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Open Source Implementations of Numerical Algorithms for Computing the Complete Elliptic Integral of the First Kind

Hong-Yan Zhang, Wen-Juan Jiang

TL;DR

This work addresses the need for license-free, open-source numerical implementations of the complete elliptic integral of the first kind, $K(k)$. It develops four top-down numerical schemes—an infinite-series expansion, AGM-based evaluation, Gauss-Chebyshev, and Gauss-Legendre quadrature—each with verification, validation, and testing to ensure correctness against the commercial ${\mathcal{K}}_{cs}(m)$. The methods are complemented by robust auxiliary components, including generation of Legendre polynomials and their derivatives, Newton-based root finding, and modular, reusable code for orthogonal-polynomial-related computations. The authors provide cross-method consistency, detailed error analyses, and open-source code on GitHub, highlighting practical impact for STEM education and research without licensing constraints.

Abstract

The complete elliptic integral of the first kind (CEI-1) plays a significant role in mathematics, physics and engineering. There is no simple formula for its computation, thus numerical algorithms are essential for coping with the practical problems involved. The commercial implementations for the numerical solutions, such as the functions \lstinline|ellipticK| and \lstinline|EllipticK| provided by MATLAB and Mathematica respectively, are based on $\mathcal{K}_{\mathrm{cs}}(m)$ instead of the usual form $K(k)$ such that $\mathcal{K}_{\mathrm{cs}}(k^2) =K(k)$ and $m=k^2$. It is necessary to develop open source implementations for the computation of the CEI-1 in order to avoid potential risks of using commercial software and possible limitations due to the unknown factors. In this paper, the infinite series method, arithmetic-geometric mean (AGM) method, Gauss-Chebyshev method and Gauss-Legendre methods are discussed in details with a top-down strategy. The four key algorithms for computing CEI-1 are designed, verified, validated and tested, which can be utilized in R\& D and be reused properly. Numerical results show that our open source implementations based on $K(k)$ are equivalent to the commercial implementation based on $\mathcal{K}_{\mathrm{cs}}(m)$. The general algorithms for computing orthogonal polynomials developed are significant byproducts in the sense of STEM education and scientific computation.

Open Source Implementations of Numerical Algorithms for Computing the Complete Elliptic Integral of the First Kind

TL;DR

This work addresses the need for license-free, open-source numerical implementations of the complete elliptic integral of the first kind, . It develops four top-down numerical schemes—an infinite-series expansion, AGM-based evaluation, Gauss-Chebyshev, and Gauss-Legendre quadrature—each with verification, validation, and testing to ensure correctness against the commercial . The methods are complemented by robust auxiliary components, including generation of Legendre polynomials and their derivatives, Newton-based root finding, and modular, reusable code for orthogonal-polynomial-related computations. The authors provide cross-method consistency, detailed error analyses, and open-source code on GitHub, highlighting practical impact for STEM education and research without licensing constraints.

Abstract

The complete elliptic integral of the first kind (CEI-1) plays a significant role in mathematics, physics and engineering. There is no simple formula for its computation, thus numerical algorithms are essential for coping with the practical problems involved. The commercial implementations for the numerical solutions, such as the functions \lstinline|ellipticK| and \lstinline|EllipticK| provided by MATLAB and Mathematica respectively, are based on instead of the usual form such that and . It is necessary to develop open source implementations for the computation of the CEI-1 in order to avoid potential risks of using commercial software and possible limitations due to the unknown factors. In this paper, the infinite series method, arithmetic-geometric mean (AGM) method, Gauss-Chebyshev method and Gauss-Legendre methods are discussed in details with a top-down strategy. The four key algorithms for computing CEI-1 are designed, verified, validated and tested, which can be utilized in R\& D and be reused properly. Numerical results show that our open source implementations based on are equivalent to the commercial implementation based on . The general algorithms for computing orthogonal polynomials developed are significant byproducts in the sense of STEM education and scientific computation.
Paper Structure (23 sections, 4 theorems, 79 equations, 11 figures, 6 tables)

This paper contains 23 sections, 4 theorems, 79 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Infinite Series Method
  4. AGM Method
  5. Gauss-Chebyshev Method
  6. Gauss-Legendre Method
  7. Numerical Algorithms
  8. Infinite Series Algorithm
  9. AGM Algorithm
  10. Gauss-Chebyshev Algorithm
  11. Gauss-Legendre Algorithm
  12. Compute CEI-1 with the Gauss-Legendre based on the roots, weights and kernel function
  13. Generating the Weights Specified by the Legendre polynomial and its Derivative
  14. Solving the $n$ Roots of the Legendre Polynomial $P_n(x)$
  15. Generating the Legendre polynomials and their derivatives
  16. ...and 8 more sections

Key Result

Theorem 1

For $f\in C^{2n}[-1,1]$, the integral ${\int^1_{-1} \frac{f(x)}{\sqrt{1-x^2}}\mathop{\mathrm{d}}\nolimits x }$ can be computed by the Gauss-Chebyshev formula where is the approximation error.

Figures (11)

  • Figure 1: Value of ${\mathcal{K}}_{\mathrm{cs}}(m)$ obtained by the MATLAB and Mathematica for $m = [0.00, 0.01, 0.20, 0.30, 0.40, 0.50]$
  • Figure 2: Four methods for computing CEI-1
  • Figure 3: Procedures involved in calculating the complete elliptic integral CEI-1 with infinite series method.
  • Figure 4: Procedures for calculating CEI-1 with AGM method.
  • Figure 5: Procedures involved in calculating complete elliptic integral CEI-1 with Gauss-Chebyshev method.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4