On convergence of W. Borchardt's algorithm
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Abstract
The W. Borchardt algorithm, which is a generalization of the arithmetic‑geometric mean algorithm, was first introduced by Borchardt and then studied by I. Hettner. In these works, the Borchardt mean was studied for real positive initial arguments. The study of the Borchardt mean for complex initial elements is devoted to the work of G. Genpert. The proof of convergence of the Borchardt algorithm is carried out by Genpert on the basis of geometric considerations. In the present paper we give an analytical proof of convergence of the Borchardt algorithm and consider cases of degeneracy of the algorithm.
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Funding:
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References
- Bohrchardt W. Monatsber. Konigl. Akad. Wiss., 1876, p. 611.
- Bohrchardt W. Abhandlung. Konigl. Akad. Wiss., 1878, p. 33.
- Hettner I. J. reine und angew. Math., 1880, vol. 89, p. 221; vol. 112, p. 89.
- Geppert H. J. reine und angew. Math., 1929, vol. 161, p. 21.