Description

The second_derivative function is used to compute the second derivative of a univariate function that maps $\mathbb{R}$ to $\mathbb{R}$. For example, x -> sin(x) is a univariate function that might be passed to the second_derivative function.

The second_derivative function comes in three core variants:

A pure function that directly computes the second derivative of a function f at a fixed numeric value of x and returns a number as a result.
A mutating function that computes the second derivative of a function f at a fixed numeric value of x and stores the result into a user-provided output array. This function returns nothing since its action is focused on mutation.
A higher-order function that constructs a new function f_prime2 that will compute the second derivative of f at any point x that is provided as an argument to f_prime2. This newly constructed function is the return value for this variant of the second_derivative function.

Primary Methods

The primary methods that most users will want to use are the following:

Use second_derivative(f::Function, x::AbstractFloat) to approximate the second derivative of f at x. This is the pure variant described above.
Use second_derivative!(output::AbstractArray, f::Function, x::AbstractFloat) to approximate the second derivative of f at x and store the result into the output array.
Use second_derivative(f::Function) to generate a new function that approximates the true second derivative function of f. The new function f_prime2 can be evaluated at any point x that is desired after it is constructed.

Detailed Method-Level Documentation

# FiniteDiff.second_derivative — Method.

second_derivative(f::Function, x::AbstractFloat)

Description

Evaluate the second derivative of f at x using finite differences. In mathematical notation, we calculate,

$\frac{f(x + \epsilon) - 2 f(x) + f(x - \epsilon)}{\epsilon^2},$

where $\epsilon$ is chosen to be small enough to approximate the second derivative, but not so small as to suffer from extreme numerical inaccuracy.

Arguments

f::Function: The function to be differentiated.
x::AbstractFloat: The point at which to evaluate the derivative of f. Its type must implement eps.

Returns

y::Real: The second derivative of f evaluated at x.

Examples

import FiniteDiff: second_derivative
y = second_derivative(sin, 1.0)

source

# FiniteDiff.second_derivative! — Method.

second_derivative!(
    output::AbstractArray,
    f::Function,
    x::AbstractFloat,
)

Description

Evaluate the second derivative of f at x using finite differences. In mathematical notation, we calculate,

$\frac{f(x + \epsilon) - 2 f(x) + f(x - \epsilon)}{\epsilon^2},$

where $\epsilon$ is chosen to be small enough to approximate the second derivative, but not so small as to suffer from extreme numerical inaccuracy.

The first argument, output, will be mutated so that the value of the second derivative is its first element.

Arguments

output::AbstractArray: An array whose first element will be mutated.
f::Function: The function to be differentiated.
x::AbstractFloat: The point at which to evaluate the derivative of f. Its type must implement eps.

Returns

void::Nothing: This function is called for its side-effects.

Examples

import FiniteDiff: second_derivative
y = Array(Float64, 1)
second_derivative!(y, sin, 1.0)

source

# FiniteDiff.second_derivative — Method.

second_derivative(f::Function; mutates::Bool=false)

Description

Construct a new function that will evaluate the second derivative of f at any point x. A keyword argument specifies whether the resulting function should be mutating or non-mutating.

Arguments

f::Function: The function to be differentiated.

Keyword Arguments

mutates::Bool = false: Determine whether the resulting function will mutate its inputs or will be a pure function.

Returns

void::Nothing: This function is called for its side-effects.

Examples

import FiniteDiff: second_derivative
f′′ = second_derivative(sin)
f′′(1.0)

source