Computing the Incomplete Gamma Function to Arbitrary Precision
I consider an arbitrary-precision computation of the incomplete Gamma function from the Legendre continued fraction. Using the method of generating functions, I compute the convergence rate of the continued fraction and find a direct estimate of the necessary number of terms. This allows to compare the performance of the continued fraction and of the power series methods. As an application, I show that the incomplete Gamma function Γ (a, z) can be computed to P digits in at most O(P) long multiplications uniformly in z for Re z > 0. The error function of the real argument, erf x, requires at most O(P 2/3) long multiplications.
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- 1.M. Abramowitz and I. Stegun, eds., Handbook of special functions, National Bureau of Standards, 1964.Google Scholar
- 10.W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical recipes in C, 2nd ed., Cambridge University Press, 1992.Google Scholar
- 13.Sh. E. Tsimring, Handbook of special functions and definite integrals: algorithms and programs for calculators, Radio and communications (publisher), Moscow, 1988 (in Russian).Google Scholar
- 16.F. W. J. Olver, Asymptotics and special functions, Academic Press, 1974.Google Scholar