Spigot Algorithm and Reliable Computation of Natural Logarithm

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.