diff API

The diff package estimates the derivative of a function at a point without a step size from the caller: the step is derived from machine precision and the local behaviour of the function. Each function returns the estimate together with an error bound.

diff_backward : (f : Func_Math, x : Double) -> (Double, Double)

Description:

Computes the numerical derivative of a function at a given point using the backward difference method with adaptive step size.

Parameters:

• f: Func_Math — A function that takes a Double and Returns: a Double. The function to be differentiated.
• x: Double — The point at which to compute the derivative.

Returns:

A tuple (Double, Double), where:

• The first value is the estimated derivative value using backward difference approximation.
• The second value is the estimated absolute error of the computation.

Example Usage:

test "diff_backward" {
  let f = fn(x : Double) { x * x } // f(x) = x^2, f'(x) = 2x
  let (derivative, error) = diff_backward(f, 2.0)
  inspect((derivative - 4.0).abs() < error, content="true")
}

Notes:

• The function evaluates the given function at three backward points using an adaptive step size.
• It applies Neville’s recursion to compute divided differences and estimate the derivative.
• The method includes an error estimation based on the second-order divided difference.
• If h is too large, accuracy may decrease; if too small, numerical precision issues may arise.

diff_forward : (f : Func_Math, x : Double) -> (Double, Double)

Description:

Computes the numerical derivative of a function at a given point using the forward difference method with adaptive step size.

Parameters:

• f: Func_Math — A function that takes a Double and Returns: a Double. The function to be differentiated.
• x: Double — The point at which to compute the derivative.

Returns:

A tuple (Double, Double), where:

• The first value is the estimated derivative value using forward difference approximation.
• The second value is the estimated absolute error of the computation.

Notes:

• The function evaluates the given function at three forward points to choose the step size, so it only needs values of f at and above x.

diff_central : (f : Func_Math, x : Double) -> (Double, Double)

Description:

Computes the numerical derivative of a function at a given point using the central difference method with adaptive step size.

Parameters:

• f: Func_Math — A function that takes a Double and Returns: a Double. The function to be differentiated.
• x: Double — The point at which to compute the derivative.

Returns:

A tuple (Double, Double), where:

• The first value is the estimated derivative value using central difference approximation.
• The second value is the estimated absolute error of the computation.

Example Usage:

test "diff_central" {
  let f = fn(x : Double) { x * x } // f(x) = x^2, f'(x) = 2x
  let (derivative, error) = diff_central(f, 2.0)
  inspect((derivative - 4.0).abs() < error, content="true")
}

Notes:

• The function evaluates the given function at four points around x to choose the step size.