Abstract
The spigot approach used in the previous paper (Reliable Computing 7 (3) (2001), pp. 247–273) for root computation is now applied to natural logarithms. The logarithm ln Q with Q∈\({\mathbb{Q}}\), Q > 1 is decomposed into a sum of two addends k 1× ln Q 1+k 2× ln Q 2 with k 1, k 2∈\({\mathbb{N}}\), then each of them is computed by the spigot algorithm and summation is carried out using integer arithmetic. The whole procedure is not literally a spigot algorithm, but advantages are the same: only integer arithmetic is needed whereas arbitrary accuracy is achieved and absolute reliability is guaranteed. The concrete procedure based on the decomposition \(Q = k \times \ln 2 + \ln \left( {1 + \frac{p}{q}} \right)\) with p, q∈ (\({\mathbb{N}}\)− {0}), p < q is simple and ready for implementation. In addition to the mentioned paper, means for determining an upper bound for the biggest integer occurring in the process of spigot computing are now provided, which is essential for the reliability of machine computation.
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References
Abdali, S. K.: Algorithm 393-Special Series Summation with Arbitrary Precision, Communications of the ACM 13(9) (1970), p. 570.
Amdeberhan, T. and Zeilberger, D.: Faster and Faster Convergent Series for ζ, The Electronic Journal of Combinatorics 3 (1995).
Amdeberhan, T. and Zeilberger, D.: Hypergeometric Series Acceleration via the WZ Method, The Electronic Journal of Combinatorics 4 (2) (1997), #R3.
Do, D.-K.: Spigot Algorithm and Root Computing, Reliable Computing 7 (3) (2001), pp. 247–273.
Rabinowitz, S. and Wagon, S.: A Spigot Algorithm for the Digits of π, American Mathematical Monthly 102(3)} (1995} ), pp. 195–
Sale, A. H. J.: The Calculation of e to Many Significant Digits, Computer Journal 11 (2) (1968), pp. 229–230.
Stoschek, E. P.: Abenteuer Algorithmus, vol. 2, Dresden University Press, 1997.
Walter, W.: Analysis1, 3 ed., Springer, 1992.
Wilf, H. S.: Accelerated Series for Universal Constants, by the WZ Method, Discrete Mathematics and Theoretical Computer Science 3 (1999), pp. 155–158.
Zelberger, D.: Closed Form, Contemporary Mathematics 143 (1993), pp. 579–607.
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Do, DK. Spigot Algorithm and Reliable Computation of Natural Logarithm. Reliable Computing 10, 489–500 (2004). https://doi.org/10.1023/B:REOM.0000047096.58634.fe
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DOI: https://doi.org/10.1023/B:REOM.0000047096.58634.fe