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add FastCholesky for any OPs
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X'W-WX == GJG', rank(G) = 2. This allows W to be Cholesky-factored in O(n^2).
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@MikaelSlevinsky
MikaelSlevinsky committed Jul 12, 2024
1 parent e5cc4c4 commit d8ccedd
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2 changes: 1 addition & 1 deletion Project.toml
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Original file line number Diff line number Diff line change
@@ -1,6 +1,6 @@
name = "FastTransforms"
uuid = "057dd010-8810-581a-b7be-e3fc3b93f78c"
version = "0.16.3"
version = "0.16.4"

[deps]
AbstractFFTs = "621f4979-c628-5d54-868e-fcf4e3e8185c"
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136 changes: 107 additions & 29 deletions src/SymmetricToeplitzPlusHankel.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -38,23 +38,23 @@ bandwidths(A::SymmetricBandedToeplitzPlusHankel{T}) where T = (A.b, A.b)

#
# X'W-W*X = G*J*G'
# This returns G and J, where J = [0 I; -I 0], respecting the skew-symmetry of the right-hand side.
# This returns G, where J = [0 1; -1 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
@inbounds @simd for j in 1:n-1
G[j, 2] = -(v[n+2-j] + v[n+j])
end
G[n, 2] = -v[2]
G
end

function cholesky(A::SymmetricToeplitzPlusHankel{T}) where T
n = size(A, 1)
G, J = compute_skew_generators(A)
G = compute_skew_generators(A)
L = zeros(T, n, n)
c = A[:, 1]
ĉ = zeros(T, n)
Expand All @@ -74,20 +74,29 @@ function STPHcholesky!(L::Matrix{T}, G, c, ĉ, l, v, row1, n) where T
for j in 1:n-k+1
v[j] = G[j, 1]*G[1, 2] - G[j, 2]*G[1, 1]
end
X21 = k == 1 ? T(2) : T(1)
ĉ[1] = (c[2] - v[1])/X21
for j in 2:n-k
ĉ[j] = (c[j+1] + c[j-1] + c[1]*row1[j] - row1[1]*c[j] - v[j])/X21
if k == 1
for j in 2:n-k
ĉ[j] = (c[j+1] + c[j-1] + c[1]*row1[j] - row1[1]*c[j] - v[j])/2
end
ĉ[n-k+1] = (c[n-k] + c[1]*row1[n-k+1] - row1[1]*c[n-k+1] - v[n-k+1])/2
cst = 2/d
for j in 1:n-k
row1[j] = -cst*l[j+1]
end
else
for j in 2:n-k
ĉ[j] = c[j+1] + c[j-1] + c[1]*row1[j] - row1[1]*c[j] - v[j]
end
ĉ[n-k+1] = c[n-k] + c[1]*row1[n-k+1] - row1[1]*c[n-k+1] - v[n-k+1]
cst = 1/d
for j in 1:n-k
row1[j] = -cst*l[j+1]
end
end
ĉ[n-k+1] = (c[n-k] + c[1]*row1[n-k+1] - row1[1]*c[n-k+1] - v[n-k+1])/X21
cst = c[2]/d
for j in 1:n-k
c[j] = ĉ[j+1] - cst*l[j+1]
end
cst = X21/d
for j in 1:n-k
row1[j] = -cst*l[j+1]
end
gd1 = G[1, 1]/d
gd2 = G[1, 2]/d
for j in 1:n-k
Expand All @@ -101,37 +110,106 @@ 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)))
L = BandedMatrix{T}(undef, (n, n), (bandwidth(A, 1), 0))
c = A[1:b+2, 1]
ĉ = zeros(T, b+3)
l = zeros(T, b+3)
row1 = zeros(T, b+2)
SBTPHcholesky!(R, c, ĉ, l, row1, n, b)
return Cholesky(R, 'U', 0)
SBTPHcholesky!(L, c, ĉ, l, row1, n, b)
return Cholesky(L, 'L', 0)
end

function SBTPHcholesky!(R::BandedMatrix{T}, c, ĉ, l, row1, n, b) where T
function SBTPHcholesky!(L::BandedMatrix{T}, c, ĉ, l, row1, n, b) where T
@inbounds @simd for k in 1:n
d = sqrt(c[1])
for j in 1:b+1
l[j] = c[j]/d
end
for j in 1:min(n-k+1, b+1)
R[k, j+k-1] = l[j]
L[j+k-1, k] = l[j]
end
X21 = k == 1 ? T(2) : T(1)
ĉ[1] = c[2]/X21
for j in 2:b+1
ĉ[j] = (c[j+1] + c[j-1] + c[1]*row1[j] - row1[1]*c[j])/X21
if k == 1
for j in 2:b+1
ĉ[j] = (c[j+1] + c[j-1] + c[1]*row1[j] - row1[1]*c[j])/2
end
ĉ[b+2] = (c[b+1] + c[1]*row1[b+2] - row1[1]*c[b+2])/2
cst = 2/d
for j in 1:b+2
row1[j] = -cst*l[j+1]
end
else
for j in 2:b+1
ĉ[j] = (c[j+1] + c[j-1] + c[1]*row1[j] - row1[1]*c[j])
end
ĉ[b+2] = (c[b+1] + c[1]*row1[b+2] - row1[1]*c[b+2])
cst = 1/d
for j in 1:b+2
row1[j] = -cst*l[j+1]
end
end
ĉ[b+2] = (c[b+1] + c[1]*row1[b+2] - row1[1]*c[b+2])/X21
cst = c[2]/d
for j in 1:b+2
c[j] = ĉ[j+1] - cst*l[j+1]
end
cst = X21/d
for j in 1:b+2
row1[j] = -cst*l[j+1]
end
end



#
# X'W-W*X = G*J*G'
# This returns G, where J = [0 1; -1 0], respecting the skew-symmetry of the right-hand side.
#
function compute_skew_generators(W::Symmetric{T, <: AbstractMatrix{T}}, X::Tridiagonal{T, Vector{T}}) where T
@assert size(W) == size(X)
m, n = size(W)
G = zeros(T, n, 2)
G[n, 1] = one(T)
G[:, 2] .= W[n-1, :]*X[n-1, n] - X'W[:, n]
return G
end

function fastcholesky(W::Symmetric{T, <: AbstractMatrix{T}}, X::Tridiagonal{T, Vector{T}}) where T
n = size(W, 1)
G = compute_skew_generators(W, X)
L = zeros(T, n, n)
c = W[:, 1]
ĉ = zeros(T, n)
l = zeros(T, n)
v = zeros(T, n)
row1 = zeros(T, n)
fastcholesky!(L, X, G, c, ĉ, l, v, row1, n)
return Cholesky(L, 'L', 0)
end


function fastcholesky!(L::Matrix{T}, X::Tridiagonal{T, Vector{T}}, G, c, ĉ, l, v, row1, n) where T
@inbounds @simd for k in 1:n-1
d = sqrt(c[k])
for j in k:n
L[j, k] = l[j] = c[j]/d
end
for j in k:n
v[j] = G[j, 1]*G[k, 2] - G[j, 2]*G[k, 1]
end
for j in k+1:n-1
ĉ[j] = (X[j-1, j]*c[j-1] + (X[j, j]-X[k, k])*c[j] + X[j+1, j]*c[j+1] + c[k]*row1[j] - row1[k]*c[j] - v[j])/X[k+1, k]
end
ĉ[n] = (X[n-1, n]*c[n-1] + (X[n, n]-X[k, k])*c[n] + c[k]*row1[n] - row1[k]*c[n] - v[n])/X[k+1, k]
cst = X[k+1, k]/d
for j in k+1:n
row1[j] = -cst*l[j]
end
cst = c[k+1]/d
for j in k:n
c[j] = ĉ[j] - cst*l[j]
end
gd1 = G[k, 1]/d
gd2 = G[k, 2]/d
for j in k:n
G[j, 1] -= l[j]*gd1
G[j, 2] -= l[j]*gd2
end
end
L[n, n] = sqrt(c[n])
end
12 changes: 12 additions & 0 deletions test/symmetrictoeplitzplushankeltests.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -37,3 +37,15 @@ end
@test R ≈ U
end
end

@testset "Fast Cholesky" begin
n = 128
for T in (Float32, Float64, BigFloat)
R = plan_leg2cheb(T, n; normcheb=true)*I
X = Tridiagonal([T(n)/(2n-1) for n in 1:n-1], zeros(T, n), [T(n)/(2n+1) for n in 1:n-1]) # Legendre X
W = Symmetric(R'R)
F = FastTransforms.fastcholesky(W, X)
@test F.L*F.L' ≈ W
@test F.U ≈ R
end
end

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Registration pull request created: JuliaRegistries/General/110978

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Release notes:

## Breaking changes

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Tagging

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git tag -a v0.16.4 -m "<description of version>" d8cceddd9e9b0829602c2a830de2588655754ea6
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