Efficient and Reliable Multiprecision Implementation of Elementary and Special Functions

• Authors: Annie Cuyt;Brigitte Verdonk;Haakon Waadeland
• Venue: SIAM Journal on Scientific Computing
• Year: 2006

Special functions are pervasive in all fields of science. The most well-known application areas are in physics, engineering, chemistry, computer science, and statistics. Because of their importance, several books and a large collection of papers have been devoted to the numerical computation of these functions. The technique for providing a floating-point implementation of a function differs substantially when going from a fixed finite precision context to a finite multiprecision context. In the former, the aim is to provide an optimal mathematical model, valid on a reduced argument range and requiring as few operations as possible. Here optimal means that, in relation to the model’s complexity, the truncation error is as small as it can get. The total relative error, including round-off error and possible argument reduction effect, should not exceed a prescribed threshold. In a finite multiprecision context, the goal is to provide a more generic technique, from which an approximant yielding the user-defined accuracy can be deduced at runtime. Hence best approximants are not an option since these models have to be recomputed every time the precision is altered and a function evaluation is requested. At the same time the generic technique should generate an approximant of as low complexity as possible. In the current approach we point out how continued fraction representations of functions can be helpful in the multiprecision context. The newly developed generic technique is based mainly on the use of sharpened a priori truncation error estimates for real continued fraction representations of a real variable, developed in section 3. As illustrated in section 4, the technique is very efficient and even quite competitive when compared to the traditional fixed precision implementations. The implementation is reliable in the sense that it allows one to return a sharp interval enclosure for the requested function evaluation, at the same cost. The paper follows a recipe style. In section 2 we gather the ingredients for the new results. In section 3 we construct or prepare, for a general function $f(x)$, a continued fraction approximant satisfying all requirements of a proper implementation. In section 4 the procedure is illustrated with results obtained for several specific functions $f(x)$.