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Computation of $ \pi $ using arithmetic-geometric mean

Author: Eugene Salamin
Journal: Math. Comp. 30 (1976), 565-570
MSC: Primary 10A30; Secondary 10A40, 33A25
MathSciNet review: 0404124
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Abstract: A new formula for $ \pi $ is derived. It is a direct consequence of Gauss' arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre's relation for elliptic integrals. The error analysis shows that its rapid convergence doubles the number of significant digits after each step. The new formula is proposed for use in a numerical computation of $ \pi $, but no actual computational results are reported here.

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Additional Information

Keywords: $ \pi $, arithmetic-geometric mean, elliptic integral, Landen's transformation, Legendre's relation, fast Fourier transform multiplication
Article copyright: © Copyright 1976 American Mathematical Society

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