Formulae of numerical differentiation

TL;DR

This work develops central differentiation formulas for the first and second derivatives using an arbitrary $(2n+1)$-point stencil, avoiding explicit interpolating polynomials. It derives explicit weight coefficients $\alpha_m^{(1)}(n)$ and $\alpha_m^{(2)}(n)$ based on $\pi_m(n)=\prod_{k=1, k\neq m}^n (1-m^2/k^2)$ and demonstrates accuracy via a $y(x)=\sin x$ example, including the $n\to\infty$ limit where $\alpha_m^{(1)}(n) \to (-1)^{m+1} 2/m$. A spectral analysis shows these weight sequences act as high-fidelity linear filters, with the first-derivative spectrum approaching $y_1(r)=2\pi r/N$ and the second-derivative spectrum approaching $y_2(r)=-(2\pi r/N)^2$, across a wide range of spatial frequencies. The methods provide accurate, explicit finite-difference coefficients suitable for high-precision numerical differentiation and potential use in lattice-quantum-field simulations.

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.

Figures (5)

• Figure 1: The coefficients $\alpha_{m}^{(1)}(1)$.
• Figure 2: The spectra of various sequences $\alpha_{m}^{(1)}(n)$ at $N=2000$.
• Figure 3: The function $\delta_1(r)=\Im\mathfrak{m}(\beta^*_1(r))-y_1(r)$ versus $r$ for different $n$.
• Figure 4: The spectra of various sequences $\alpha_{m}^{(2)}(n)$ at $N=2000$.
• Figure 5: The function $\delta_2(r)=\Re\mathfrak{e}(\beta^*_2(r))-y_2(r)$ versus $r$ for different $n$.