NAG Library Routine Document

s14baf (gamma_incomplete)


s14baf computes values for the incomplete gamma functions Pa,x and Qa,x.


Fortran Interface
Subroutine s14baf ( a, x, tol, p, q, ifail)
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: a, x, tol
Real (Kind=nag_wp), Intent (Out):: p, q
C Header Interface
#include <nagmk26.h>
void  s14baf_ (const double *a, const double *x, const double *tol, double *p, double *q, Integer *ifail)


s14baf evaluates the incomplete gamma functions in the normalized form
Pa,x = 1Γa 0x ta-1 e-t dt ,  
Qa,x = 1Γ a x ta- 1 e-t dt ,  
with x0 and a>0, to a user-specified accuracy. With this normalization, Pa,x+Qa,x=1.
Several methods are used to evaluate the functions depending on the arguments a and x, the methods including Taylor expansion for Pa,x, Legendre's continued fraction for Qa,x, and power series for Qa,x. When both a and x are large, and ax, the uniform asymptotic expansion of Temme (1987) is employed for greater efficiency – specifically, this expansion is used when a20 and 0.7ax1.4a.
Once either P or Q is computed, the other is obtained by subtraction from 1. In order to avoid loss of relative precision in this subtraction, the smaller of P and Q is computed first.
This routine is derived from the subroutine GAM in Gautschi (1979b).


Gautschi W (1979a) A computational procedure for incomplete gamma functions ACM Trans. Math. Software 5 466–481
Gautschi W (1979b) Algorithm 542: Incomplete gamma functions ACM Trans. Math. Software 5 482–489
Temme N M (1987) On the computation of the incomplete gamma functions for large values of the parameters Algorithms for Approximation (eds J C Mason and M G Cox) Oxford University Press


1:     a – Real (Kind=nag_wp)Input
On entry: the argument a of the functions.
Constraint: a>0.0.
2:     x – Real (Kind=nag_wp)Input
On entry: the argument x of the functions.
Constraint: x0.0.
3:     tol – Real (Kind=nag_wp)Input
On entry: the relative accuracy required by you in the results. If s14baf is entered with tol greater than 1.0 or less than machine precision, then the value of machine precision is used instead.
4:     p – Real (Kind=nag_wp)Output
5:     q – Real (Kind=nag_wp)Output
On exit: the values of the functions Pa,x and Qa,x respectively.
6:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, a=value.
Constraint: a>0.0.
On entry, x=value.
Constraint: x0.0.
Algorithm fails to terminate in value iterations.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


There are rare occasions when the relative accuracy attained is somewhat less than that specified by argument tol. However, the error should never exceed more than one or two decimal places. Note also that there is a limit of 18 decimal places on the achievable accuracy, because constants in the routine are given to this precision.

Parallelism and Performance

s14baf is not threaded in any implementation.

Further Comments

The time taken for a call of s14baf depends on the precision requested through tol, and also varies slightly with the input arguments a and x.


This example reads values of the argument a and x from a file, evaluates the function and prints the results.

Program Text

Program Text (s14bafe.f90)

Program Data

Program Data (s14bafe.d)

Program Results

Program Results (s14bafe.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 a x 0 0.2 0.4 0.6 0.8 1 Example Program Incomplete Gamma Functions gnuplot_plot_1 P(a,x) gnuplot_plot_2 Q(a,x) 0 5 10 15 20 0 5 10 15 20