Skip to content

major refactor in choice of epsilon for all finite_difference methods #17

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
merged 3 commits into from
Aug 20, 2013
Merged

major refactor in choice of epsilon for all finite_difference methods #17

Changes from all commits
Commits
Show all changes
3 commits
Select commit Hold shift + click to select a range
18425a7
major refactor in choice of epsilon for all finite_difference methods
eriktaubeneck Aug 15, 2013
610c62b
added special abs fuction so complex step finitie difference can work…
eriktaubeneck Aug 15, 2013
c737736
bug fix in complex_differentiable_abs fuction
eriktaubeneck Aug 15, 2013
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
106 changes: 78 additions & 28 deletions src/finite_difference.jl
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -2,13 +2,6 @@
##
## Graveyard of Alternative Functions
##
## In Nodedal and Wright, epsilon was calculated as
##
## epsilon = sqrt(eps())
##
## In the examples I've tested, this strategy is best when x is small
## and becomes very bad when x is larger
##
## The proper setting of epsilon depends on whether we're doing central
## differencing or forward differencing. See, for example, section 5.7 of
## Numerical Recipes.
Expand All @@ -21,28 +14,83 @@
##
##############################################################################

macro forwardrule(x, T, e)
x, T, e = esc(x), esc(T), esc(e)
quote
$e = sqrt(eps($T)) * max(one($T), abs($x))
end
end

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

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

macro complexrule(x, T, e)
x, T, e = esc(x), esc(T), esc(e)
quote
$e = eps($x)
end
end

function finite_difference{T <: Number}(f::Function,
x::T,
dtype::Symbol = :central)
if dtype == :forward
epsilon = sqrt(eps(T))*max(one(T),abs(x))
@forwardrule x T epsilon
xplusdx = x + epsilon
# use machine-representable numbers
return (f(xplusdx) - f(x)) / epsilon
elseif dtype == :central
epsilon = cbrt(eps(T))*max(one(T), abs(x))
@centralrule x T epsilon
xplusdx, xminusdx = x + epsilon, x - epsilon
# Is it better to do 2 * epsilon or xplusdx - xminusdx?
return (f(xplusdx) - f(xminusdx)) / (2 * epsilon)
return (f(xplusdx) - f(xminusdx)) / (epsilon + epsilon)
elseif dtype == :complex
epsilon = eps(x)
@complexrule x T epsilon
xplusdx = x + epsilon * im
return imag(f(xplusdx)) / epsilon
else
error("dtype must be :forward, :central or :complex")
end
end

##############################################################################
##
## Complex Step Finite Differentiation Tools
##
## Martins, Sturdza, and Alonso (2003) suggest the only non-analytic
## fuction of which complex step finite difference approximation
## will fail and finite difference will not is abs().
## They suggest redefining as follows for z = x + im*y
##
## if x < 0
## -x - im * y
## else
## x + im * y
##
## This is provided below as complex_differentiable_abs (renaming encouraged!)
##
## Also, if your fuctions has control flow using < or >, you must compare
## real(z) for your control flow.
##
##############################################################################

function complex_differentiable_abs{T <: Complex}(z::T)
if real(z) < 0
return -real(z) - im * imag(z)
else
return real(z) + im * imag(z)
end
end

##############################################################################
##
## Gradient of f: R^n -> R
Expand All @@ -63,7 +111,7 @@ function finite_difference!{S <: Number, T <: Number}(f::Function,
# Establish a baseline value of f(x).
f_x = f(x)
for i = 1:n
epsilon = sqrt(eps(max(one(T), abs(x[i]))))
@forwardrule x[i] S epsilon
oldx = x[i]
x[i] = oldx + epsilon
f_xplusdx = f(x)
Expand All @@ -72,14 +120,14 @@ function finite_difference!{S <: Number, T <: Number}(f::Function,
end
elseif dtype == :central
for i = 1:n
epsilon = cbrt(eps(max(one(T), abs(x[i]))))
@centralrule x[i] S epsilon
oldx = x[i]
x[i] = oldx + epsilon
f_xplusdx = f(x)
x[i] = oldx - epsilon
f_xminusdx = f(x)
x[i] = oldx
g[i] = (f_xplusdx - f_xminusdx) / (2 * epsilon)
g[i] = (f_xplusdx - f_xminusdx) / (epsilon + epsilon)
end
else
error("dtype must be :forward or :central")
Expand Down Expand Up @@ -119,7 +167,7 @@ function finite_difference_jacobian!{R <: Number,
# Iterate over each dimension of the gradient separately.
if dtype == :forward
for i = 1:n
epsilon = sqrt(eps(max(one(T), abs(x[i]))))
@forwardrule x[i] R epsilon
oldx = x[i]
x[i] = oldx + epsilon
f_xplusdx = f(x)
Expand All @@ -128,14 +176,14 @@ function finite_difference_jacobian!{R <: Number,
end
elseif dtype == :central
for i = 1:n
epsilon = cbrt(eps(max(one(T), abs(x[i]))))
@centralrule x[i] R epsilon
oldx = x[i]
x[i] = oldx + epsilon
f_xplusdx = f(x)
x[i] = oldx - epsilon
f_xminusdx = f(x)
x[i] = oldx
J[:, i] = (f_xplusdx - f_xminusdx) / (2 * epsilon)
J[:, i] = (f_xplusdx - f_xminusdx) / (epsilon + epsilon)
end
else
error("dtype must :forward or :central")
Expand Down Expand Up @@ -165,8 +213,9 @@ end
##
##############################################################################

function finite_difference_hessian{T <: Number}(f::Function, x::T)
epsilon = eps(max(one(T), abs(x)))^(1/4)
function finite_difference_hessian{T <: Number}(f::Function,
x::T)
@hessianrule x T epsilon
(f(x + epsilon) - 2*f(x) + f(x - epsilon))/epsilon^2
end
function finite_difference_hessian(f::Function,
Expand All @@ -189,21 +238,22 @@ function finite_difference_hessian!{S <: 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]
epsilon = eps(max(one(T), abs(x[i])))^(1/4)
@hessianrule x[i] S epsilon
xpp[i], xmm[i] = xi + epsilon, xi - epsilon
H[i, i] = (f(xpp) - 2*fx + f(xmm)) / epsilon^2
epsiloni = cbrt(eps(max(one(T), abs(x[i]))))
@centralrule x[i] S 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]
epsilonj = cbrt(eps(max(one(T), abs(x[j]))))
@centralrule x[j] S epsilonj
xp = xj + epsilonj
xm = xj - epsilonj
xpp[j], xpm[j], xmp[j], xmm[j] = xp, xm, xp, xm
Expand All @@ -228,10 +278,10 @@ function finite_difference_hessian{T <: Number}(f::Function,
# Return the Hessian
return H
end
function finite_difference_hessian{T}(f::Function,
g::Function,
x::Vector{T},
dtype::Symbol = :central)
function finite_difference_hessian{T <: Number}(f::Function,
g::Function,
x::Vector{T},
dtype::Symbol = :central)
finite_difference_jacobian(g, x, dtype)
end

Expand Down