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Change Taylor finite difference arguments #15

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Merged
merged 1 commit into from
Aug 9, 2013
Merged

Change Taylor finite difference arguments #15

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62 changes: 24 additions & 38 deletions src/finite_difference.jl
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Original file line number Diff line number Diff line change
@@ -1,10 +1,3 @@
##############################################################################
##
## TODO: dtype == :complex
## See minFunc code for examples of how this could be done
##
##############################################################################

##############################################################################
##
## Graveyard of Alternative Functions
Expand All @@ -24,13 +17,13 @@

##############################################################################
##
## Derivative of f: R -> R
## Derivative of f: C -> R
##
##############################################################################

function finite_difference{T <: Number}(f::Function,
x::T,
dtype::Symbol)
dtype::Symbol = :central)
if dtype == :forward
epsilon = sqrt(eps(max(one(T), abs(x))))
xplusdx = x + epsilon
Expand All @@ -41,13 +34,14 @@ function finite_difference{T <: Number}(f::Function,
xplusdx, xminusdx = x + epsilon, x - epsilon
# Is it better to do 2 * epsilon or xplusdx - xminusdx?
return (f(xplusdx) - f(xminusdx)) / (2 * epsilon)
elseif dtype == :complex
epsilon = eps(x)
xplusdx = x + epsilon * im
return imag(f(xplusdx)) / epsilon
else
error("dtype must be :forward or :central")
error("dtype must be :forward, :central or :complex")
end
end
function finite_difference(f::Function, x::Number)
finite_difference(f, x, :central)
end

##############################################################################
##
Expand Down Expand Up @@ -95,7 +89,7 @@ function finite_difference!{S <: Number, T <: Number}(f::Function,
end
function finite_difference{T <: Number}(f::Function,
x::Vector{T},
dtype::Symbol)
dtype::Symbol = :central)
# Allocate memory for gradient
g = Array(Float64, length(x))

Expand All @@ -105,9 +99,6 @@ function finite_difference{T <: Number}(f::Function,
# Return mutated gradient
return g
end
function finite_difference{T <: Number}(f::Function, x::Vector{T})
finite_difference(f, x, :central)
end

##############################################################################
##
Expand All @@ -121,7 +112,7 @@ function finite_difference_jacobian!{R <: Number,
x::Vector{R},
f_x::Vector{S},
J::Array{T},
dtype::Symbol)
dtype::Symbol = :central)
# What is the dimension of x?
m, n = size(J)

Expand Down Expand Up @@ -154,7 +145,7 @@ function finite_difference_jacobian!{R <: Number,
end
function finite_difference_jacobian{T <: Number}(f::Function,
x::Vector{T},
dtype::Symbol)
dtype::Symbol = :central)
# Establish a baseline for f_x
f_x = f(x)

Expand All @@ -181,14 +172,9 @@ end
function finite_difference_hessian(f::Function,
g::Function,
x::Number,
dtype::Symbol)
dtype::Symbol = :central)
finite_difference(g, x, dtype)
end
function finite_difference_hessian(f::Function,
g::Function,
x::Number)
finite_difference(g, x, :central)
end

##############################################################################
##
Expand Down Expand Up @@ -242,8 +228,12 @@ function finite_difference_hessian{T <: Number}(f::Function,
# Return the Hessian
return H
end
finite_difference_hessian{T}(f::Function, g::Function, x::Vector{T}, dtype::Symbol) = finite_difference_jacobian(g, x, dtype)
finite_difference_hessian{T}(f::Function, g::Function, x::Vector{T}) = finite_difference_jacobian(g, x, :central)
function finite_difference_hessian{T}(f::Function,
g::Function,
x::Vector{T},
dtype::Symbol = :central)
finite_difference_jacobian(g, x, dtype)
end

##############################################################################
##
Expand All @@ -255,7 +245,10 @@ finite_difference_hessian{T}(f::Function, g::Function, x::Vector{T}) = finite_di

# Higher precise finite difference method based on Taylor series approximation.
# h is the stepsize
function taylor_finite_difference(f::Function, x::Float64, h::Float64, dtype::Symbol)
function taylor_finite_difference(f::Function,
x::Real,
dtype::Symbol = :central,
h::Real = 10e-4)
if dtype == :forward
f_x = f(x)
d = 2^3 * (2^2 * (f(x + h) - f_x) - (f(x + 2 * h) - f_x))
Expand All @@ -270,17 +263,10 @@ function taylor_finite_difference(f::Function, x::Float64, h::Float64, dtype::Sy
end
return d
end
function taylor_finite_difference(f::Function, x::Float64, h::Float64)
taylor_finite_difference(f, x, h, :central)
end
function taylor_finite_difference(f::Function, x::Float64, dtype::Symbol)
taylor_finite_difference(f, x, 10e-4, dtype)
end
function taylor_finite_difference(f::Function, x::Float64)
taylor_finite_difference(f, x, 10e-4)
end

function taylor_finite_difference_hessian(f::Function, x::Float64, h::Float64)
function taylor_finite_difference_hessian(f::Function,
x::Real,
h::Real)
f_x = f(x)
d = 4^6 * (2^4 * (f(x + h) + f(x - h) - 2 * f_x) - (f(x + 2 * h) + f(x - 2 * h) - 2 * f_x))
d += - (2^4 * (f(x + 4 * h) + f(x - 4 * h) - 2 * f_x) - (f(x + 8 * h) + f(x - 8 * h) - 2 * f_x))
Expand Down
4 changes: 2 additions & 2 deletions test/finite_difference.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -72,5 +72,5 @@ gx = gradient(fx)
@elapsed Calculus.finite_difference(x -> x^2, 1.0, :forward)
@elapsed Calculus.finite_difference(x -> x^2, 1.0, :central)
@elapsed Calculus.finite_difference(x -> x^2, 1.0)
@elapsed Calculus.taylor_finite_difference(x -> x^2, 1.0, 10e-4, :forward)
@elapsed Calculus.taylor_finite_difference(x -> x^2, 1.0, 10e-4, :central)
@elapsed Calculus.taylor_finite_difference(x -> x^2, 1.0, :forward, 10e-4)
@elapsed Calculus.taylor_finite_difference(x -> x^2, 1.0, :central, 10e-4)