Spigot Algorithm and Reliable Computation of Natural Logarithm

Spigot Algorithm and Reliable Computation of Natural Logarithm 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

Spigot Algorithm and Reliable Computation of Natural Logarithm

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2004 by Kluwer Academic Publishers
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/B:REOM.0000047096.58634.fe
Publisher site
See Article on Publisher Site

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.

Journal

Reliable ComputingSpringer Journals

Published: Nov 19, 2004

References

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