(B::AbstractVecOrMat, ir_dest::UnitRange{Int}, jr_dest::UnitRange{Int}, tM::Char, M::AbstractVecOrMat, ir_src::UnitRange{Int}, jr_src::UnitRange{Int})
if tM == 'N'
copy!(B, ir_dest, jr_dest, M, ir_src, jr_src)
else
Base.copy_transpose!(B, ir_dest, jr_dest, M, jr_src, ir_src)
tM == 'C' && conj!(B)
end
B
end
function copy_transpose!(B::AbstractMatrix, ir_dest::UnitRange{Int}, jr_dest::UnitRange{Int}, tM::Char, M::AbstractVecOrMat, ir_src::UnitRange{Int}, jr_src::UnitRange{Int})
if tM == 'N'
Base.copy_transpose!(B, ir_dest, jr_dest, M, ir_src, jr_src)
else
copy!(B, ir_dest, jr_dest, M, jr_src, ir_src)
tM == 'C' && conj!(B)
end
B
end
# TODO: It will be faster for large matrices to convert to float,
# call BLAS, and convert back to required type.
# NOTE: the generic version is also called as fallback for
# strides != 1 cases
function generic_matvecmul!(C::AbstractVector{R}, tA, A::AbstractVecOrMat, B::AbstractVector) where R
mB = length(B)
mA, nA = lapack_size(tA, A)
if mB != nA
throw(DimensionMismatch("matrix A has dimensions ($mA,$nA), vector B has length $mB"))
end
if mA != length(C)
throw(DimensionMismatch("result C has length $(length(C)), needs length $mA"))
end
Astride = size(A, 1)
if tA == 'T' # fastest case
for k = 1:mA
aoffs = (k-1)*Astride
if mB == 0
s = zero(R)
else
s = zero(A[aoffs + 1]*B[1] + A[aoffs + 1]*B[1])
end
for i = 1:nA
s += A[aoffs+i].'B[i]
end
C[k] = s
end
elseif tA == 'C'
for k = 1:mA
aoffs = (k-1)*Astride
if mB == 0
s = zero(R)
else
s = zero(A[aoffs + 1]*B[1] + A[aoffs + 1]*B[1])
end
for i = 1:nA
s += A[aoffs + i]'B[i]
end
C[k] = s
end
else # tA == 'N'
for i = 1:mA
if mB == 0
C[i] = zero(R)
else
C[i] = zero(A[i]*B[1] + A[i]*B[1])
end
end
for k = 1:mB
aoffs = (k-1)*Astride
b = B[k]
for i = 1:mA
C[i] += A[aoffs + i] * b
end
end
end
C
end
function generic_matmatmul(tA, tB, A::AbstractVecOrMat{T}, B::AbstractMatrix{S}) where {T,S}
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
C = similar(B, promote_op(matprod, T, S), mA, nB)
generic_matmatmul!(C, tA, tB, A, B)
end
const tilebufsize = 10800 # Approximately 32k/3
const Abuf = Vector{UInt8}(tilebufsize)
const Bbuf = Vector{UInt8}(tilebufsize)
const Cbuf = Vector{UInt8}(tilebufsize)
function generic_matmatmul!(C::AbstractMatrix, tA, tB, A::AbstractMatrix, B::AbstractMatrix)
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
mC, nC = size(C)
if mA == nA == mB == nB == mC == nC == 2
return matmul2x2!(C, tA, tB, A, B)
end
if mA == nA == mB == nB == mC == nC == 3
return matmul3x3!(C, tA, tB, A, B)
end
_generic_matmatmul!(C, tA, tB, A, B)
end
generic_matmatmul!(C::AbstractVecOrMat, tA, tB, A::AbstractVecOrMat, B::AbstractVecOrMat) = _generic_matmatmul!(C, tA, tB, A, B)
function _generic_matmatmul!(C::AbstractVecOrMat{R}, tA, tB, A::AbstractVecOrMat{T}, B::AbstractVecOrMat{S}) where {T,S,R}
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
if mB != nA
throw(DimensionMismatch("matrix A has dimensions ($mA,$nA), matrix B has dimensions ($mB,$nB)"))
end
if size(C,1) != mA || size(C,2) != nB
throw(DimensionMismatch("result C has dimensions $(size(C)), needs ($mA,$nB)"))
end
if isempty(A) || isempty(B)
return fill!(C, zero(R))
end
tile_size = 0
if isbits(R) && isbits(T) && isbits(S) && (tA == 'N' || tB != 'N')
tile_size = floor(Int, sqrt(tilebufsize / max(sizeof(R), sizeof(S), sizeof(T))))
end
@inbounds begin
if tile_size > 0
sz = (tile_size, tile_size)
Atile = unsafe_wrap(Array, convert(Ptr{T}, pointer(Abuf)), sz)
Btile = unsafe_wrap(Array, convert(Ptr{S}, pointer(Bbuf)), sz)
z1 = zero(A[1, 1]*B[1, 1] + A[1, 1]*B[1, 1])
z = convert(promote_type(typeof(z1), R), z1)
if mA < tile_size && nA < tile_size && nB < tile_size
Base.copy_transpose!(Atile, 1:nA, 1:mA, tA, A, 1:mA, 1:nA)
copy!(Btile, 1:mB, 1:nB, tB, B, 1:mB, 1:nB)
for j = 1:nB
boff = (j-1)*tile_size
for i = 1:mA
aoff = (i-1)*tile_size
s = z
for k = 1:nA
s += Atile[aoff+k] * Btile[boff+k]
end
C[i,j] = s
end
end
else
Ctile = unsafe_wrap(Array, convert(Ptr{R}, pointer(Cbuf)), sz)
for jb = 1:tile_size:nB
jlim = min(jb+tile_size-1,nB)
jlen = jlim-jb+1
for ib = 1:tile_size:mA
ilim = min(ib+tile_size-1,mA)
ilen = ilim-ib+1
fill!(Ctile, z)
for kb = 1:tile_size:nA
klim = min(kb+tile_size-1,mB)
klen = klim-kb+1
Base.copy_transpose!(Atile, 1:klen, 1:ilen, tA, A, ib:ilim, kb:klim)
copy!(Btile, 1:klen, 1:jlen, tB, B, kb:klim, jb:jlim)
for j=1:jlen
bcoff = (j-1)*tile_size
for i = 1:ilen
aoff = (i-1)*tile_size
s = z
for k = 1:klen
s += Atile[aoff+k] * Btile[bcoff+k]
end
Ctile[bcoff+i] += s
end
end
end
copy!(C, ib:ilim, jb:jlim, Ctile, 1:ilen, 1:jlen)
end
end
end
else
# Multiplication for non-plain-data uses the naive algorithm
if tA == 'N'
if tB == 'N'
for i = 1:mA, j = 1:nB
z2 = zero(A[i, 1]*B[1, j] + A[i, 1]*B[1, j])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[i, k]*B[k, j]
end
C[i,j] = Ctmp
end
elseif tB == 'T'
for i = 1:mA, j = 1:nB
z2 = zero(A[i, 1]*B[j, 1] + A[i, 1]*B[j, 1])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[i, k]*B[j, k].'
end
C[i,j] = Ctmp
end
else
for i = 1:mA, j = 1:nB
z2 = zero(A[i, 1]*B[j, 1] + A[i, 1]*B[j, 1])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[i, k]*B[j, k]'
end
C[i,j] = Ctmp
end
end
elseif tA == 'T'
if tB == 'N'
for i = 1:mA, j = 1:nB
z2 = zero(A[1, i]*B[1, j] + A[1, i]*B[1, j])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[k, i].'B[k, j]
end
C[i,j] = Ctmp
end
elseif tB == 'T'
for i = 1:mA, j = 1:nB
z2 = zero(A[1, i]*B[j, 1] + A[1, i]*B[j, 1])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[k, i].'B[j, k].'
end
C[i,j] = Ctmp
end
else
for i = 1:mA, j = 1:nB
z2 = zero(A[1, i]*B[j, 1] + A[1, i]*B[j, 1])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[k, i].'B[j, k]'
end
C[i,j] = Ctmp
end
end
else
if tB == 'N'
for i = 1:mA, j = 1:nB
z2 = zero(A[1, i]*B[1, j] + A[1, i]*B[1, j])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[k, i]'B[k, j]
end
C[i,j] = Ctmp
end
elseif tB == 'T'
for i = 1:mA, j = 1:nB
z2 = zero(A[1, i]*B[j, 1] + A[1, i]*B[j, 1])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[k, i]'B[j, k].'
end
C[i,j] = Ctmp
end
else
for i = 1:mA, j = 1:nB
z2 = zero(A[1, i]*B[j, 1] + A[1, i]*B[j, 1])
Ctmp = convert(promote_type(R, typeof(z2)), z2)
for k = 1:nA
Ctmp += A[k, i]'B[j, k]'
end
C[i,j] = Ctmp
end
end
end
end
end # @inbounds
C
end
# multiply 2x2 matrices
function matmul2x2(tA, tB, A::AbstractMatrix{T}, B::AbstractMatrix{S}) where {T,S}
matmul2x2!(similar(B, promote_op(matprod, T, S), 2, 2), tA, tB, A, B)
end
function matmul2x2!(C::AbstractMatrix, tA, tB, A::AbstractMatrix, B::AbstractMatrix)
if !(size(A) == size(B) == size(C) == (2,2))
throw(DimensionMismatch("A has size $(size(A)), B has size $(size(B)), C has size $(size(C))"))
end
@inbounds begin
if tA == 'T'
A11 = transpose(A[1,1]); A12 = transpose(A[2,1]); A21 = transpose(A[1,2]); A22 = transpose(A[2,2])
elseif tA == 'C'
A11 = ctranspose(A[1,1]); A12 = ctranspose(A[2,1]); A21 = ctranspose(A[1,2]); A22 = ctranspose(A[2,2])
else
A11 = A[1,1]; A12 = A[1,2]; A21 = A[2,1]; A22 = A[2,2]
end
if tB == 'T'
B11 = transpose(B[1,1]); B12 = transpose(B[2,1]); B21 = transpose(B[1,2]); B22 = transpose(B[2,2])
elseif tB == 'C'
B11 = ctranspose(B[1,1]); B12 = ctranspose(B[2,1]); B21 = ctranspose(B[1,2]); B22 = ctranspose(B[2,2])
else
B11 = B[1,1]; B12 = B[1,2]; B21 = B[2,1]; B22 = B[2,2]
end
C[1,1] = A11*B11 + A12*B21
C[1,2] = A11*B12 + A12*B22
C[2,1] = A21*B11 + A22*B21
C[2,2] = A21*B12 + A22*B22
end # inbounds
C
end
# Multiply 3x3 matrices
function matmul3x3(tA, tB, A::AbstractMatrix{T}, B::AbstractMatrix{S}) where {T,S}
matmul3x3!(similar(B, promote_op(matprod, T, S), 3, 3), tA, tB, A, B)
end
function matmul3x3!(C::AbstractMatrix, tA, tB, A::AbstractMatrix, B::AbstractMatrix)
if !(size(A) == size(B) == size(C) == (3,3))
throw(DimensionMismatch("A has size $(size(A)), B has size $(size(B)), C has size $(size(C))"))
end
@inbounds begin
if tA == 'T'
A11 = transpose(A[1,1]); A12 = transpose(A[2,1]); A13 = transpose(A[3,1])
A21 = transpose(A[1,2]); A22 = transpose(A[2,2]); A23 = transpose(A[3,2])
A31 = transpose(A[1,3]); A32 = transpose(A[2,3]); A33 = transpose(A[3,3])
elseif tA == 'C'
A11 = ctranspose(A[1,1]); A12 = ctranspose(A[2,1]); A13 = ctranspose(A[3,1])
A21 = ctranspose(A[1,2]); A22 = ctranspose(A[2,2]); A23 = ctranspose(A[3,2])
A31 = ctranspose(A[1,3]); A32 = ctranspose(A[2,3]); A33 = ctranspose(A[3,3])
else
A11 = A[1,1]; A12 = A[1,2]; A13 = A[1,3]
A21 = A[2,1]; A22 = A[2,2]; A23 = A[2,3]
A31 = A[3,1]; A32 = A[3,2]; A33 = A[3,3]
end
if tB == 'T'
B11 = transpose(B[1,1]); B12 = transpose(B[2,1]); B13 = transpose(B[3,1])
B21 = transpose(B[1,2]); B22 = transpose(B[2,2]); B23 = transpose(B[3,2])
B31 = transpose(B[1,3]); B32 = transpose(B[2,3]); B33 = transpose(B[3,3])
elseif tB == 'C'
B11 = ctranspose(B[1,1]); B12 = ctranspose(B[2,1]); B13 = ctranspose(B[3,1])
B21 = ctranspose(B[1,2]); B22 = ctranspose(B[2,2]); B23 = ctranspose(B[3,2])
B31 = ctranspose(B[1,3]); B32 = ctranspose(B[2,3]); B33 = ctranspose(B[3,3])
else
B11 = B[1,1]; B12 = B[1,2]; B13 = B[1,3]
B21 = B[2,1]; B22 = B[2,2]; B23 = B[2,3]
B31 = B[3,1]; B32 = B[3,2]; B33 = B[3,3]
end
C[1,1] = A11*B11 + A12*B21 + A13*B31
C[1,2] = A11*B12 + A12*B22 + A13*B32
C[1,3] = A11*B13 + A12*B23 + A13*B33
C[2,1] = A21*B11 + A22*B21 + A23*B31
C[2,2] = A21*B12 + A22*B22 + A23*B32
C[2,3] = A21*B13 + A22*B23 + A23*B33
C[3,1] = A31*B11 + A32*B21 + A33*B31
C[3,2] = A31*B12 + A32*B22 + A33*B32
C[3,3] = A31*B13 + A32*B23 + A33*B33
end # inbounds
C
end