numericalderivative

What is it?

The goal of this project is to compute the derivative of a function using finite difference formulas. The difficulty with these formulas is that it must use a step which must be neither too large (otherwise the truncation error dominates the error) nor too small (otherwise the condition error dominates). For this purpose, it provides exact methods (based on the value of higher derivatives) and approximate methods (based on function values). Furthermore, the module provides finite difference formulas for the first, second, third or any arbitrary order derivative of a function. Finally, this package provides 15 benchmark problems for numerical differentiation.

This module makes it possible to do this:

import math
import numericalderivative as nd

def scaled_exp(x):
    alpha = 1.0e6
    return math.exp(-x / alpha)


h0 = 1.0e5  # This is the initial step size
x = 1.0e0
algorithm = nd.SteplemanWinarsky(scaled_exp, x)
h_optimal, iterations = algorithm.compute_step(h0)
f_prime_approx = algorithm.compute_first_derivative(h_optimal)

Documentation & references

• Package documentation

Authors

• Michaël Baudin, 2024

Installation

To install from Github:

git clone https://github.com/mbaudin47/numerical_derivative.git
cd numerical_derivative
python setup.py install

To install from Pip:

pip install numericalderivative

References

• Gill, P. E., Murray, W., Saunders, M. A., & Wright, M. H. (1983). Computing forward-difference intervals for numerical optimization. SIAM Journal on Scientific and Statistical Computing, 4(2), 310-321.
• Adaptive numerical differentiation R. S. Stepleman and N. D. Winarsky Journal: Math. Comp. 33 (1979), 1257-1264
• Dumontet, J., & Vignes, J. (1977). Détermination du pas optimal dans le calcul des dérivées sur ordinateur. RAIRO. Analyse numérique, 11 (1), 13-25.

Roadmap

• GillMurraySaundersWright: update to setup from absolute precision instead of relative precision. This is because the current computation based on the relative precision cannot work if the function value at point x is zero.

• Implement the method of:

  Shi, H. J. M., Xie, Y., Xuan, M. Q., & Nocedal, J. (2022). Adaptive finite-difference interval estimation for noisy derivative-free optimization. SIAM Journal on Scientific Computing, 44(4), A2302-A2321.

• Implement a method to compute the absolute error of evaluation of the function f, for example :

  J. J. Moré and S. M. Wild, Estimating computational noise, SIAM Journal on Scientific Computing, 33 (2011), pp. 1292–1314.

• Make sure that the API help page of each method has a paragraph on the cases of failure. Also, add a paragraph on an alternative method using compute_step of F.D. formula, with some extra assumption on a higher derivative.