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Remove lower bound on size of epsilon #131

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Remove lower bound on size of epsilon #131

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2b11abb
fix - jacobian's x type consistency
briochemc Apr 8, 2019
dc167df
fix size of epsilon to be only relative to `x`
briochemc Apr 9, 2019
e32533a
fix Vector -> AbstractVector
briochemc Apr 9, 2019
a022d07
fix hessian to use hessianrule?
briochemc Apr 10, 2019
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24 changes: 12 additions & 12 deletions src/derivative.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -2,7 +2,7 @@ function derivative(f, ftype::Symbol, dtype::Symbol)
if ftype == :scalar
return x::Number -> finite_difference(f, float(x), dtype)
elseif ftype == :vector
return x::Vector -> finite_difference(f, float(x), dtype)
return x::AbstractVector -> finite_difference(f, float(x), dtype)
else
error("ftype must :scalar or :vector")
end
Expand Down Expand Up @@ -35,11 +35,11 @@ else
Base.ctranspose(f::Function) = derivative(f)
end

function jacobian(f, x::Vector{T}, dtype::Symbol) where T <: Number
function jacobian(f, x::AbstractVector{T}, dtype::Symbol) where T <: Number
finite_difference_jacobian(f, x, dtype)
end
function jacobian(f, dtype::Symbol)
g(x::Vector) = finite_difference_jacobian(f, x, dtype)
g(x::AbstractVector) = finite_difference_jacobian(f, x, dtype)
return g
end
jacobian(f) = jacobian(f, :central)
Expand All @@ -48,21 +48,21 @@ function second_derivative(f, g, ftype::Symbol, dtype::Symbol)
if ftype == :scalar
return x::Number -> finite_difference_hessian(f, g, x, dtype)
elseif ftype == :vector
return x::Vector -> finite_difference_hessian(f, g, x, dtype)
return x::AbstractVector -> finite_difference_hessian(f, g, x, dtype)
else
error("ftype must :scalar or :vector")
end
end
function second_derivative(f, g, x::Union{T, Vector{T}}, dtype::Symbol) where T <: Number
function second_derivative(f, g, x::Union{T, AbstractVector{T}}, dtype::Symbol) where T <: Number
finite_difference_hessian(f, g, x, dtype)
end
function hessian(f, g, x::Union{T, Vector{T}}, dtype::Symbol) where T <: Number
function hessian(f, g, x::Union{T, AbstractVector{T}}, dtype::Symbol) where T <: Number
finite_difference_hessian(f, g, x, dtype)
end
function second_derivative(f, g, x::Union{T, Vector{T}}) where T <: Number
function second_derivative(f, g, x::Union{T, AbstractVector{T}}) where T <: Number
finite_difference_hessian(f, g, x, :central)
end
function hessian(f, g, x::Union{T, Vector{T}}) where T <: Number
function hessian(f, g, x::Union{T, AbstractVector{T}}) where T <: Number
finite_difference_hessian(f, g, x, :central)
end
function second_derivative(f, x::Number, dtype::Symbol)
Expand All @@ -71,10 +71,10 @@ end
function hessian(f, x::Number, dtype::Symbol)
finite_difference_hessian(f, derivative(f), x, dtype)
end
function second_derivative(f, x::Vector{T}, dtype::Symbol) where T <: Number
function second_derivative(f, x::AbstractVector{T}, dtype::Symbol) where T <: Number
finite_difference_hessian(f, gradient(f), x, dtype)
end
function hessian(f, x::Vector{T}, dtype::Symbol) where T <: Number
function hessian(f, x::AbstractVector{T}, dtype::Symbol) where T <: Number
finite_difference_hessian(f, gradient(f), x, dtype)
end
function second_derivative(f, x::Number)
Expand All @@ -83,10 +83,10 @@ end
function hessian(f, x::Number)
finite_difference_hessian(f, derivative(f), x, :central)
end
function second_derivative(f, x::Vector{T}) where T <: Number
function second_derivative(f, x::AbstractVector{T}) where T <: Number
finite_difference_hessian(f, gradient(f), x, :central)
end
function hessian(f, x::Vector{T}) where T <: Number
function hessian(f, x::AbstractVector{T}) where T <: Number
finite_difference_hessian(f, gradient(f), x, :central)
end
second_derivative(f, g, dtype::Symbol) = second_derivative(f, g, :scalar, dtype)
Expand Down
57 changes: 25 additions & 32 deletions src/finite_difference.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -17,21 +17,21 @@
macro forwardrule(x, e)
x, e = esc(x), esc(e)
quote
$e = sqrt(eps(eltype($x))) * max(one(eltype($x)), abs($x))
$e = sqrt(eps($x^2))
end
end

macro centralrule(x, e)
x, e = esc(x), esc(e)
quote
$e = cbrt(eps(eltype($x))) * max(one(eltype($x)), abs($x))
$e = cbrt(eps($x^3))
end
end

macro hessianrule(x, e)
x, e = esc(x), esc(e)
quote
$e = eps(eltype($x))^(1/4) * max(one(eltype($x)), abs($x))
$e = eps($x^4)^(1/4)
end
end

Expand Down Expand Up @@ -231,38 +231,31 @@ end
##############################################################################

function finite_difference_hessian!(f,
x::AbstractVector{S},
H::Array{T}) where {S <: Number,
H::Array{T},
x::AbstractVector{S}) where {S <: Number,
T <: Number}
# What is the dimension of x?
n = length(x)

epsilon = NaN
# TODO: Remove all these copies
xpp, xpm, xmp, xmm = copy(x), copy(x), copy(x), copy(x)
fx = f(x)
for i = 1:n
xi = x[i]
@hessianrule x[i] epsilon
xpp[i], xmm[i] = xi + epsilon, xi - epsilon
H[i, i] = (f(xpp) - 2*fx + f(xmm)) / epsilon^2
@centralrule x[i] epsiloni
xp = xi + epsiloni
xm = xi - epsiloni
xpp[i], xpm[i], xmp[i], xmm[i] = xp, xp, xm, xm
for j = i+1:n
xj = x[j]
@centralrule x[j] epsilonj
xp = xj + epsilonj
xm = xj - epsilonj
xpp[j], xpm[j], xmp[j], xmm[j] = xp, xm, xp, xm
H[i, j] = (f(xpp) - f(xpm) - f(xmp) + f(xmm))/(4*epsiloni*epsilonj)
xpp[j], xpm[j], xmp[j], xmm[j] = xj, xj, xj, xj
n = size(H)[1]
e(j) = [i == j for i in 1:n] # j-th basis vector
hⱼ, hₖ = NaN, NaN
f₀ = f(x)
for j = 1:n
@hessianrule x[j] hⱼ # According to Numerical Recipes 5.7
f₊ = f(x + hⱼ * e(j))
f₋ = f(x - hⱼ * e(j))
H[j,j] = (f₋ - 2f₀ + f₊) / hⱼ^2
for k = j+1:n # Off-diagonal terms
@hessianrule x[k] hₖ # According to Numerical Recipes 5.7
f₊₊ = f(x + hⱼ * e(j) + hₖ * e(k))
f₊₋ = f(x + hⱼ * e(j) - hₖ * e(k))
f₋₊ = f(x - hⱼ * e(j) + hₖ * e(k))
f₋₋ = f(x - hⱼ * e(j) - hₖ * e(k))
H[j,k] = (f₊₊ - f₋₊ - f₊₋ + f₋₋) / (4 * hⱼ * hₖ)
j ≠ k ? H[k,j] = H[j,k] : nothing
end
xpp[i], xpm[i], xmp[i], xmm[i] = xi, xi, xi, xi
end
Compat.LinearAlgebra.copytri!(H,'U')
return H
end

function finite_difference_hessian(f,
x::AbstractVector{T}) where T <: Number
# What is the dimension of x?
Expand All @@ -272,7 +265,7 @@ function finite_difference_hessian(f,
H = Matrix{Float64}(undef, n, n)

# Mutate the allocated Hessian
finite_difference_hessian!(f, x, H)
finite_difference_hessian!(f, H, x)

# Return the Hessian
return H
Expand Down
48 changes: 28 additions & 20 deletions test/check_derivative.jl
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Original file line number Diff line number Diff line change
@@ -1,23 +1,31 @@
@test check_derivative(x -> sin(x), x -> cos(x), 0.0) < 10e-4
@test check_derivative(x -> sin(x), x -> cos(x), 1.0) < 10e-4
@test check_derivative(x -> sin(x), x -> cos(x), 10.0) < 10e-4
@test check_derivative(x -> sin(x), x -> cos(x), 100.0) < 10e-4
@test check_derivative(x -> sin(x), x -> cos(x), 1000.0) < 10e-4
@testset "check_derivative: f(x) = sin(x)" begin
@test check_derivative(x -> sin(x), x -> cos(x), 0.0) < 10e-4
@test check_derivative(x -> sin(x), x -> cos(x), 1.0) < 10e-4
@test check_derivative(x -> sin(x), x -> cos(x), 10.0) < 10e-4
@test check_derivative(x -> sin(x), x -> cos(x), 100.0) < 10e-4
@test check_derivative(x -> sin(x), x -> cos(x), 1000.0) < 10e-4
end

@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [0.0, 0.0]) < 10e-4
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [1.0, 1.0]) < 10e-4
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [10.0, 10.0]) < 10e-4
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [100.0, 100.0]) < 10e-4
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [1000.0, 1000.0]) < 10e-4
@testset "check_gradient: f(x) = sin(x[1]) + cos(x[2])" begin
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [0.1, 0.1]) < 10e-4
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [1.0, 1.0]) < 10e-4
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [10.0, 10.0]) < 10e-4
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [100.0, 100.0]) < 10e-4
@test check_gradient(x -> sin(x[1]) + cos(x[2]), x -> [cos(x[1]), -sin(x[2])], [1000.0, 1000.0]) < 10e-4
end

@test check_second_derivative(x -> sin(x), x -> -sin(x), 0.0) < 10e-4
@test check_second_derivative(x -> sin(x), x -> -sin(x), 1.0) < 10e-4
@test check_second_derivative(x -> sin(x), x -> -sin(x), 10.0) < 10e-4
@test check_second_derivative(x -> sin(x), x -> -sin(x), 100.0) < 10e-4
@test check_second_derivative(x -> sin(x), x -> -sin(x), 1000.0) < 10e-4
@testset "check_second_derivative: f(x) = sin(x)" begin
@test check_second_derivative(x -> sin(x), x -> -sin(x), 0.0) < 10e-4
@test check_second_derivative(x -> sin(x), x -> -sin(x), 1.0) < 10e-4
@test check_second_derivative(x -> sin(x), x -> -sin(x), 10.0) < 10e-4
@test check_second_derivative(x -> sin(x), x -> -sin(x), 100.0) < 10e-4
@test check_second_derivative(x -> sin(x), x -> -sin(x), 1000.0) < 10e-4
end

@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [0.0, 0.0]) < 10e-4
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [1.0, 1.0]) < 10e-4
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [10.0, 10.0]) < 10e-4
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [100.0, 100.0]) < 10e-4
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [1000.0, 1000.0]) < 10e-4
@testset "check_hessian: f(x) = sin(x[1]) + cos(x[2])" begin
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [0.1, 0.1]) < 10e-4
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [1.0, 1.0]) < 10e-4
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [10.0, 10.0]) < 10e-4
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [100.0, 100.0]) < 10e-4
@test check_hessian(x -> sin(x[1]) + cos(x[2]), x -> [-sin(x[1]) 0.0; 0.0 -cos(x[2])], [1000.0, 1000.0]) < 10e-4
end
70 changes: 45 additions & 25 deletions test/derivative.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -2,55 +2,75 @@
# derivative()
#

f1(x::Real) = sin(x)
@test norm(derivative(f1, :scalar, :forward)(0.0) - cos(0.0)) < 10e-4
@test norm(derivative(f1, :scalar, :central)(0.0) - cos(0.0)) < 10e-4
@test norm(derivative(f1, :forward)(0.0) - cos(0.0)) < 10e-4
@test norm(derivative(f1, :central)(0.0) - cos(0.0)) < 10e-4
@test norm(derivative(f1)(0.0) - cos(0.0)) < 10e-4
@testset "derivative()" begin
@testset "f(x::Real) = sin(x)" begin
f1(x::Real) = sin(x)
@test norm(derivative(f1, :scalar, :forward)(0.0) - cos(0.0)) < 10e-4
@test norm(derivative(f1, :scalar, :central)(0.0) - cos(0.0)) < 10e-4
@test norm(derivative(f1, :forward)(0.0) - cos(0.0)) < 10e-4
@test norm(derivative(f1, :central)(0.0) - cos(0.0)) < 10e-4
@test norm(derivative(f1)(0.0) - cos(0.0)) < 10e-4
end

f2(x::Vector) = sin(x[1])
@test norm(derivative(f2, :vector, :forward)([0.0]) .- cos(0.0)) < 10e-4
@test norm(derivative(f2, :vector, :central)([0.0]) .- cos(0.0)) < 10e-4
@testset "f(x::Vector) = sin(x[1])" begin
f2(x::Vector) = sin(x[1])
@test norm(derivative(f2, :vector, :forward)([0.0]) .- cos(0.0)) < 10e-4
@test norm(derivative(f2, :vector, :central)([0.0]) .- cos(0.0)) < 10e-4
end
end

#
# adjoint overloading
#

f3(x::Real) = sin(x)
for x in Compat.range(0.0, stop=0.1, length=11) # seq()
@test norm(f3'(x) - cos(x)) < 10e-4
@testset "adjoint overloading" begin
@testset "f(x::Vector) = sin(x[1])" begin
f3(x::Real) = sin(x)
for x in Compat.range(0.0, stop=0.1, length=11) # seq()
@test norm(f3'(x) - cos(x)) < 10e-4
end
end
end

#
# gradient()
#

f4(x::Vector) = (100.0 - x[1])^2 + (50.0 - x[2])^2
@test norm(Calculus.gradient(f4, :forward)([100.0, 50.0]) - [0.0, 0.0]) < 10e-4
@test norm(Calculus.gradient(f4, :central)([100.0, 50.0]) - [0.0, 0.0]) < 10e-4
@test norm(Calculus.gradient(f4)([100.0, 50.0]) - [0.0, 0.0]) < 10e-4
@testset "gradient()" begin
f4(x::Vector) = (100.0 - x[1])^2 + (50.0 - x[2])^2
@test norm(Calculus.gradient(f4, :forward)([100.0, 50.0]) - [0.0, 0.0]) < 10e-4
@test norm(Calculus.gradient(f4, :central)([100.0, 50.0]) - [0.0, 0.0]) < 10e-4
@test norm(Calculus.gradient(f4)([100.0, 50.0]) - [0.0, 0.0]) < 10e-4
end

#
# jacobian()
#

@test norm(Calculus.jacobian(identity, rand(3), :forward) - Matrix(1.0I, 3, 3)) < 10e-4
@test norm(Calculus.jacobian(identity, rand(3), :central) - Matrix(1.0I, 3, 3)) < 10e-4
@testset "jacobian()" begin
@test norm(Calculus.jacobian(identity, rand(3), :forward) - Matrix(1.0I, 3, 3)) < 10e-4
@test norm(Calculus.jacobian(identity, rand(3), :central) - Matrix(1.0I, 3, 3)) < 10e-4
end

#
# second_derivative()
#

@test norm(second_derivative(x -> x^2)(0.0) - 2.0) < 10e-4
@test norm(second_derivative(x -> x^2)(1.0) - 2.0) < 10e-4
@test norm(second_derivative(x -> x^2)(10.0) - 2.0) < 10e-4
@test norm(second_derivative(x -> x^2)(100.0) - 2.0) < 10e-4
@testset "second_derivative()" begin
@test norm(second_derivative(x -> x^2)(0.0) - 2.0) < 10e-4
@test norm(second_derivative(x -> x^2)(1.0) - 2.0) < 10e-4
@test norm(second_derivative(x -> x^2)(10.0) - 2.0) < 10e-4
@test norm(second_derivative(x -> x^2)(100.0) - 2.0) < 10e-4
end

#
# hessian()
#

f5(x) = sin(x[1]) + cos(x[2])
@test norm(Calculus.gradient(f5)([0.0, 0.0]) - [cos(0.0), -sin(0.0)]) < 10e-4
@test norm(hessian(f5)([0.0, 0.0]) - [-sin(0.0) 0.0; 0.0 -cos(0.0)]) < 10e-4
@testset "hessian()" begin
f5(x) = sin(x[1]) + cos(x[2])
x₁, x₂ = 1.0, 1.0
x = [x₁, x₂]
@test norm(Calculus.gradient(f5)(x) - [cos(x₁), -sin(x₂)]) < 10e-4
@test norm(hessian(f5)(x) - [-sin(x₁) 0.0; 0.0 -cos(x₂)]) < 10e-4
end
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