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matmul.jl
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matmul.jl
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# matmul.jl: Everything to do with dense matrix multiplication
# multiply by diagonal matrix as vector
function diagmm!(C::Matrix, A::Matrix, b::Vector)
m, n = size(A)
if n != length(b)
error("argument dimensions do not match")
end
for j = 1:n
bj = b[j]
for i = 1:m
C[i,j] = A[i,j]*bj
end
end
return C
end
function diagmm!(C::Matrix, b::Vector, A::Matrix)
m, n = size(A)
if m != length(b)
error("argument dimensions do not match")
end
for j=1:n
for i=1:m
C[i,j] = A[i,j]*b[i]
end
end
return C
end
diagmm(A::Matrix, b::Vector) =
diagmm!(Array(promote_type(eltype(A),eltype(b)),size(A)), A, b)
diagmm(b::Vector, A::Matrix) =
diagmm!(Array(promote_type(eltype(A),eltype(b)),size(A)), b, A)
# Dot products
function dot{T<:Union(Vector{Float64}, Vector{Float32})}(x::T, y::T)
length(x) != length(y) ? error("Inputs should be of same length") : true
Blas.dot(length(x), x, 1, y, 1)
end
function dot{T<:Union(Float64, Float32), TI<:Integer}(x::Vector{T}, rx::Union(Range1{TI},Range{TI}), y::Vector{T}, ry::Union(Range1{TI},Range{TI}))
length(rx) != length(ry) ? error("Ranges should be of same length") : true
if min(rx) < 1 || max(rx) > length(x) || min(ry) < 1 || max(ry) > length(y)
throw(BoundsError())
end
Blas.dot(length(rx), pointer(x)+(first(rx)-1)*sizeof(T), step(rx), pointer(y)+(first(ry)-1)*sizeof(T), step(ry))
end
Ac_mul_B(x::Vector, y::Vector) = [dot(x, y)]
At_mul_B{T<:Real}(x::Vector{T}, y::Vector{T}) = [dot(x, y)]
function dot(x::Vector, y::Vector)
s = zero(eltype(x))
for i=1:length(x)
s += conj(x[i])*y[i]
end
s
end
# Matrix-vector multiplication
function (*){T<:LapackType}(A::StridedMatrix{T},
X::StridedVector{T})
Y = similar(A, size(A,1))
gemv(Y, 'N', A, X)
end
A_mul_B{T<:LapackType}(y::StridedVector{T}, A::StridedMatrix{T}, x::StridedVector{T}) = gemv(y, 'N', A, x)
A_mul_B(y::StridedVector, A::StridedMatrix, x::StridedVector) = generic_matvecmul(y, 'N', A, x)
function At_mul_B{T<:LapackType}(A::StridedMatrix{T}, x::StridedVector{T})
y = similar(A, size(A, 2))
gemv(y, 'T', A, x)
end
At_mul_B{T<:LapackType}(y::StridedVector{T}, A::StridedMatrix{T}, x::StridedVector{T}) = gemv(y, 'T', A, x)
At_mul_B(y::StridedVector, A::StridedMatrix, x::StridedVector) = generic_matvecmul(y, 'T', A, x)
# Matrix-matrix multiplication
(*){T<:LapackType}(A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper('N', 'N', A, B)
A_mul_B{T<:LapackType}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper(C, 'N', 'N', A, B)
A_mul_B{T,S,R}(C::StridedMatrix{R}, A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul(C, 'N', 'N', A, B)
function At_mul_B{T<:LapackType}(A::StridedMatrix{T},
B::StridedMatrix{T})
if is(A, B) && size(A,1)>=500
syrk_wrapper('T', A)
else
gemm_wrapper('T', 'N', A, B)
end
end
At_mul_B{T<:LapackType}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper(C, 'T', 'N', A, B)
At_mul_B{T,S}(A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul('T', 'N', A, B)
At_mul_B{T,S,R}(C::StridedMatrix{R}, A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul(C, 'T', 'N', A, B)
function A_mul_Bt{T<:LapackType}(A::StridedMatrix{T},
B::StridedMatrix{T})
if is(A, B) && size(A,2)>=500
syrk_wrapper('N', A)
else
gemm_wrapper('N', 'T', A, B)
end
end
A_mul_Bt{T<:LapackType}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper(C, 'N', 'T', A, B)
A_mul_Bt{T,S}(A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul('N', 'T', A, B)
A_mul_Bt{T,S,R}(C::StridedMatrix{R}, A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul(C, 'N', 'T', A, B)
At_mul_Bt{T<:LapackType}(A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper('T', 'T', A, B)
At_mul_Bt{T<:LapackType}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper(C, 'T', 'T', A, B)
At_mul_Bt{T,S}(A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul('T', 'T', A, B)
At_mul_Bt{T,S,R}(C::StridedMatrix{R}, A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul(C, 'T', 'T', A, B)
Ac_mul_B{T<:Union(Float64,Float32)}(A::StridedMatrix{T}, B::StridedMatrix{T}) = At_mul_B(A, B)
Ac_mul_B{T<:Union(Float64,Float32)}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = At_mul_B(C, A, B)
function Ac_mul_B{T<:Union(Complex128,Complex64)}(A::StridedMatrix{T},
B::StridedMatrix{T})
if is(A, B) && size(A,1)>=500
herk_wrapper('C', A)
else
gemm_wrapper('C', 'N', A, B)
end
end
Ac_mul_B{T<:Union(Complex128,Complex64)}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper('C', 'N', A, B)
Ac_mul_B{T,S}(A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul('C', 'N', A, B)
Ac_mul_B{T,S,R}(C::StridedMatrix{R}, A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul(C, 'C', 'N', A, B)
A_mul_Bc{T<:Union(Float64,Float32)}(A::StridedMatrix{T}, B::StridedMatrix{T}) = A_mul_Bt(A, B)
A_mul_Bc{T<:Union(Float64,Float32)}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = A_mul_Bt(C, A, B)
function A_mul_Bc{T<:Union(Complex128,Complex64)}(A::StridedMatrix{T},
B::StridedMatrix{T})
if is(A, B) && size(A,2)>=500
herk_wrapper('N', A)
else
gemm_wrapper('N', 'C', A, B)
end
end
A_mul_Bc{T<:Union(Complex128,Complex64)}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper(C, 'N', 'C', A, B)
A_mul_Bc{T,S}(A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul('N', 'C', A, B)
A_mul_Bc{T,S,R}(C::StridedMatrix{R}, A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul(C, 'N', 'C', A, B)
Ac_mul_Bc{T<:LapackType}(A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper('C', 'C', A, B)
Ac_mul_Bc{T<:LapackType}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::StridedMatrix{T}) = gemm_wrapper(C, 'C', 'C', A, B)
Ac_mul_Bt{T,S}(A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul('C', 'C', A, B)
Ac_mul_Bt{T,S,R}(C::StridedMatrix{R}, A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul(C, 'C', 'C', A, B)
# Supporting functions for matrix multiplication
function symmetrize!(A::StridedMatrix, upper::Bool)
m, n = size(A)
if m != n error("symmetrize: Matrix must be square") end
if upper
for i = 1:(n-1)
for j = (i+1):n
A[j,i] = A[i,j]
end
end
else
for i = 1:(n-1)
for j = (i+1):n
A[i,j] = A[j,i]
end
end
end
return A
end
symmetrize!(A) = symmetrize!(A, true)
function symmetrize_conj!(A::StridedMatrix, upper::Bool)
m, n = size(A)
if m != n error("symmetrize: Matrix must be square") end
if upper
for i = 1:(n-1)
for j = (i+1):n
A[j,i] = conj(A[i,j])
end
end
else
for i = 1:(n-1)
for j = (i+1):n
A[i,j] = conj(A[j,i])
end
end
end
return A
end
symmetrize_conj!(A) = symmetrize_conj!(A, true)
function gemv{T<:LapackType}(y::StridedVector{T},
tA,
A::StridedMatrix{T},
x::StridedVector{T})
if stride(A, 1) != 1
return generic_matvecmul(y, tA, A, x)
end
if tA != 'N'
(nA, mA) = size(A)
else
(mA, nA) = size(A)
end
if nA != length(x); error("*: argument shapes do not match"); end
if mA != length(y); error("*: output size is incorrect"); end
Blas.gemv!(tA, one(T), A, x, zero(T), y)
end
function syrk_wrapper{T<:LapackType}(tA, A::StridedMatrix{T})
if tA == 'T'
(nA, mA) = size(A)
tAt = 'N'
else
(mA, nA) = size(A)
tAt = 'T'
end
if mA == 2 && nA == 2; return matmul2x2(tA,tAt,A,A); end
if mA == 3 && nA == 3; return matmul3x3(tA,tAt,A,A); end
if stride(A, 1) != 1
return generic_matmatmul(tA, tAt, A, A)
end
symmetrize!(Blas.syrk('U', tA, one(T), A))
end
function herk_wrapper{T<:LapackType}(tA, A::StridedMatrix{T})
if tA == 'C'
(nA, mA) = size(A)
tAt = 'N'
else
(mA, nA) = size(A)
tAt = 'C'
end
if mA == 2 && nA == 2; return matmul2x2(tA,tAt,A,A); end
if mA == 3 && nA == 3; return matmul3x3(tA,tAt,A,A); end
if stride(A, 1) != 1
return generic_matmatmul(tA, tAt, A, A)
end
# Result array does not need to be initialized as long as beta==0
# C = Array(T, mA, mA)
symmetrize_conj!(Blas.herk('U', tA, one(T), A))
end
function gemm_wrapper{T<:LapackType}(tA, tB,
A::StridedMatrix{T},
B::StridedMatrix{T})
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
C = Array(T, mA, nB)
gemm_wrapper(C, tA, tB, A, B)
end
function gemm_wrapper{T<:LapackType}(C::StridedMatrix{T}, tA, tB,
A::StridedMatrix{T},
B::StridedMatrix{T})
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
if nA != mB; error("*: argument shapes do not match"); end
if mA == 0 || nA == 0 || nB == 0; return zeros(T, mA, nB); 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) != 1 || stride(B, 1) != 1
return generic_matmatmul(C, tA, tB, A, B)
end
Blas.gemm!(tA, tB, one(T), A, B, zero(T), C)
end
# blas.jl defines matmul for floats; other integer and mixed precision
# cases are handled here
lapack_size(t::Char, M::StridedVecOrMat) = (t == 'N') ? (size(M, 1), size(M, 2)) : (size(M,2), size(M, 1))
function copy_to{R,S}(B::Matrix{R}, ir_dest::Range1{Int}, jr_dest::Range1{Int}, tM::Char, M::StridedMatrix{S}, ir_src::Range1{Int}, jr_src::Range1{Int})
if tM == 'N'
copy_to(B, ir_dest, jr_dest, M, ir_src, jr_src)
else
copy_to_transpose(B, ir_dest, jr_dest, M, jr_src, ir_src)
if tM == 'C'
conj!(B)
end
end
end
function copy_to_transpose{R,S}(B::Matrix{R}, ir_dest::Range1{Int}, jr_dest::Range1{Int}, tM::Char, M::StridedMatrix{S}, ir_src::Range1{Int}, jr_src::Range1{Int})
if tM == 'N'
copy_to_transpose(B, ir_dest, jr_dest, M, ir_src, jr_src)
else
copy_to(B, ir_dest, jr_dest, M, jr_src, ir_src)
if tM == 'C'
conj!(B)
end
end
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 in libalg_blas.jl
(*){T,S}(A::StridedMatrix{T}, B::StridedVector{S}) = generic_matvecmul('N', A, B)
function generic_matvecmul{T,S}(tA, A::StridedMatrix{T}, B::StridedVector{S})
if tA == 'N'
C = Array(promote_type(T,S), size(A, 1))
else
C = Array(promote_type(T,S), size(A, 2))
end
generic_matvecmul(C, tA, A, B)
end
function generic_matvecmul{T,S,R}(C::StridedVector{R}, tA, A::StridedMatrix{T}, B::StridedVector{S})
mB = length(B)
mA, nA = lapack_size(tA, A)
if nA != mB; error("*: argument shapes do not match"); end
if length(C) != mA; error("*: output size does not match"); end
z = zero(R)
Astride = size(A, 1)
if tA == 'T' # fastest case
for k = 1:mA
aoffs = (k-1)*Astride
s = z
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
s = z
for i = 1:nA
s += conj(A[aoffs+i]) * B[i]
end
C[k] = s
end
else # tA == 'N'
fill!(C, z)
for k = 1:mB
aoffs = (k-1)*Astride
b = B[k]
for i = 1:mA
C[i] += A[aoffs+i] * b
end
end
end
return C
end
(*){T,S}(A::Vector{S}, B::Matrix{T}) = reshape(A,length(A),1)*B
# NOTE: the generic version is also called as fallback for strides != 1 cases
# in libalg_blas.jl
(*){T,S}(A::StridedMatrix{T}, B::StridedMatrix{S}) = generic_matmatmul('N', 'N', A, B)
function generic_matmatmul{T,S}(tA, tB, A::StridedMatrix{T}, B::StridedMatrix{S})
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
C = Array(promote_type(T,S), mA, nB)
generic_matmatmul(C, tA, tB, A, B)
end
const tilebufsize = 10800 # Approximately 32k/3
const Abuf = Array(Uint8, tilebufsize)
const Bbuf = Array(Uint8, tilebufsize)
const Cbuf = Array(Uint8, tilebufsize)
function generic_matmatmul{T,S,R}(C::StridedMatrix{R}, tA, tB, A::StridedMatrix{T}, B::StridedMatrix{S})
mA, nA = lapack_size(tA, A)
mB, nB = lapack_size(tB, B)
if nA != mB; error("*: argument shapes do not match"); end
if size(C,1) != mA || size(C,2) != nB; error("*: output size is incorrect"); end
if mA == nA == nB == 2; return matmul2x2(C, tA, tB, A, B); end
if mA == nA == nB == 3; return matmul3x3(C, tA, tB, A, B); end
if isa(R, BitsKind)
tile_size = int(ifloor(sqrt(tilebufsize/sizeof(R))))
sz = (tile_size, tile_size)
Atile = pointer_to_array(convert(Ptr{R}, pointer(Abuf)), sz)
Btile = pointer_to_array(convert(Ptr{R}, pointer(Bbuf)), sz)
z = zero(R)
if mA < tile_size && nA < tile_size && nB < tile_size
copy_to_transpose(Atile, 1:nA, 1:mA, tA, A, 1:mA, 1:nA)
copy_to(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 = pointer_to_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
copy_to_transpose(Atile, 1:klen, 1:ilen, tA, A, ib:ilim, kb:klim)
copy_to(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_to(C, ib:ilim, jb:jlim, Ctile, 1:ilen, 1:jlen)
end
end
end
else
# Multiplication for non-BitsKind uses the naive algorithm
if tA == 'N'
if tB == 'N'
for i = 1:mA
for j = 1:nB
Ctmp = A[i, 1]*B[1, j]
for k = 2:nA
Ctmp += A[i, k]*B[k, j]
end
C[i,j] = Ctmp
end
end
elseif tB == 'T'
for i = 1:mA
for j = 1:nB
Ctmp = A[i, 1]*B[j, 1]
for k = 2:nA
Ctmp += A[i, k]*B[j, k]
end
C[i,j] = Ctmp
end
end
else
for i = 1:mA
for j = 1:nB
Ctmp = A[i, 1]*conj(B[j, 1])
for k = 2:nA
Ctmp += A[i, k]*conj(B[j, k])
end
C[i,j] = Ctmp
end
end
end
elseif tA == 'T'
if tB == 'N'
for i = 1:mA
for j = 1:nB
Ctmp = A[1, i]*B[1, j]
for k = 2:nA
Ctmp += A[k, i]*B[k, j]
end
C[i,j] = Ctmp
end
end
elseif tB == 'T'
for i = 1:mA
for j = 1:nB
Ctmp = A[1, i]*B[j, 1]
for k = 2:nA
Ctmp += A[k, i]*B[j, k]
end
C[i,j] = Ctmp
end
end
else
for i = 1:mA
for j = 1:nB
Ctmp = A[1, i]*conj(B[j, 1])
for k = 2:nA
Ctmp += A[k, i]*conj(B[j, k])
end
C[i,j] = Ctmp
end
end
end
else
if tB == 'N'
for i = 1:mA
for j = 1:nB
Ctmp = conj(A[1, i])*B[1, j]
for k = 2:nA
Ctmp += conj(A[k, i])*B[k, j]
end
C[i,j] = Ctmp
end
end
elseif tB == 'T'
for i = 1:mA
for j = 1:nB
Ctmp = conj(A[1, i])*B[j, 1]
for k = 2:nA
Ctmp += conj(A[k, i])*B[j, k]
end
C[i,j] = Ctmp
end
end
else
for i = 1:mA
for j = 1:nB
Ctmp = conj(A[1, i])*conj(B[j, 1])
for k = 2:nA
Ctmp += conj(A[k, i])*conj(B[j, k])
end
C[i,j] = Ctmp
end
end
end
end
end
return C
end
# multiply 2x2 matrices
function matmul2x2{T,S}(tA, tB, A::StridedMatrix{T}, B::StridedMatrix{S})
R = promote_type(T,S)
C = Array(R, 2, 2)
matmul2x2(C, tA, tB, A, B)
end
function matmul2x2{T,S,R}(C::StridedMatrix{R}, tA, tB, A::StridedMatrix{T}, B::StridedMatrix{S})
if tA == 'T'
A11 = A[1,1]; A12 = A[2,1]; A21 = A[1,2]; A22 = A[2,2]
elseif tA == 'C'
A11 = conj(A[1,1]); A12 = conj(A[2,1]); A21 = conj(A[1,2]); A22 = conj(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 = B[1,1]; B12 = B[2,1]; B21 = B[1,2]; B22 = B[2,2]
elseif tB == 'C'
B11 = conj(B[1,1]); B12 = conj(B[2,1]); B21 = conj(B[1,2]); B22 = conj(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
return C
end
# Multiply 3x3 matrices
function matmul3x3{T,S}(tA, tB, A::StridedMatrix{T}, B::StridedMatrix{S})
R = promote_type(T,S)
C = Array(R, 3, 3)
matmul3x3(C, tA, tB, A, B)
end
function matmul3x3{T,S,R}(C::StridedMatrix{R}, tA, tB, A::StridedMatrix{T}, B::StridedMatrix{S})
if tA == 'T'
A11 = A[1,1]; A12 = A[2,1]; A13 = A[3,1];
A21 = A[1,2]; A22 = A[2,2]; A23 = A[3,2];
A31 = A[1,3]; A32 = A[2,3]; A33 = A[3,3];
elseif tA == 'C'
A11 = conj(A[1,1]); A12 = conj(A[2,1]); A13 = conj(A[3,1]);
A21 = conj(A[1,2]); A22 = conj(A[2,2]); A23 = conj(A[3,2]);
A31 = conj(A[1,3]); A32 = conj(A[2,3]); A33 = conj(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 = B[1,1]; B12 = B[2,1]; B13 = B[3,1];
B21 = B[1,2]; B22 = B[2,2]; B23 = B[3,2];
B31 = B[1,3]; B32 = B[2,3]; B33 = B[3,3];
elseif tB == 'C'
B11 = conj(B[1,1]); B12 = conj(B[2,1]); B13 = conj(B[3,1]);
B21 = conj(B[1,2]); B22 = conj(B[2,2]); B23 = conj(B[3,2]);
B31 = conj(B[1,3]); B32 = conj(B[2,3]); B33 = conj(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
return C
end