NAG CL Interface
s14bac (gamma_​incomplete)

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1 Purpose

s14bac computes values for the incomplete gamma functions P(a,x) and Q(a,x).

2 Specification

#include <nag.h>
void  s14bac (double a, double x, double tol, double *p, double *q, NagError *fail)
The function may be called by the names: s14bac, nag_specfun_gamma_incomplete or nag_incomplete_gamma.

3 Description

s14bac evaluates the incomplete gamma functions in the normalized form
P(a,x) = 1Γ(a) 0x ta-1 e-t dt ,  
Q(a,x) = 1Γ (a) x ta- 1 e-t dt ,  
with x0 and a>0, to a user-specified accuracy. With this normalization, P(a,x)+Q(a,x)=1.
Several methods are used to evaluate the functions depending on the arguments a and x, the methods including Taylor expansion for P(a,x), Legendre's continued fraction for Q(a,x), and power series for Q(a,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).

4 References

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

5 Arguments

1: a double Input
On entry: the argument a of the functions.
Constraint: a>0.0.
2: x double Input
On entry: the argument x of the functions.
Constraint: x0.0.
3: tol double Input
On entry: the relative accuracy required by you in the results. If 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 P(a,x) and Q(a,x) respectively.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Algorithm fails to terminate in value iterations.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, a=value.
Constraint: a>0.0.
On entry, x=value.
Constraint: x0.0.

7 Accuracy

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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s14bac is not threaded in any implementation.

9 Further Comments

The time taken for a call of s14bac depends on the precision requested through tol, and also varies slightly with the input arguments a and x.

10 Example

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

10.1 Program Text

Program Text (s14bace.c)

10.2 Program Data

Program Data (s14bace.d)

10.3 Program Results

Program Results (s14bace.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 a x 0 0.2 0.4 0.6 0.8 1 gnuplot_plot_1 P(a,x) gnuplot_plot_2 Q(a,x) 0 5 10 15 20 0 5 10 15 20 Example Program Incomplete Gamma Functions