add Symmetric(Banded)ToeplitzPlusHankel

Author: MikaelSlevinsky

Project.toml

@@ -1,9 +1,10 @@
 name = "FastTransforms"
 uuid = "057dd010-8810-581a-b7be-e3fc3b93f78c"
-version = "0.15.16"
+version = "0.16.0"
 
 [deps]
 AbstractFFTs = "621f4979-c628-5d54-868e-fcf4e3e8185c"
+BandedMatrices = "aae01518-5342-5314-be14-df237901396f"
 FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
 FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"
 FastTransforms_jll = "34b6f7d7-08f9-5794-9e10-3819e4c7e49a"
@@ -17,6 +18,7 @@ ToeplitzMatrices = "c751599d-da0a-543b-9d20-d0a503d91d24"
 
 [compat]
 AbstractFFTs = "1.0"
+BandedMatrices = "1.5"
 FFTW = "1.7"
 FastGaussQuadrature = "0.4, 0.5, 1"
 FastTransforms_jll = "0.6.2"

src/FastTransforms.jl

@@ -1,6 +1,6 @@
 module FastTransforms
 
-using FastGaussQuadrature, FillArrays, LinearAlgebra,
+using BandedMatrices, FastGaussQuadrature, FillArrays, LinearAlgebra,
       Reexport, SpecialFunctions, ToeplitzMatrices
 
 @reexport using AbstractFFTs
@@ -19,14 +19,16 @@ import AbstractFFTs: Plan, ScaledPlan,
                      fftshift, ifftshift, rfft_output_size, brfft_output_size,
                      normalization
 
+import BandedMatrices: bandwidths
+
 import FFTW: dct, dct!, idct, idct!, plan_dct!, plan_idct!,
              plan_dct, plan_idct, fftwNumber
 
 import FastGaussQuadrature: unweightedgausshermite
 
 import FillArrays: AbstractFill, getindex_value
 
-import LinearAlgebra: mul!, lmul!, ldiv!
+import LinearAlgebra: mul!, lmul!, ldiv!, cholesky
 
 import GenericFFT: interlace # imported in downstream packages
 
@@ -112,6 +114,8 @@ include("specialfunctions.jl")
 include("toeplitzplans.jl")
 include("toeplitzhankel.jl")
 
+include("SymmetricToeplitzPlusHankel.jl")
+
 # following use libfasttransforms by default
 for f in (:jac2jac,
     :lag2lag, :jac2ultra, :ultra2jac, :jac2cheb,

src/SymmetricToeplitzPlusHankel.jl

@@ -0,0 +1,135 @@
+struct SymmetricToeplitzPlusHankel{T} <: AbstractMatrix{T}
+    v::Vector{T}
+    n::Int
+end
+
+function SymmetricToeplitzPlusHankel(v::Vector{T}) where T
+    n = (length(v)+1)÷2
+    SymmetricToeplitzPlusHankel{T}(v, n)
+end
+
+size(A::SymmetricToeplitzPlusHankel{T}) where T = (A.n, A.n)
+getindex(A::SymmetricToeplitzPlusHankel{T}, i::Integer, j::Integer) where T = A.v[abs(i-j)+1] + A.v[i+j-1]
+
+struct SymmetricBandedToeplitzPlusHankel{T} <: BandedMatrices.AbstractBandedMatrix{T}
+    v::Vector{T}
+    n::Int
+    b::Int
+end
+
+function SymmetricBandedToeplitzPlusHankel(v::Vector{T}, n::Integer) where T
+    SymmetricBandedToeplitzPlusHankel{T}(v, n, length(v)-1)
+end
+
+size(A::SymmetricBandedToeplitzPlusHankel{T}) where T = (A.n, A.n)
+function getindex(A::SymmetricBandedToeplitzPlusHankel{T}, i::Integer, j::Integer) where T
+    v = A.v
+    if abs(i-j) < length(v)
+        if i+j-1 ≤ length(v)
+            v[abs(i-j)+1] + v[i+j-1]
+        else
+            v[abs(i-j)+1]
+        end
+    else
+        zero(T)
+    end
+end
+bandwidths(A::SymmetricBandedToeplitzPlusHankel{T}) where T = (A.b, A.b)
+
+#
+# Jac*W-W*Jac' = G*J*G'
+# This returns G and J, where J = [0 I; -I 0], respecting the skew-symmetry of the right-hand side.
+#
+function compute_skew_generators(A::SymmetricToeplitzPlusHankel{T}) where T
+    v = A.v
+    n = size(A, 1)
+    J = [zero(T) one(T); -one(T) zero(T)]
+    G = zeros(T, n, 2)
+    G[n, 1] = one(T)
+    u2 = reverse(v[2:n+1])
+    u2[1:n-1] .+= v[n+1:2n-1]
+    G[:, 2] .= -u2
+    G, J
+end
+
+function cholesky(A::SymmetricToeplitzPlusHankel{T}) where T
+    n = size(A, 1)
+    G, J = compute_skew_generators(A)
+    L = zeros(T, n, n)
+    r = A[:, 1]
+    r2 = zeros(T, n)
+    l = zeros(T, n)
+    v = zeros(T, n)
+    col1 = zeros(T, n)
+    STPHcholesky!(L, G, r, r2, l, v, col1, n)
+    return Cholesky(L, 'L', 0)
+end
+
+function STPHcholesky!(L::Matrix{T}, G, r, r2, l, v, col1, n) where T
+    @inbounds @simd for k in 1:n-1
+        x = sqrt(r[1])
+        for j in 1:n-k+1
+            L[j+k-1, k] = l[j] = r[j]/x
+        end
+        for j in 1:n-k+1
+            v[j] = G[j, 1]*G[1,2]-G[j,2]*G[1,1]
+        end
+        F12 = k == 1 ? T(2) : T(1)
+        r2[1] = (r[2] - v[1])/F12
+        for j in 2:n-k
+            r2[j] = (r[j+1]+r[j-1] + r[1]*col1[j] - col1[1]*r[j] - v[j])/F12
+        end
+        r2[n-k+1] = (r[n-k] + r[1]*col1[n-k+1] - col1[1]*r[n-k+1] - v[n-k+1])/F12
+        cst = r[2]/x
+        for j in 1:n-k
+            r[j] = r2[j+1] - cst*l[j+1]
+        end
+        for j in 1:n-k
+            col1[j] = -F12/x*l[j+1]
+        end
+        c1 = G[1, 1]
+        c2 = G[1, 2]
+        for j in 1:n-k
+            G[j, 1] = G[j+1, 1] - l[j+1]*c1/x
+            G[j, 2] = G[j+1, 2] - l[j+1]*c2/x
+        end
+    end
+    L[n, n] = sqrt(r[1])
+end
+
+function cholesky(A::SymmetricBandedToeplitzPlusHankel{T}) where T
+    n = size(A, 1)
+    b = A.b
+    R = BandedMatrix{T}(undef, (n, n), (0, bandwidth(A, 2)))
+    r = A[1:b+2, 1]
+    r2 = zeros(T, b+3)
+    l = zeros(T, b+3)
+    col1 = zeros(T, b+2)
+    SBTPHcholesky!(R, r, r2, l, col1, n, b)
+    return Cholesky(R, 'U', 0)
+end
+
+function SBTPHcholesky!(R::BandedMatrix{T}, r, r2, l, col1, n, b) where T
+    @inbounds @simd for k in 1:n
+        x = sqrt(r[1])
+        for j in 1:b+1
+            l[j] = r[j]/x
+        end
+        for j in 1:min(n-k+1, b+1)
+            R[k, j+k-1] = l[j]
+        end
+        F12 = k == 1 ? T(2) : T(1)
+        r2[1] = r[2]/F12
+        for j in 2:b+1
+            r2[j] = (r[j+1]+r[j-1] + r[1]*col1[j] - col1[1]*r[j])/F12
+        end
+        r2[b+2] = (r[b+1] + r[1]*col1[b+2] - col1[1]*r[b+2])/F12
+        cst = r[2]/x
+        for j in 1:b+2
+            r[j] = r2[j+1] - cst*l[j+1]
+        end
+        for j in 1:b+2
+            col1[j] = -F12/x*l[j+1]
+        end
+    end
+end

test/runtests.jl

@@ -10,4 +10,5 @@ include("gaunttests.jl")
 include("hermitetests.jl")
 include("clenshawtests.jl")
 include("toeplitzplanstests.jl")
-include("toeplitzhankeltests.jl")
+include("toeplitzhankeltests.jl")
+include("symmetrictoeplitzplushankeltests.jl")

test/symmetrictoeplitzplushankeltests.jl

@@ -0,0 +1,39 @@
+using BandedMatrices, FastTransforms, LinearAlgebra, ToeplitzMatrices, Test
+
+import FastTransforms: SymmetricToeplitzPlusHankel, SymmetricBandedToeplitzPlusHankel
+
+@testset "SymmetricToeplitzPlusHankel" begin
+    n = 128
+    for T in (Float32, Float64, BigFloat)
+        μ = -FastTransforms.chebyshevlogmoments1(T, 2n-1)
+        μ[1] += 1
+        W = SymmetricToeplitzPlusHankel(μ/2)
+        SMW = Symmetric(Matrix(W))
+        @test W ≈ SymmetricToeplitz(μ[1:(length(μ)+1)÷2]/2) + Hankel(μ/2)
+        L = cholesky(W).L
+        R = cholesky(SMW).U
+        @test L*L' ≈ W
+        @test L' ≈ R
+    end
+end
+
+@testset "SymmetricBandedToeplitzPlusHankel" begin
+    n = 1024
+    for T in (Float32, Float64)
+        μ = T[1.875; 0.00390625; 0.5; 0.0009765625; 0.0625]
+        W = SymmetricBandedToeplitzPlusHankel(μ/2, n)
+        SBW = Symmetric(BandedMatrix(W))
+        W1 = SymmetricToeplitzPlusHankel([μ/2; zeros(2n-1-length(μ))])
+        SMW = Symmetric(Matrix(W))
+        U = cholesky(SMW).U
+        L = cholesky(W1).L
+        UB = cholesky(SBW).U
+        R = cholesky(W).U
+        @test L*L' ≈ W
+        @test UB'UB ≈ W
+        @test R'R ≈ W
+        @test UB ≈ U
+        @test L' ≈ U
+        @test R ≈ U
+    end
+end