An algorithm for computing logarithms and arctangents

Author:
B. C. Carlson

Journal:
Math. Comp. **26** (1972), 543-549

MSC:
Primary 65D20

DOI:
https://doi.org/10.1090/S0025-5718-1972-0307438-2

MathSciNet review:
0307438

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Abstract | References | Similar Articles | Additional Information

Abstract: An iterative algorithm with fast convergence can be used to compute logarithms, inverse circular functions, or inverse hyperbolic functions according to the choice of initial conditions. Only rational operations and square roots are required. The method consists in adding an auxiliary recurrence relation to Borchardt's algorithm to speed the convergence.

**[1]***Handbook of mathematical functions, with formulas, graphs, and mathematical tables*, Edited by Milton Abramowitz and Irene A. Stegun. Third printing, with corrections. National Bureau of Standards Applied Mathematics Series, vol. 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1965. MR**0177136****[2]**B. C. Carlson,*Algorithms involving arithmetic and geometric means*, Amer. Math. Monthly**78**(1971), 496–505. MR**0283246**, https://doi.org/10.2307/2317754**[3]**B. C. Carlson,*The logarithmic mean*, Amer. Math. Monthly**79**(1972), 615–618. MR**0302842**, https://doi.org/10.2307/2317088**[4]**J. F. Hart et al.,*Computer Approximations*, Wiley, New York, 1968.**[5]**Eduard L. Stiefel,*An introduction to numerical mathematics*, Translated by Werner C. Rheinboldt and Cornelie J. Rheinboldt, Academic Press, New York-London, 1963. MR**0181077****[6]**Henry C. Thacher Jr.,*Iterated square root expansions for the inverse cosine and inverse hyperbolic cosine*, Math. Comp.**15**(1961), 399–403. MR**0135228**, https://doi.org/10.1090/S0025-5718-1961-0135228-5

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0307438-2

Keywords:
Logarithms,
arctangents,
iteration,
Borchardt's algorithm,
inverse circular functions,
inverse hyperbolic functions,
acceleration of convergence

Article copyright:
© Copyright 1972
American Mathematical Society