NAG Library Routine Document
s14baf (gamma_incomplete)
1
Purpose
s14baf computes values for the incomplete gamma functions $P\left(a,x\right)$ and $Q\left(a,x\right)$.
2
Specification
Fortran Interface
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) 

3
Description
s14baf evaluates the incomplete gamma functions in the normalized form
with
$x\ge 0$ and
$a>0$, to a userspecified accuracy. With this normalization,
$P\left(a,x\right)+Q\left(a,x\right)=1$.
Several methods are used to evaluate the functions depending on the arguments
$a$ and
$x$, the methods including Taylor expansion for
$P\left(a,x\right)$, Legendre's continued fraction for
$Q\left(a,x\right)$, and power series for
$Q\left(a,x\right)$. When both
$a$ and
$x$ are large, and
$a\simeq x$, the uniform asymptotic expansion of
Temme (1987) is employed for greater efficiency – specifically, this expansion is used when
$a\ge 20$ and
$0.7a\le x\le 1.4a$.
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).
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: $\mathbf{a}$ – Real (Kind=nag_wp)Input

On entry: the argument $a$ of the functions.
Constraint:
${\mathbf{a}}>0.0$.
 2: $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the argument $x$ of the functions.
Constraint:
${\mathbf{x}}\ge 0.0$.
 3: $\mathbf{tol}$ – 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: $\mathbf{p}$ – Real (Kind=nag_wp)Output
 5: $\mathbf{q}$ – Real (Kind=nag_wp)Output

On exit: the values of the functions $P\left(a,x\right)$ and $Q\left(a,x\right)$ respectively.
 6: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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 $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{a}}\le 0.0$. 
 ${\mathbf{ifail}}=2$

On entry,  ${\mathbf{x}}<0.0$. 
 ${\mathbf{ifail}}=3$

Convergence of the Taylor series or Legendre continued fraction fails within
$600$ iterations. This error is extremely unlikely to occur; if it does, contact
NAG.
 ${\mathbf{ifail}}=99$
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.
 ${\mathbf{ifail}}=399$
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.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
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 routine are given to this precision.
8
Parallelism and Performance
s14baf is not threaded in any implementation.
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
$x$.
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 (s14bafe.f90)
10.2
Program Data
Program Data (s14bafe.d)
10.3
Program Results
Program Results (s14bafe.r)