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Algodiff: numerical differentiation module

This module provides APIs for performing numerical differentiation of any user defined functions. The module is part of Algodiff module, and only supports float numbers at the moment.

Numerical differentiation is subject to truncation error and round-off error. If you need more precise results, please consider to use Algodiff.AD module for algorithmic differentiation.

Core APIs of differentiation
val diff : (float -> float) -> float -> float

diff f x returns the numrical derivative of a function f : float -> float at point x. Simply calling diff f will return its derivative function g of the same type, i.e. g : float -> float.

Keep calling this function will give you higher-order derivatives of f, i.e. f |> diff |> diff |> diff |> .... However, the error grows quickly due to the numerical method of differentiation.

val diff2 : (float -> float) -> float -> float

diff2 f x returns second-order derivative of f : float -> float at x.

val grad : (vec -> float) -> vec -> vec

grad f x returns the gradient of f at point x. Note the type of the function is f : mat -> float, i.e. f takes an one-row matrix as input and outputs a float scalar. An exception will be raised if the passed in paramter is not an one-row matrix.

val jacobian : (vec -> vec) -> vec -> mat

jacobian f x returns the jacobian of f at point x. Again, the mat type here refers to one-row matrix. I am still thinking maybe it is necessary to introduce vector as a separate type in Owl? Currently, the returned result uses numerator-layout. I.e., if x is n-dimensional and f x is y-dimensional, then the returned jacobian is an m x n matrix.

val jacobianT : (vec -> vec) -> vec -> mat

jacobianT f returns the transposed jacobian of f at point x.

Enhanced APIs with more returned information
val diff' : (float -> float) -> float -> float * float

diff' f x is similar to diff but also returns the function value f x.

val diff2' : (float -> float) -> float -> float * float

diff2' f x is similar to diff2 but also returns the function value f x.

val grad' : (vec -> float) -> vec -> vec * vec

grad' f x is similar to grad but also returns the function value f x.

val jacobian' : (vec -> vec) -> vec -> vec * mat

jacobian' f x is similar to jacobian but also returns the function value f x.

val jacobianT' : (vec -> vec) -> vec -> vec * mat

jacobianT' f x is similar to jacobianT but also returns the function value f x.