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test_derivative.jl
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using SemiclassicalOrthogonalPolynomials, ClassicalOrthogonalPolynomials, LazyArrays, Test
import ClassicalOrthogonalPolynomials: recurrencecoefficients, _BandedMatrix, _p0, Weighted
import LazyArrays: Accumulate, AccumulateAbstractVector
import SemiclassicalOrthogonalPolynomials: MulAddAccumulate, HalfWeighted, toclassical
@testset "Derivative" begin
@testset "basics" begin
t = 2
P = SemiclassicalJacobi(t, -0.5, -0.5, -0.5)
Q = SemiclassicalJacobi(t, 0.5, 0.5, 0.5, P)
x = axes(P,1)
D = Derivative(x)
D = Q \ (D*P)
@test (D*(P \ exp.(x)))[1:50] ≈ (Q \ exp.(x))[1:50]
end
@testset "Derivation" begin
t = 2
P = SemiclassicalJacobi(t, -0.5, -0.5, -0.5)
P̃ = toclassical(P)
Q = SemiclassicalJacobi(t, 0.5, 0.5, 0.5, P)
@test (Q \ P̃)[1:10,1:10] ≈ 0.6175596179729587*(Q \ P)[1:10,1:10]
x = axes(P,1)
D = Derivative(x)
for n = 3:10
u = (D * (P̃ * [[zeros(n);1]; zeros(∞)]))
@test norm((Q \ u)[1:n-2]) ≤ 3E-12
end
L = Q \ (D * P̃);
# L is bidiagonal
@test norm(triu(L[1:10,1:10],3)) ≤ 3E-12
@test L[:,5][1:10] ≈ L[1:10,5]
A,B,C = recurrencecoefficients(P);
α,β,γ = recurrencecoefficients(Q);
k = cumprod(A);
κ = cumprod(α);
j = Vector{Float64}(undef, 100)
j[1] = B[1]
for n = 1:length(j)-1
j[n+1] = A[n+1]*j[n] + B[n+1]*k[n]
end
ξ = Vector{Float64}(undef, 100)
ξ[1] = β[1]
for n = 1:length(ξ)-1
ξ[n+1] = α[n+1]*ξ[n] + β[n+1]*κ[n]
end
for n = 3:5
@test Base.unsafe_getindex(P̃,100,n+1) ≈ (k[n]*100^n + j[n]*100^(n-1)) * P̃[0.1,1] rtol=0.001
@test Base.unsafe_getindex(Q,100,n+1) ≈ (κ[n]*100^n + ξ[n]*100^(n-1)) rtol=0.001
end
n = 1
@test k[1]*P̃[0.1,1] ≈ L[1,2]
n = 2
@test L[n,n+1] ≈ n*k[n]/κ[n-1]*P̃[0.1,1]
@test L[n-1,n+1] ≈ ((n-1)*j[n] - n*k[n]*ξ[n-1]/κ[n-1])*P̃[0.1,1]
for n = 3:6
@test L[n,n+1] ≈ n*k[n]/κ[n-1]*P̃[0.1,1]
@test L[n-1,n+1] ≈ ((n-1)*j[n]/κ[n-2] - n*k[n]*ξ[n-1]/(κ[n-2]κ[n-1]))*P̃[0.1,1]
end
dv = n -> isone(n) ? A[1] : k[n]/κ[n-1]
ev1 = n -> j[n]/k[n]
ev2 = n -> n ==1 ? 1/κ[n] : ξ[n-1]/κ[n]
ev3 = n -> n == 1 ? k[n+1] : k[n+1]/κ[n-1]
# ev = n -> (n-1)*j[n]/κ[n-2] - n*k[n]*ξ[n-1]/(κ[n-2]κ[n-1])
@test ev2(1) ≈ 1/α[1]
n = 1
@test ev3(n) ≈ A[n]A[n+1]
@test ev2(n+1) ≈ β[n]/(α[n]α[n+1])
n = 3
@test dv(n+1) ≈ dv(n) * A[n+1]/α[n]
@test ev1(n+1) ≈ ev1(n) + B[n+1]/A[n+1]
@test ev2(n+1) ≈ ev2(n)*α[n]/α[n+1] + β[n]/(α[n]α[n+1])
@test ev3(n+1) ≈ ev3(n) * A[n+2]/α[n]
@test L[n,n+1]/P̃[0.1,1] ≈ n * dv(n)
@test L[n-1,n+1]/P̃[0.1,1] ≈ ((n-1)*j[n]/k[n] - n*ξ[n-1]/κ[n-1]) * k[n]/κ[n-2] ≈
((n-1)*(ev1(n-1) + B[n]/A[n]) - n*ξ[n-1]/κ[n-1]) * ev3(n-1) ≈
((n-1)*(ev1(n-1) + B[n]/A[n]) - n*(α[n-1]*ev2(n-1) + β[n-1]/α[n-1])) * ev3(n-1)
n = 1
@test L[n,n+2]/P̃[0.1,1] ≈ (ev1(1) + B[2]/A[2] - 2*(β[1]/α[1]))*ev3(1)
n = 2
@test L[n,n+2]/P̃[0.1,1] ≈ (n*(ev1(n) + B[n+1]/A[n+1]) - (n+1)*(α[n]*ev2(n) + β[n]/α[n])) * ev3(n)
n = 3
@test L[n,n+2]/P̃[0.1,1] ≈ (n*(ev1(n) + B[n+1]/A[n+1]) - (n+1)*(α[n]*ev2(n) + β[n]/α[n])) * ev3(n)
d = AccumulateAbstractVector(*, A ./ Vcat(1,α))
v1 = AccumulateAbstractVector(+, B ./ A)
v2 = MulAddAccumulate(Vcat(0,0,α[2:∞]) ./ α, Vcat(0,β ./ α) ./ α);
v3 = AccumulateAbstractVector(*, Vcat(A[1]A[2], A[3:∞] ./ α))
@test d[1:10] ≈ dv.(1:10)
@test v1[1:10] ≈ ev1.(1:10)
@test v2[1] ≈ 0
@test v2[2:10] ≈ ev2.(2:10)
@test v3[1:10] ≈ ev3.(1:10)
@test [L[n,n+1] for n=1:10]/P̃[0.1,1] ≈ ((1:∞) .* d)[1:10]
@test [L[n,n+2] for n=1:10]/P̃[0.1,1] ≈ (((1:∞) .* (v1 .+ B[2:end]./A[2:end]) .- (2:∞) .* (α .* v2 .+ β ./ α)) .* v3)[1:10]
D_M = _BandedMatrix(Vcat(((1:∞) .* d)', (((1:∞) .* (v1 .+ B[2:end]./A[2:end]) .- (2:∞) .* (α .* v2 .+ β ./ α)) .* v3)'), ∞, 2,-1)' * _p0(P̃)
@test D_M[1:10,1:10] ≈ L[1:10,1:10]
end
@testset "Weighted" begin
t = 2
P = SemiclassicalJacobi(t, 0, 0, 0)
Q = SemiclassicalJacobi(t, 1, 1, 1, P)
D = Derivative(axes(P,1))
@test (D * Weighted(Q))[0.1,1:5]' ≈ (D * P)[0.1,1:8]'* (P \ Weighted(Q))[1:8,1:5]
P = SemiclassicalJacobi(t, -1/2, -1/2, -1/2)
Q = SemiclassicalJacobi(t, 1/2, 1/2, 1/2, P)
h = 0.00001
@test (D * Weighted(Q))[0.1,1:5] ≈ (Weighted(Q)[0.1+h,1:5] - Weighted(Q)[0.1,1:5])/h atol=100h
end
@testset "HalfWeighted" begin
t = 2
P = SemiclassicalJacobi(t, 1, 1, 1)
P̃ = Normalized(jacobi(1,1,0..1))
x = axes(P,1)
D = Derivative(x)
@test HalfWeighted{:a}(P)[0.1,1:10] ≈ 0.1*P[0.1,1:10]
@test HalfWeighted{:b}(P)[0.1,1:10] ≈ (1-0.1)*P[0.1,1:10]
@test HalfWeighted{:c}(P)[0.1,1:10] ≈ (t-0.1)*P[0.1,1:10]
t,a,b,c = 2,0.1,0.2,0.3
P = SemiclassicalJacobi(t, a+1, b, c)
# Q = SemiclassicalJacobi(t,a,b+1,c+1)
HP = HalfWeighted{:a}(P)
h = 0.000001
@test (D * HP)[0.1,1:10] ≈ (HP[0.1+h,1:10]-HP[0.1,1:10])/h atol=200h
P = SemiclassicalJacobi(t, a, b+1, c)
# Q = SemiclassicalJacobi(t,a+1,b,c+1)
HP = HalfWeighted{:b}(P)
@test (D * HP)[0.1,1:10] ≈ (HP[0.1+h,1:10]-HP[0.1,1:10])/h atol=1000h
P = SemiclassicalJacobi(t, a, b, c+1)
# Q = SemiclassicalJacobi(t,a+1,b+1,c)
HP = HalfWeighted{:c}(P)
@test (D * HP)[0.1,1:10] ≈ (HP[0.1+h,1:10]-HP[0.1,1:10])/h atol=2000h
end
@testset "Double-HalfWeighted" begin
t = 2
P = SemiclassicalJacobi(t, 1, 1, 1)
x = axes(P,1)
D = Derivative(x)
@test HalfWeighted{:ab}(P)[0.1,1:10] ≈ 0.1*(1-0.1)*P[0.1,1:10]
@test HalfWeighted{:bc}(P)[0.1,1:10] ≈ (1-0.1)*(t-0.1)*P[0.1,1:10]
@test HalfWeighted{:ac}(P)[0.1,1:10] ≈ 0.1*(t-0.1)*P[0.1,1:10]
h = 0.000001
@test (D * HalfWeighted{:ab}(P))[0.1,1:10] ≈ (HalfWeighted{:ab}(P)[0.1+h,1:10]-HalfWeighted{:ab}(P)[0.1,1:10])/h atol=2000h
@test (D * HalfWeighted{:bc}(P))[0.1,1:10] ≈ (HalfWeighted{:bc}(P)[0.1+h,1:10]-HalfWeighted{:bc}(P)[0.1,1:10])/h atol=2000h
@test (D * HalfWeighted{:ac}(P))[0.1,1:10] ≈ (HalfWeighted{:ac}(P)[0.1+h,1:10]-HalfWeighted{:ac}(P)[0.1,1:10])/h atol=2000h
end
end