论文信息 - Computing the Incomplete Gamma Function to Arbitrary Precision

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(P2/3) long multiplications.

Serge Winitzki

[1] Estimates of the speed of convergence of continued fraction expansions of functions , 1977 .

[2] Walter Gautschi,et al. A note on the recursive calculation of incomplete gamma functions , 1999, TOMS.

[3] William H. Press,et al. Numerical recipes in C , 2002 .

[4] William B. Jones,et al. Numerical stability in evaluating continued fractions , 1974 .

[5] J. H. McCabe,et al. A continued fraction expansion, with a truncation error estimate, for Dawson’s integral , 1974 .

[6] Armido R. Didonato,et al. Computation of the incomplete gamma function ratios and their inverse , 1986, TOMS.

[7] David M. Smith. Algorithm 814: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions , 2001, TOMS.

[8] William H. Press,et al. Numerical Recipes in C, 2nd Edition , 1992 .

[9] David M. Smith,et al. Efficient multiple-precision evaluation of elementary functions , 1989 .

[10] Walter Gautschi,et al. A Computational Procedure for Incomplete Gamma Functions , 1979, TOMS.

[11] Alan H. Karp,et al. High-precision division and square root , 1997, TOMS.

[12] Kevin Barraclough,et al. I and i , 2001, BMJ : British Medical Journal.

[13] N. Temme. The asymptotic expansion of the incomplete gamma functions : (preprint) , 1977 .

[14] F. Olver. Asymptotics and Special Functions , 1974 .