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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Iterated square root expansions for the inverse cosine and inverse hyperbolic cosine


Author: Henry C. Thacher
Journal: Math. Comp. 15 (1961), 399-403
MSC: Primary 65.20
MathSciNet review: 0135228
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Abstract: Let $ {R_1} = \sqrt {2 + 2x} $, $ {R_{k + 1}} = \sqrt {\vert{2 + {R_k}}\vert} $. Then $ {2^k}\sqrt {\vert{2 - {R_k}}\vert} $ and $ {2^k}{\{ \vert{6 - 2\sqrt {3 + 3{R_k}} }\vert\} ^{1/2}}$ both converge to $ \arccos \,x$ if $ \vert x\vert \leqq 1$ and to $ \operatorname{arccosh}\,x$ if $ 1 \leqq x < \infty $. Truncation errors for the two expressions are of the order of 2$ ^{-2k}$ and 2$ ^{-4k}$, respectively.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1961-0135228-5
PII: S 0025-5718(1961)0135228-5
Article copyright: © Copyright 1961 American Mathematical Society