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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
DOI: https://doi.org/10.1090/S0025-5718-1961-0135228-5
MathSciNet review: 0135228
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Abstract: Let ${R_1} = \sqrt {2 + 2x}$, ${R_{k + 1}} = \sqrt {|{2 + {R_k}}|}$. Then ${2^k}\sqrt {|{2 - {R_k}}|}$ and ${2^k}{\{ |{6 - 2\sqrt {3 + 3{R_k}} }|\} ^{1/2}}$ both converge to $\arccos x$ if $|x| \leqq 1$ 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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Article copyright: © Copyright 1961 American Mathematical Society