We present an efficient algorithm for the validated high precision
computation of real continued fractions, accurate to the last digit.
The algorithm proceeds in two stages. In the first stage, computations
are done in double precision. A forward error analysis and some
heuristics are used to obtain an a priori error estimate. This estimate
is used in the second stage to compute the fraction to the requested
accuracy in high precision (adaptively incrementing the precision for
reasons of efficiency). A running error analysis and techniques from
interval arithmetic are used to validate the result.
As an application, we use this algorithm to compute Gauss and confluent
hypergeometric functions when one of the numerator parameters is a
positive integer. |