An Arithmetic-Geometric Mean of a Third Kind!

  • Semjon AdlajEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


The concept of the generalized arithmetic-geometric mean (GAGM) embraces both the arithmetic-geometric mean (AGM) and the modified arithmetic-geometric mean (MAGM) as two special concepts. The GAGM is applied for attaining a unifying formula for calculating complete elliptic integrals (CEI), including those of the third kind, thereby providing a conceptual basis for their exploration and exact evaluation, bypassing typical troubles of common software in calculating CEI. Detailed clarifying examples are provided.


Generalized arithmetic-geometric mean Linear fractional transformation Quadratic convergence Complete elliptic integral 


Acknowledgment and Notification

The author supports an unrestricted access to knowledge, and grants his permission for using his algorithms and formulas to persons and non-profit-seeking organizations. Profit-seeking organizations, including commercial software companies and their representatives, must address the author for an explicit written permission, without which they are never permitted to use any formulas, algorithms or methods based on the concept of MAGM or GAGM.

Supplementary material


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Authors and Affiliations

  1. 1.Federal Research Center “Informatics and Control” of the Russian Academy of SciencesMoscowRussia

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