Simple and accurate complete elliptic integrals for the full range of modulus

TL;DR

This paper introduces simple, highly accurate approximations for complete elliptic integrals of the first and second kind, effective across the entire range of the modulus, with practical applications in physics and engineering.

Contribution

The paper proposes novel asymptotic-based approximations for elliptic integrals that are accurate in all regimes and introduces an inverse function with reliable initialization, enhancing computational efficiency.

Findings

Achieves 0.06% and 0.01% average error in mid-range for K and E

Provides an inverse of K with reliable Newton initialization

Offers an algorithm for exact integrals and derivatives suitable for limited platforms

Abstract

The complete elliptic integral of the first and second kind, K(k) and E(k), appear in a multitude of physics and engineering applications. Because there is no known closed-form, the exact values have to be computed numerically. Here, approximations for the integrals are proposed based on their asymptotic behaviors. An inverse of K is also presented. As a result, the proposed K(k) and E(k) reproduce the exact analytical forms both in the zero and asymptotic limits, while in the mid-range of modulus maintain average error of 0.06% and 0.01% respectively. The key finding is the ability to compute the integrals with exceptional accuracy on both limits of elliptical conditions. An accuracy of 1 in 1,000 should be sufficient for practical or prototyping engineering and architecture designs. The simplicity should facilitate discussions of advanced physics topics in introductory physics classes, and enable broader collaborations among researchers from other fields of expertise. For example, the phase space of energy-conserving nonlinear pendulum using only elementary functions is discussed. The proposed inverse of K is shown to be Never Failing Newton Initialization and is an important step for the computation of the exact inverse. An algorithm based on Arithmetic-Geometric Mean for computing exact integrals and their derivatives are also presented, which should be useful in a platform that special functions are not accessible such as web-based and firmware developments. Comparisons with sharp bounds from the mathematical inequalities literature further highlight the competitiveness of the proposed approximations.