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Some observations on algorithms of the Gauss-Borchardt type

Published online by Cambridge University Press:  20 January 2009

Jaak Peetre
Jaak Peetre
Affiliation:
Matematiska InstitutionenStockholms UniversitetBox 6701S-113 85 StockholmSweden
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Abstract

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A one parameter family of algorithms is studied, which contains both the arithmetic-geometric mean of Gauss and its generalization by Borchardt, recently studied by J. and P. Borwein. We prove that the presence of an asymptotic formula for such an algorithm is, in view of the Poisson summation formula, equivalent to the vanishing of certain integrals. In the case of Gauss and Borchardt the latter involve theta functions. Finally, we investigate the question of convergence of the algorithm for complex values, thereby generalizing the corresponding result of Gauss.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 3 , October 1991 , pp. 415 - 431
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

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