Add Butterfly Factorization

.vscode/settings.json

@@ -0,0 +1,3 @@
+{
+    "julia.environmentPath": "c:\\Users\\sivasubl\\OneDrive - UMass Chan Medical School\\Documents\\GitHub\\RecursiveFactorization.jl"
+}

Project.toml

@@ -10,6 +10,7 @@ Polyester = "f517fe37-dbe3-4b94-8317-1923a5111588"
 PrecompileTools = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
 StrideArraysCore = "7792a7ef-975c-4747-a70f-980b88e8d1da"
 TriangularSolve = "d5829a12-d9aa-46ab-831f-fb7c9ab06edf"
+VectorizedRNG = "33b4df10-0173-11e9-2a0c-851a7edac40e"
 
 [compat]
 LinearAlgebra = "1.5"
@@ -18,6 +19,7 @@ Polyester = "0.3.2,0.4.1, 0.5, 0.6, 0.7"
 PrecompileTools = "1"
 StrideArraysCore = "0.5.5"
 TriangularSolve = "0.2"
+VectorizedRNG = "0.2.25"
 julia = "1.10"
 
 [extras]

src/RecursiveFactorization.jl

@@ -4,6 +4,7 @@ if isdefined(Base, :Experimental) &&
     @eval Base.Experimental.@max_methods 1
 end
 include("./lu.jl")
+include("./butterflylu.jl")
 
 import PrecompileTools
 

src/butterflylu.jl

@@ -0,0 +1,133 @@
+using VectorizedRNG
+using LinearAlgebra: Diagonal, I
+using LoopVectorization
+using RecursiveFactorization
+
+@inline exphalf(x) = exp(x) * oftype(x, 0.5)
+function 🦋!(wv, ::Val{SEED} = Val(888)) where {SEED}
+    T = eltype(wv)
+    mrng = VectorizedRNG.MutableXoshift(SEED)
+    GC.@preserve mrng begin rand!(exphalf, VectorizedRNG.Xoshift(mrng), wv, static(0),
+                                  T(-0.05), T(0.1)) end
+end
+
+function 🦋generate_random!(A, ::Val{SEED} = Val(888)) where {SEED}
+    Usz = 2 * size(A, 1)
+    Vsz = 2 * size(A, 2)
+    uv = similar(A, Usz + Vsz)
+    🦋!(uv, Val(SEED))
+    (uv,)
+end
+
+function 🦋workspace(A, ::Val{SEED} = Val(888)) where {SEED}
+    A = pad!(A)
+    B = similar(A);
+    ws = 🦋generate!(B)
+    🦋mul!(copyto!(B, A), ws)
+    U, V, B = materializeUV(B, ws)
+    F = RecursiveFactorization.lu!(B, Val(false))
+    A, B, U, V, F
+end
+
+const butterfly_workspace = 🦋workspace;
+
+function 🦋mul_level!(A, u, v)
+    # for now, assume... 
+    M, N = size(A)
+    Ml = M >>> 1
+    Nl = N >>> 1
+    Mh = M - Ml
+    Nh = N - Nl
+    @turbo for n in 1:Nl
+        for m in 1:Ml
+            A11 = A[m, n]
+            A21 = A[m + Mh, n]
+            A12 = A[m, n + Nh]
+            A22 = A[m + Mh, n + Nh]
+
+            T1 = A11 + A12
+            T2 = A21 + A22
+            T3 = A11 - A12
+            T4 = A21 - A22
+            C11 = T1 + T2
+            C21 = T1 - T2
+            C12 = T3 + T4
+            C22 = T3 - T4
+
+            u1 = u[m]
+            u2 = u[m + Mh]
+            v1 = v[n]
+            v2 = v[n + Nh]
+
+            A[m, n] = u1 * C11 * v1
+            A[m + Mh, n] = u2 * C21 * v1
+            A[m, n + Nh] = u1 * C12 * v2
+            A[m + Mh, n + Nh] = u2 * C22 * v2
+        end
+    end
+end
+
+function 🦋mul!(A, (uv,))
+    M, N = size(A)
+    Mh = M >>> 1
+    Nh = N >>> 1
+    U₁ = @view(uv[1:Mh])
+    U₂ = @view(uv[(1 + Mh + Nh):(2 * Mh + Nh)])
+    V₁ = @view(uv[(Mh + 1):(Mh + Nh)])
+    V₂ = @view(uv[(1 + 2 * Mh + Nh):(2 * Mh + 2 * Nh)])
+    🦋mul_level!(@view(A[1:Mh, 1:Nh]), U₁, V₁)
+    🦋mul_level!(@view(A[(1 + Mh):M, 1:Nh]), U₂, V₁)
+    🦋mul_level!(@view(A[1:Mh, (1 + Nh):N]), U₁, V₂)
+    🦋mul_level!(@view(A[(1 + Mh):M, (1 + Nh):N]), U₂, V₂)
+    U = @view(uv[(1 + 2 * Mh + 2 * Nh):(2 * Mh + 2 * Nh + M)])
+    V = @view(uv[(1 + 2 * Mh + 2 * Nh + M):(2 * Mh + 2 * Nh + M + N)])
+    🦋mul_level!(@view(A[1:M, 1:N]), U, V)
+    A
+end
+
+function diagnegbottom(x)
+    N = length(x)
+    y = similar(x, N >>> 1)
+    z = similar(x, N >>> 1)
+    for n in 1:(N >>> 1)
+        y[n] = x[n]
+    end
+    for n in 1:(N >>> 1)
+        z[n] = x[n + (N >>> 1)]
+    end
+    Diagonal(y), Diagonal(z)
+end
+🦋(A, B) = [A B
+           A -B]
+
+function materializeUV(A, (uv,))
+    M, N = size(A)
+    Mh = M >>> 1
+    Nh = N >>> 1
+
+    U₁u, U₁l = diagnegbottom(@view(uv[1:Mh]))
+    U₂u, U₂l = diagnegbottom(@view(uv[(1 + Mh + Nh):(2 * Mh + Nh)]))
+    V₁u, V₁l = diagnegbottom(@view(uv[(Mh + 1):(Mh + Nh)]))
+    V₂u, V₂l = diagnegbottom(@view(uv[(1 + 2 * Mh + Nh):(2 * Mh + 2 * Nh)]))
+    Uu, Ul = diagnegbottom(@view(uv[(1 + 2 * Mh + 2 * Nh):(2 * Mh + 2 * Nh + M)]))
+    Vu, Vl = diagnegbottom(@view(uv[(1 + 2 * Mh + 2 * Nh + M):(2 * Mh + 2 * Nh + M + N)]))
+
+    Bu2 = [🦋(U₁u, U₁l) 0*I
+           0*I 🦋(U₂u, U₂l)]
+    Bu1 = 🦋(Uu, Ul)
+
+    Bv2 = [🦋(V₁u, V₁l) 0*I
+           0*I 🦋(V₂u, V₂l)]
+    Bv1 = 🦋(Vu, Vl)
+
+    (Bu2 * Bu1)', Bv2 * Bv1, A
+end
+
+function pad!(A)
+    M, N = size(A)
+    xn = 4 - M % 4
+    A = [A   zeros(N, xn)
+    zeros(xn, N) I(xn)
+    ]
+    A
+end

test/runtests.jl

@@ -64,3 +64,28 @@ testlu(A::Union{Transpose, Adjoint}, MF, BF, p) = testlu(parent(A), parent(MF),
         end
     end
 end
+
+function wilkinson(N)
+    A = zeros(N, N)
+    A[1:(N+1):N*N] .= 1
+    A[:, end] .= 1
+    for n in 1:(N - 1)
+        for r in (n + 1):N
+            @inbounds A[r, n] = -1
+        end
+    end
+    A
+end
+
+@testset "🦋" begin
+    for i in 790 : 810
+        A = wilkinson(i)
+        b = rand(i)
+        A_ext, B, U, V, F = RecursiveFactorization.🦋workspace(A)
+        M, N = size(A)
+        xn = 4 - M % 4
+        b_ext = [b; rand(xn)]
+        x = V * (F \ (U * b_ext))    
+        @test norm(A * x[1:M] .- b) <= 1e-10
+    end
+end