Computation of incomplete beta function ratios Ix(a,b) with Deuflhard's algorithm.

Gautschi proposed a method for computing incomplete beta functions I x (a , b) using Miller's algorithm with a three-term recurrence relation and showed a computation program in ALGOL. In this paper, first, Miller's algorithm using the recurrence relation satisfied by f k (x) = I x (a + k , b) is described. Next, another solution that is first-order independent of f k (x) of the recurrence relation is given, and its general solution can be expressed as a linear sum of these. Using this general solution, an error analysis for the function I x (a , b) is performed for the first time. The relative error of the function values is then expressed in a new formula to a trend of the error behavior. Also, Miller's algorithm with a normalizing sum is explained and its error analysis is performed. Since Miller's algorithm requires a predefined number of iterations of the recurrence relation, it is necessary to repeat the computation of the recurrence relation with increasing number of iterations until the required accuracy will be met. Therefore, in this paper, we apply Deuflhard's algorithm, which can automatically obtain the function value with the required accuracy. This algorithm requires far fewer iterations than Gautschi's algorithm to obtain the same accuracy. [ABSTRACT FROM AUTHOR]