Validated Evaluation of Special Mathematical Functions

Because of the importance of special functions, several books and a large collection of papers have been devoted to the numerical computation of these functions, the most well-known being the Abramowitz and Stegun handbook [1]. But up to this date, no environment offers routines for the provable correct evaluation of these special functions. We point out how series and limit-periodic continued fraction representations of the functions can be helpful in this respect. Our scalable precision technique is mainly based on the use of sharpened a priori truncation and round-off error upper bounds, in case of real arguments. The implementation is validated in the sense that it returns a sharp interval enclosure for the requested function evaluation, at the same cost as the evaluation.