Open source implementations of numerical algorithms for computing the complete elliptic integral of the first kind

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 n...

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Veröffentlicht in:Results in applied mathematics Jg. 23; S. 100479
Hauptverfasser: Zhang, Hong-Yan, Jiang, Wen-Juan
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier B.V 01.08.2024
Elsevier
Schlagworte:
Algorithm design
Complete elliptic integral of the first kind (CEI-1)
Computer programming
Orthogonal polynomials
STEM education
Verification-validation-testing (VVT)
Computer programming
STEM education
Algorithm design
Verification-validation-testing (VVT)
Orthogonal polynomials
Complete elliptic integral of the first kind (CEI-1)
ISSN:2590-0374, 2590-0374
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Zusammenfassung: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 ellipticK and EllipticK provided by MATLAB and Mathematica respectively, are based on Kcs(m) instead of the usual form K(k) such that Kcs(k2)=K(k) and m=k2. 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 the 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 Kcs(m). The general algorithms for computing orthogonal polynomials developed are valuable for the STEM education and scientific computation. [Display omitted] •The complete elliptic integral of the first kind (CEI-1) plays a significant role in STEM.•The open source version for the CEI-1 K(k) are equivalent to the commercial versions Kcs(m).•Four reliable numerical solvers for K(k) are designed and implemented with details.•The algorithms for computing orthogonal polynomials developed are valuable for STEM education.
ISSN:2590-0374
2590-0374
DOI:10.1016/j.rinam.2024.100479