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validation_2.jl
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@testset verbose = true "TEST 2: Single-variable functions" begin
k = 10 # order
eps = 1e-15 # tolerance for comparing real numbers
@testset "2.1 Power function" begin
for n = 1:10
@testset "2.1 Power function (n=$(n))" begin
DACE.init(k, 1)
x = DA(1, 1)
f = (1 + x)^n
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if i <= n
a_exact = binomial(n, i)
else
a_exact = 0.0
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
end
end
@testset "2.2 Division" begin
DACE.init(k, 1)
x = DA(1, 1)
f = 1 / (1 - x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
a_exact = 1.0
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.3 nth root function" begin
for n = 2:5
@testset "2.3 nth root function (n=$(n))" begin
DACE.init(k, 1)
x = DA(1, 1)
f = DACE.root(1 + x, n)
jj = Vector{UInt32}(undef, 1)
args = Vector{Float64}(undef, 0)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if i == 0
a_exact = 1.0
else
push!(args, 1.0 / n - (i - 1))
a_exact = 1.0 / factorial(i) * prod(args)
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
end
end
@testset "2.4 Exponential function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = exp(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
a_exact = 1 / factorial(i)
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.5 Logarithmic function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = log(1 + x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if i == 0
a_exact = 0.0
else
a_exact = (-1)^(i+1) * 1.0 / i
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.6 Sine function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = sin(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 0.0
else
a_exact = (-1)^((i - 1) / 2) / factorial(i)
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.7 Cosine function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = cos(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = (-1)^(i / 2) / factorial(i)
else
a_exact = 0.0
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.8 Tangent function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = tan(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 0.0
else
a_exact = (-1)^((i-1)/2) * 2^(i+1) * (2^(i+1) - 1) * bernoulli(i+1) / factorial(i+1)
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.9 Arcsine function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = asin(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 0.0
else
a_exact = factorial(i - 1) / (4.0^((i - 1) / 2) * factorial(div(i - 1, 2))^2 * i)
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.10 Arccosine function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = acos(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if i == 0
a_exact = pi / 2.0
elseif iseven(i)
a_exact = 0.0
else
a_exact = - factorial(i - 1) / (4.0^((i - 1) / 2) * factorial(div(i - 1, 2))^2 * i)
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.11 Arctangent function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = atan(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 0.0
else
a_exact = (-1)^((i-1)/2) / i
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.12 Hyperbolic sine function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = sinh(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 0.0
else
a_exact = 1.0 / factorial(i)
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.13 Hyperbolic cosine function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = cosh(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 1.0 / factorial(i)
else
a_exact = 0.0
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.14 Hyperbolic tangent function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = tanh(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 0.0
@test isapprox(a_dace, a_exact, atol=eps)
else
# TODO: I think there is a problem with a_exact (the values from DACE, i.e. a_dace, seem to be correct)
a_exact = bernoulli(i+1) * 4.0^((i+1)/2) * 4.0^((i+1)/2-1) / factorial(i+1)
@test_broken isapprox(a_dace, a_exact, atol=eps)
# possibly correct formula based on https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Tangent_Function
a_exact = 2^(i+1) * (2^(i+1) - 1) * bernoulli(i+1) / factorial(i+1)
@test isapprox(a_dace, a_exact, atol=eps)
end
end
end
@testset "2.15 Hyperbolic arcsine function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = asinh(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 0.0
else
a_exact = (-1)^((i-1)/2) * factorial(i-1) / (4^((i-1)/2) * factorial(div(i-1, 2))^2 * i)
end
@test isapprox(a_dace, a_exact, atol=eps)
end
end
@testset "2.16 Hyperbolic arctangent function" begin
DACE.init(k, 1)
x = DA(1, 1)
f = atanh(x)
jj = Vector{UInt32}(undef, 1)
for i in 0:k
jj[1] = i
a_dace = DACE.getCoefficient(f, jj)
if iseven(i)
a_exact = 0.0
else
a_exact = 1 / i
end
# NOTE: using looser toleration, noted in validation doc that this test didn't meet the required tolerance
# but was deemed close enough to pass anyway
@test isapprox(a_dace, a_exact, atol=1e-12)
end
end
end