NAG Library Function Document

nag_incomplete_gamma (s14bac)


    1  Purpose
    7  Accuracy


nag_incomplete_gamma (s14bac) computes values for the incomplete gamma functions Pa,x and Qa,x.


#include <nag.h>
#include <nags.h>
void  nag_incomplete_gamma (double a, double x, double tol, double *p, double *q, NagError *fail)


nag_incomplete_gamma (s14bac) evaluates the incomplete gamma functions in the normalized form
Pa,x=1Γa 0xta-1e-tdt,  
Qa,x=1Γ a xta- 1e-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 function is derived from the function 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 doubleInput
On entry: the argument a of the functions.
Constraint: a>0.0.
2:     x doubleInput
On entry: the argument x of the functions.
Constraint: x0.0.
3:     tol doubleInput
On entry: the relative accuracy required by you in the results. If nag_incomplete_gamma (s14bac) is entered with tol greater than 1.0 or less than machine precision, then the value of machine precision is used instead.
4:     p double *Output
5:     q double *Output
On exit: the values of the functions Pa,x and Qa,x respectively.
6:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Algorithm fails to terminate in value iterations.
Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, a=value.
Constraint: a>0.0.
On entry, x=value.
Constraint: x0.0.


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 function are given to this precision.

Parallelism and Performance

nag_incomplete_gamma (s14bac) is not threaded in any implementation.

Further Comments

The time taken for a call of nag_incomplete_gamma (s14bac) 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 (s14bace.c)

Program Data

Program Data (s14bace.d)

Program Results

Program Results (s14bace.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
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017