Add 3-arg * methods (JuliaLang#37898)

This addresses the simplest part of JuliaLang#12065 (optimizing * for optimal matrix order), by adding some methods for * with 3 arguments, where this can be done more efficiently than working left-to-right.

Co-authored-by: Daniel Karrasch <[email protected]>

stdlib/LinearAlgebra/src/matmul.jl

@@ -1081,3 +1081,141 @@ function matmul3x3!(C::AbstractMatrix, tA, tB, A::AbstractMatrix, B::AbstractMat
  end # inbounds
  C
 end
+
+const RealOrComplex = Union{Real,Complex}
+
+# Three-argument *
+"""
+ *(A, B::AbstractMatrix, C)
+ A * B * C * D
+
+Chained multiplication of 3 or 4 matrices is done in the most efficient sequence,
+based on the sizes of the arrays. That is, the number of scalar multiplications needed
+for `(A * B) * C` (with 3 dense matrices) is compared to that for `A * (B * C)`
+to choose which of these to execute.
+
+If the last factor is a vector, or the first a transposed vector, then it is efficient
+to deal with these first. In particular `x' * B * y` means `(x' * B) * y`
+for an ordinary column-major `B::Matrix`. Unlike `dot(x, B, y)`, this
+allocates an intermediate array.
+
+If the first or last factor is a number, this will be fused with the matrix
+multiplication, using 5-arg [`mul!`](@ref).
+
+See also [`muladd`](@ref), [`dot`](@ref).
+
+!!! compat "Julia 1.7"
+ These optimisations require at least Julia 1.7.
+"""
+*(A::AbstractMatrix, B::AbstractMatrix, x::AbstractVector) = A * (B*x)
+
+*(tu::AdjOrTransAbsVec, B::AbstractMatrix, v::AbstractVector) = (tu*B) * v
+*(tu::AdjOrTransAbsVec, B::AdjOrTransAbsMat, v::AbstractVector) = tu * (B*v)
+
+*(A::AbstractMatrix, x::AbstractVector, γ::Number) = mat_vec_scalar(A,x,γ)
+*(A::AbstractMatrix, B::AbstractMatrix, γ::Number) = mat_mat_scalar(A,B,γ)
+*(α::RealOrComplex, B::AbstractMatrix{<:RealOrComplex}, C::AbstractVector{<:RealOrComplex}) =
+ mat_vec_scalar(B,C,α)
+*(α::RealOrComplex, B::AbstractMatrix{<:RealOrComplex}, C::AbstractMatrix{<:RealOrComplex}) =
+ mat_mat_scalar(B,C,α)
+
+*(α::Number, u::AbstractVector, tv::AdjOrTransAbsVec) = broadcast(*, α, u, tv)
+*(u::AbstractVector, tv::AdjOrTransAbsVec, γ::Number) = broadcast(*, u, tv, γ)
+*(u::AbstractVector, tv::AdjOrTransAbsVec, C::AbstractMatrix) = u * (tv*C)
+
+*(A::AbstractMatrix, B::AbstractMatrix, C::AbstractMatrix) = _tri_matmul(A,B,C)
+*(tv::AdjOrTransAbsVec, B::AbstractMatrix, C::AbstractMatrix) = (tv*B) * C
+
+function _tri_matmul(A,B,C,δ=nothing)
+ n,m = size(A)
+ # m,k == size(B)
+ k,l = size(C)
+ costAB_C = n*m*k + n*k*l # multiplications, allocations n*k + n*l
+ costA_BC = m*k*l + n*m*l # m*l + n*l
+ if costA_BC < costAB_C
+ isnothing(δ) ? A * (B*C) : A * mat_mat_scalar(B,C,δ)
+ else
+ isnothing(δ) ? (A*B) * C : mat_mat_scalar(A*B, C, δ)
+ end
+end
+
+# Fast path for two arrays * one scalar is opt-in, via mat_vec_scalar and mat_mat_scalar.
+
+mat_vec_scalar(A, x, γ) = A * (x .* γ) # fallback
+mat_vec_scalar(A::StridedMaybeAdjOrTransMat, x::StridedVector, γ) = _mat_vec_scalar(A, x, γ)
+mat_vec_scalar(A::AdjOrTransAbsVec, x::StridedVector, γ) = (A * x) * γ
+
+function _mat_vec_scalar(A, x, γ)
+ T = promote_type(eltype(A), eltype(x), typeof(γ))
+ C = similar(A, T, axes(A,1))
+ mul!(C, A, x, γ, false)
+end
+
+mat_mat_scalar(A, B, γ) = (A*B) .* γ # fallback
+mat_mat_scalar(A::StridedMaybeAdjOrTransMat, B::StridedMaybeAdjOrTransMat, γ) =
+ _mat_mat_scalar(A, B, γ)
+
+function _mat_mat_scalar(A, B, γ)
+ T = promote_type(eltype(A), eltype(B), typeof(γ))
+ C = similar(A, T, axes(A,1), axes(B,2))
+ mul!(C, A, B, γ, false)
+end
+
+mat_mat_scalar(A::AdjointAbsVec, B, γ) = (γ' .* (A * B)')' # preserving order, adjoint reverses
+mat_mat_scalar(A::AdjointAbsVec{<:RealOrComplex}, B::StridedMaybeAdjOrTransMat{<:RealOrComplex}, γ::RealOrComplex) =
+ mat_vec_scalar(B', A', γ')'
+
+mat_mat_scalar(A::TransposeAbsVec, B, γ) = transpose(γ .* transpose(A * B))
+mat_mat_scalar(A::TransposeAbsVec{<:RealOrComplex}, B::StridedMaybeAdjOrTransMat{<:RealOrComplex}, γ::RealOrComplex) =
+ transpose(mat_vec_scalar(transpose(B), transpose(A), γ))
+
+
+# Four-argument *, by type
+*(α::Number, β::Number, C::AbstractMatrix, x::AbstractVector) = (α*β) * C * x
+*(α::Number, β::Number, C::AbstractMatrix, D::AbstractMatrix) = (α*β) * C * D
+*(α::Number, B::AbstractMatrix, C::AbstractMatrix, x::AbstractVector) = α * B * (C*x)
+*(α::Number, vt::AdjOrTransAbsVec, C::AbstractMatrix, x::AbstractVector) = α * (vt*C*x)
+*(α::RealOrComplex, vt::AdjOrTransAbsVec{<:RealOrComplex}, C::AbstractMatrix{<:RealOrComplex}, D::AbstractMatrix{<:RealOrComplex}) =
+ (α*vt*C) * D # solves an ambiguity
+
+*(A::AbstractMatrix, x::AbstractVector, γ::Number, δ::Number) = A * x * (γ*δ)
+*(A::AbstractMatrix, B::AbstractMatrix, γ::Number, δ::Number) = A * B * (γ*δ)
+*(A::AbstractMatrix, B::AbstractMatrix, x::AbstractVector, δ::Number, ) = A * (B*x*δ)
+*(vt::AdjOrTransAbsVec, B::AbstractMatrix, x::AbstractVector, δ::Number) = (vt*B*x) * δ
+*(vt::AdjOrTransAbsVec, B::AbstractMatrix, C::AbstractMatrix, δ::Number) = (vt*B) * C * δ
+
+*(A::AbstractMatrix, B::AbstractMatrix, C::AbstractMatrix, x::AbstractVector) = A * B * (C*x)
+*(vt::AdjOrTransAbsVec, B::AbstractMatrix, C::AbstractMatrix, D::AbstractMatrix) = (vt*B) * C * D
+*(vt::AdjOrTransAbsVec, B::AbstractMatrix, C::AbstractMatrix, x::AbstractVector) = vt * B * (C*x)
+
+# Four-argument *, by size
+*(A::AbstractMatrix, B::AbstractMatrix, C::AbstractMatrix, δ::Number) = _tri_matmul(A,B,C,δ)
+*(α::RealOrComplex, B::AbstractMatrix{<:RealOrComplex}, C::AbstractMatrix{<:RealOrComplex}, D::AbstractMatrix{<:RealOrComplex}) =
+ _tri_matmul(B,C,D,α)
+*(A::AbstractMatrix, B::AbstractMatrix, C::AbstractMatrix, D::AbstractMatrix) =
+ _quad_matmul(A,B,C,D)
+
+function _quad_matmul(A,B,C,D)
+ c1 = _mul_cost((A,B),(C,D))
+ c2 = _mul_cost(((A,B),C),D)
+ c3 = _mul_cost(A,(B,(C,D)))
+ c4 = _mul_cost((A,(B,C)),D)
+ c5 = _mul_cost(A,((B,C),D))
+ cmin = min(c1,c2,c3,c4,c5)
+ if c1 == cmin
+ (A*B) * (C*D)
+ elseif c2 == cmin
+ ((A*B) * C) * D
+ elseif c3 == cmin
+ A * (B * (C*D))
+ elseif c4 == cmin
+ (A * (B*C)) * D
+ else
+ A * ((B*C) * D)
+ end
+end
+@inline _mul_cost(A::AbstractMatrix) = 0
+@inline _mul_cost((A,B)::Tuple) = _mul_cost(A,B)
+@inline _mul_cost(A,B) = _mul_cost(A) + _mul_cost(B) + *(_mul_sizes(A)..., last(_mul_sizes(B)))
+@inline _mul_sizes(A::AbstractMatrix) = size(A)
+@inline _mul_sizes((A,B)::Tuple) = first(_mul_sizes(A)), last(_mul_sizes(B))

stdlib/LinearAlgebra/test/matmul.jl

@@ -766,4 +766,103 @@ end
  @test Matrix{Int}(undef, 2, 0) * Matrix{Int}(undef, 0, 3) == zeros(Int, 2, 3)
 end
 
+@testset "3-arg *, order by type" begin
+ x = [1, 2im]
+ y = [im, 20, 30+40im]
+ z = [-1, 200+im, -3]
+ A = [1 2 3im; 4 5 6+im]
+ B = [-10 -20; -30 -40]
+ a = 3 + im * round(Int, 10^6*(pi-3))
+ b = 123
+
+ @test x'*A*y == (x'*A)*y == x'*(A*y)
+ @test y'*A'*x == (y'*A')*x == y'*(A'*x)
+ @test y'*transpose(A)*x == (y'*transpose(A))*x == y'*(transpose(A)*x)
+
+ @test B*A*y == (B*A)*y == B*(A*y)
+
+ @test a*A*y == (a*A)*y == a*(A*y)
+ @test A*y*a == (A*y)*a == A*(y*a)
+
+ @test a*B*A == (a*B)*A == a*(B*A)
+ @test B*A*a == (B*A)*a == B*(A*a)
+
+ @test a*y'*z == (a*y')*z == a*(y'*z)
+ @test y'*z*a == (y'*z)*a == y'*(z*a)
+
+ @test a*y*z' == (a*y)*z' == a*(y*z')
+ @test y*z'*a == (y*z')*a == y*(z'*a)
+
+ @test a*x'*A == (a*x')*A == a*(x'*A)
+ @test x'*A*a == (x'*A)*a == x'*(A*a)
+ @test a*x'*A isa Adjoint{<:Any, <:Vector}
+
+ @test a*transpose(x)*A == (a*transpose(x))*A == a*(transpose(x)*A)
+ @test transpose(x)*A*a == (transpose(x)*A)*a == transpose(x)*(A*a)
+ @test a*transpose(x)*A isa Transpose{<:Any, <:Vector}
+
+ @test x'*B*A == (x'*B)*A == x'*(B*A)
+ @test x'*B*A isa Adjoint{<:Any, <:Vector}
+
+ @test y*x'*A == (y*x')*A == y*(x'*A)
+ y31 = reshape(y,3,1)
+ @test y31*x'*A == (y31*x')*A == y31*(x'*A)
+
+ vm = [rand(1:9,2,2) for _ in 1:3]
+ Mm = [rand(1:9,2,2) for _ in 1:3, _ in 1:3]
+
+ @test vm' * Mm * vm == (vm' * Mm) * vm == vm' * (Mm * vm)
+ @test Mm * Mm' * vm == (Mm * Mm') * vm == Mm * (Mm' * vm)
+ @test vm' * Mm * Mm == (vm' * Mm) * Mm == vm' * (Mm * Mm)
+ @test Mm * Mm' * Mm == (Mm * Mm') * Mm == Mm * (Mm' * Mm)
+end
+
+@testset "3-arg *, order by size" begin
+ M44 = randn(4,4)
+ M24 = randn(2,4)
+ M42 = randn(4,2)
+ @test M44*M44*M44 ≈ (M44*M44)*M44 ≈ M44*(M44*M44)
+ @test M42*M24*M44 ≈ (M42*M24)*M44 ≈ M42*(M24*M44)
+ @test M44*M42*M24 ≈ (M44*M42)*M24 ≈ M44*(M42*M24)
+end
+
+@testset "4-arg *, by type" begin
+ y = [im, 20, 30+40im]
+ z = [-1, 200+im, -3]
+ a = 3 + im * round(Int, 10^6*(pi-3))
+ b = 123
+ M = rand(vcat(1:9, im.*[1,2,3]), 3,3)
+ N = rand(vcat(1:9, im.*[1,2,3]), 3,3)
+
+ @test a * b * M * y == (a*b) * (M*y)
+ @test a * b * M * N == (a*b) * (M*N)
+ @test a * M * N * y == (a*M) * (N*y)
+ @test a * y' * M * z == (a*y') * (M*z)
+ @test a * y' * M * N == (a*y') * (M*N)
+
+ @test M * y * a * b == (M*y) * (a*b)
+ @test M * N * a * b == (M*N) * (a*b)
+ @test M * N * y * a == (a*M) * (N*y)
+ @test y' * M * z * a == (a*y') * (M*z)
+ @test y' * M * N * a == (a*y') * (M*N)
+
+ @test M * N * conj(M) * y == (M*N) * (conj(M)*y)
+ @test y' * M * N * conj(M) == (y'*M) * (N*conj(M))
+ @test y' * M * N * z == (y'*M) * (N*z)
+end
+
+@testset "4-arg *, by size" begin
+ for shift in 1:5
+ s1,s2,s3,s4,s5 = circshift(3:7, shift)
+ a=randn(s1,s2); b=randn(s2,s3); c=randn(s3,s4); d=randn(s4,s5)
+
+ # _quad_matmul
+ @test *(a,b,c,d) ≈ (a*b) * (c*d)
+
+ # _tri_matmul(A,B,B,δ)
+ @test *(11.1,b,c,d) ≈ (11.1*b) * (c*d)
+ @test *(a,b,c,99.9) ≈ (a*b) * (c*99.9)
+ end
+end
+
 end # module TestMatmul