Formulae of numerical differentiation

TL;DR

This paper derives new central difference formulas for first and second derivatives from discrete data points, avoiding polynomial interpolation, and analyzes their accuracy and spectral properties.

Contribution

It introduces novel formulas for numerical differentiation that do not require interpolating polynomials, enhancing computational efficiency and accuracy.

Findings

Derived formulas for first and second derivatives at arbitrary points

Analyzed the spectral characteristics of the weight coefficients

Validated the method with functions of known derivatives

Abstract

We derived the formulae of central differentiation for the finding of the first and second derivatives of functions given in discrete points, with the number of points being arbitrary. The obtained formulae for the derivative calculation do not require direct construction of the interpolating polynomial. As an example of the use of the developed method we calculated the first derivative of the function having known analytical value of the derivative. The result was examined in the limiting case of infinite number of points. We studied the spectral characteristics of the weight coefficients sequence of the numerical differentiation formulae. The performed investigation enabled one to analyze the accuracy of the numerical differentiation carried out with the use of the developed technique.