# NAG Library Routine Document

## 1Purpose

s14baf computes values for the incomplete gamma functions $P\left(a,x\right)$ and $Q\left(a,x\right)$.

## 2Specification

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

## 3Description

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 $x\ge 0$ and $a>0$, to a user-specified 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).

## 4References

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

## 5Arguments

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 $0$, . 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  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  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).

## 6Error 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, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 0.0$.
${\mathbf{ifail}}=3$
Algorithm fails to terminate in $〈\mathit{\text{value}}〉$ iterations.
${\mathbf{ifail}}=-99$
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.

## 7Accuracy

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.

## 8Parallelism 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 $x$.

## 10Example

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

### 10.1Program Text

Program Text (s14bafe.f90)

### 10.2Program Data

Program Data (s14bafe.d)

### 10.3Program Results

Program Results (s14bafe.r)