deriv API

The deriv package estimates the derivative of a function at a point from a caller-supplied initial step size h. Each function returns the estimate together with an error bound.

deriv_central : (f: Func_Math, x: Double, h: Double) -> (Double, Double)

Description: Computes the derivative of the function f at point x using an adaptive central difference method, which balances truncation and round-off errors for improved accuracy.

Parameters:

• f: Func_Math — A mathematical function that takes a Double as input and Returns: a Double as output.
• x: Double — The point at which the derivative is evaluated.
• h: Double — The initial step size for numerical differentiation, which will be adaptively adjusted if necessary.

Returns:

A tuple (Double, Double), where:

• The first value is the estimated derivative at x.
• The second value is the total estimated error, combining truncation and round-off errors.

Example Usage:

test "deriv_central" {
  let f = fn(x : Double) { x * x } // f(x) = x², f'(x) = 2x
  let (deriv, error) = deriv_central(f, 2.0, 0.1)
  inspect((deriv - 4.0).abs() < error, content="true")
}

Notes:

• The method automatically adjusts the step size to achieve the best accuracy.
• Works well for smooth functions but may be less accurate near discontinuities.
• Compared to a simple 2-point method, this approach significantly reduces numerical errors.

deriv_forward : (f: Func_Math, x: Double, h: Double) -> (Double, Double)

Description: Computes the derivative of the function f at point x using an adaptive forward difference method, which optimizes step size to minimize truncation and round-off errors.

Parameters:

• f: Func_Math — A mathematical function that takes a Double as input and Returns: a Double as output.
• x: Double — The point at which the derivative is evaluated.
• h: Double — The initial step size for numerical differentiation, which will be adaptively adjusted if necessary.

Returns:

A tuple (Double, Double), where:

• The first value is the estimated derivative at x.
• The second value is the total estimated error, combining truncation and round-off errors.

Example Usage:

test "deriv_forward" {
  let f = fn(x : Double) { x * x } // f(x) = x², f'(x) = 2x
  let (deriv, error) = deriv_forward(f, 2.0, 0.1)
  inspect((deriv - 4.0).abs() < error, content="true")
}

Notes:

• The method automatically refines the step size for better accuracy.
• Compared to basic forward difference methods, this approach significantly reduces numerical errors.
• Works well for smooth functions but may be less accurate for functions with sharp changes or discontinuities.
• This method is especially useful when evaluating derivatives at boundaries where central differences are not feasible.

deriv_backward : (f: Func_Math, x: Double, h: Double) -> (Double, Double)

Description: Computes the derivative of the function f at point x using an adaptive backward difference method, which mirrors forward differentiation but with a negative step size.

Parameters:

• f: Func_Math — A mathematical function that takes a Double as input and Returns: a Double as output.
• x: Double — The point at which the derivative is evaluated.
• h: Double — The initial step size for numerical differentiation, which will be adaptively adjusted if necessary.

Returns: A tuple (Double, Double), where:

• The first value is the estimated derivative at x.
• The second value is the total estimated error, combining truncation and round-off errors.

Example Usage:

test "deriv_backward" {
  let f = fn(x : Double) { x * x } // f(x) = x², f'(x) = 2x
  let (deriv, error) = deriv_backward(f, 2.0, 0.1)
  inspect((deriv - 4.0).abs() < error, content="true")
}

Notes:

• The method automatically refines the step size to optimize accuracy.
• Suitable for cases where backward differences are preferred, such as differentiation near the upper boundary of a dataset.
• Shares the same numerical advantages as deriv_forward, significantly reducing numerical errors compared to simple finite differences.