A Note on the Recursive Calculation of Incomplete Gamma Functions

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1 A Note on the Recursive Calculation of Incomplete Gamma Functions WALTER GAUTSCHI Purdue University It is known that the recurrence relation for incomplete gamma functions a n, x, 0 a 1, n 0,1,2,..., when x is positive, is unstable more so the larger x. Nevertheless, the recursion can be used in the range 0 n x practically without error growth, and in larger ranges 0 n N with a loss of accuracy that can be controlled by suitably limiting N. Categories and Subject Descriptors: G.1.0 [Numerical Analysis]: General stability (and instability); G.1.2 [Numerical Analysis]: Approximation General Terms: Algorithms, Reliability Additional Key Words and Phrases: Incomplete gamma functions, recursive calculation 1. INTRODUCTION Our concern is with the incomplete gamma function Pa, x a, x a, a, x 0 x e t t a1 dt a a, x (1.1) for nonnegative values of the parameter a, and positive x. More precisely, we are interested in the generation of P n Pa n, x, 0 a 1, for n 0,1,2,... by means of the well-known recurrence relation P n P n1 xan1 e x For a 0 we define P 0 lim a30 Pa, x 1. a n, n 1, 2,... ; P 0 Pa, x. (1.2) Author s address: Department of Computer Sciences, Purdue University, West Lafayette, IN 47907; wxg@cs.purdue.edu. Permission to make digital / hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and / or a fee ACM /99/ $5.00 ACM Transactions on Mathematical Software, Vol. 25, No. 1, March 1999, Pages

2 102 W. Gautschi It is known (cf., e.g., Gautschi [1997, Sect. 2.4] or Gautschi [1998, Sect. 6.2]) that, for x 0, this recursion is technically unstable in the sense of rounding errors being amplified by factors that grow monotonically to infinity as n 3. Since, however, serious error growth is delayed when x is large, the recursion can still be used without significant loss of accuracy, or with a controlled loss of accuracy, if the range of n is suitably limited. This is studied in Section 2 by means of uniform asymptotics and is illustrated in Section 3 by numerical examples. 2. ANALYSIS OF ERROR PROPAGATION The error propagation in the recursion (1.2) can be described by amplification factors n that determine the amplification s3t at n t of a small error introduced at n s, assuming no additional errors. Indeed (cf. Gautschi [1997, Eq. (2.11)] or Gautschi [1998, Eq. (6.2)]), s3t t s, (2.1) where, if t s, we are dealing with forward recursion and, if t s, with backward recursion. In the case of (1.2) we have 1 Pa, x n n a, x, n 0,1,2,..., (2.2) Pa n, x which, for 0 a 1, x 0, increases monotonically from 0 1 to. This means that forward recurrence is unstable, and backward recurrence very stable, so much so that Pa n, x for any fixed n can be obtained to any prescribed accuracy by recurring backward from some sufficiently large down to n, using an arbitrary starting value, say zero. On the other hand, when x is large, the growth of n is initially very slow, so that we may get away with recurring forward up to some n 0 n 0 a, x without suffering any significant loss of accuracy. We will show that n 0 x for any a with 0 a The Case a 0 If a 0, then n 0, x 1 Pn, x n, x. (2.3) 1 In Gautschi [1997; 1998] different normalizations are used for the incomplete gamma function which, however, do not affect the amplification factors, since they are invariant with respect to scaling.

3 A Note on the Recursive Calculation of Incomplete Gamma Functions 103 We are interested in estimating n such that n c 10 d, where 1 c 10. This means that, recurring from 0 to n, we must expect a loss of about d or d 1 decimal digits of accuracy. To obtain a realistic estimate of n, we make use of Temme s uniform asymptotic approximation [Temme 1975] in terms of the complementary error function erfc [Abramowitz et al. 1965, Chap. 7], a, x 1 erfc a/2, (2.4) a 2 where 21ln, x/a, and the plus or minus sign holds depending on whether 1 or 1. (The validity of (2.4) is uniform in x/a 0 as a 3 and/or x 3.) The equation n c 10 d then becomes Fn 0 (2.5) with Fn erfc x n nlnx/n 2 c 10d if n x, erfc x n nlnx/n 2 c 10d if n x. (2.6) We solve this equation numerically by the method of bisections, for x and d 0, c 2; d 0, c 5; and d 1110, c 1, using rational approximations of Cody [1969] to evaluate the complementary error function. The results are shown graphically in the upper left frame of Figure 1, where the higher lines correspond to larger values of d respectively c. The plot suggests that n 1 is relatively small for n x. We show indeed that n 0, x 2 whenever n x. (2.7) Since n, x is decreasing with increasing x, it suffices to prove (2.7) for x n. In this case, by (2.3), n 0, n n, n n, n 1, n, n 1 and the known inequalities n, n 1, n 1, 2, 3,..., (2.8) 2 attributed by Vietoris [1983] to G. Lochs, establish (2.7).

4 104 W. Gautschi Fig. 1. Lines of constant amplification factors for a 0,0.25,0.5, The Case 0 a 1 Rewriting (2.2) as a, x a n n n a, x a a n, x, (2.9) we can use (2.4) again to estimate the value of n for which n c 10 d. The relevant equation now is (2.5) with erfc x a n a nlnx/a n erfc x a alnx/a 1 c 10d if a n x, Fn (2.10) erfc x a n a nlnx/a n erfc x a alnx/a 1 c 10d if a n x. The solution, for a , is shown in the last three frames of Figure 1 for the same values of x, c, and d as before in Section 2.1. The

5 A Note on the Recursive Calculation of Incomplete Gamma Functions 105 curves are all quite similar and in particular suggest that forward recursion is safe if n does not exceed x. We can show this analytically if we use the fact that a n/a n, x in (2.9) is increasing as a function of a [Tricomi 1950] and a, x as a function of x. Then, for a 1, whenever n 1 x, we have n a, x n 1 n 1, x n 1 n 1, n 1 n 1 n 1 1, n 1 1 2, n 1, n 1 1 n 1 where (2.8) has been used again. Thus, similarly as in (2.7), n a, x 2 if 0 a 1, n x 1. (2.11) We can say, therefore, that for large x forward recursion in (1.2) can safely be used when n x and backward recursion, as indicated earlier, if n x. The graphs in Figure 1 moreover show the extent accuracy deteriorates when forward recursion is used up to an n that is significantly larger than x. 3. NUMERICAL EXAMPLES We conclude with some numerical examples that illustrate the results obtained in Section 2. For a and x , weletn 0 x and first generate P b n for n 0, 1,..., n 0, and for n n k 1 k/10n 0, k 1, 2,..., 5, by backward recursion, P b b 0, P n1 P b n xan1 e x, n, 1,..., 1, (3.1) a n b choosing large enough so that P nk (and hence also all P b n, n n k ) approximate the true values accurately to 12 significant decimal digits. We then generate P f n for the same values of n by forward recursion (1.2) and compute the relative errors e n P n f P n b /P n b (3.2) for n 0,1,..., n 0 and for n n k, k 1,2,..., 5. The results are similar for all values of a and are shown in Table I for a 0. The starting value Pa, x required in (1.2) is computed by a double-precision version of the routine gam in Gautschi [1979]. The fractional term on the right of (1.2)

6 106 W. Gautschi Table I. Errors in Forward Recursion Beyond n x, x 20(20)100 x n k P nk e nk max 0nn0 e n D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D Table II. Errors in Forward Recursion Beyond n x, x 500, 1000 x n k P nk e nk max 0nn0 e n D D D D D D D D D D D D D D D D and (3.1), when x and n are large, is prone to overflow if computed as written. It is better to first take logarithms and then exponentiate, i.e., to compute this term as x an1 e x expa n 1lnx x lna n. (3.3) a n

7 A Note on the Recursive Calculation of Incomplete Gamma Functions 107 One can see from Table I that the maximum relative error in the range 0 n n 0 is consistent with the one requested, whereas the errors for n n k, k 1,2,....5, eventually become significantly larger, as predicted by the graphs in Figure 1. For still larger values of x, the effect is more pronounced, as is shown in Table II for x 500 and x It is interesting to observe that the paradigm of forward recursion for n x and backward recursion for n x is valid, though for different reasons, also when x is negative and Tricomi s definition of the incomplete gamma function is used (cf. Gautschi [1998, Sect. 6.2]). REFERENCES ABRAMOWITZ, M. AND STEGUN, I. A., Eds Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Dover Publications, Inc., Mineola, NY. CODY, W. J Rational Chebyshev approximations for the error function. Math. Comput. 23, GAUTSCHI, W Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Softw. 5, 4 (Dec.), GAUTSCHI, W The computation of special functions by linear difference equations. In Advances in Difference Equations, S. Elaydi, I. Győri, and G. Ladas, Eds. Gordon and Breach Science Publishers, Inc., Langhorn, PA, GAUTSCHI, W The incomplete gamma functions since Tricomi. In Tricomi s Ideas and Contemporary Applied Mathematics. Atti dei Convegni Lincei, vol Accademia Nazionale dei Lincei, Rome, Italy, TEMME, N. M Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function. Math. Comput. 29, TRICOMI, F. G Sulla funzione gamma incompleta. Ann. Mat. Pura Appl. 4, 31, VIETORIS, L Dritter Beweis der die unvollständige Gammafunktion betreffenden Lochsschen Ungleichungen. Österreich. Akad. Wiss. Math. Natur. Kl. Sitzungsber. 2, 192, Received: September 1998; revised: October 1998; accepted: November 1998