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NAG C Library Manual

# NAG Library Function Documentnag_incomplete_gamma (s14bac)

## 1  Purpose

nag_incomplete_gamma (s14bac) computes values for the incomplete gamma functions $P\left(a,x\right)$ and $Q\left(a,x\right)$.

## 2  Specification

 #include #include
 void nag_incomplete_gamma (double a, double x, double tol, double *p, double *q, NagError *fail)

## 3  Description

nag_incomplete_gamma (s14bac) evaluates the incomplete gamma functions in the normalized form
 $P a,x = 1 Γ a ∫ 0 x t a-1 e -t dt Q a,x = 1 Γ a ∫ x ∞ t a-1 e -t dt ,$
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 of $P$ 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 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

On entry: the argument $a$ of the functions.
Constraint: ${\mathbf{a}}>0.0$.
2:     xdoubleInput
On entry: the argument $x$ of the functions.
Constraint: ${\mathbf{x}}\ge 0.0$.
3:     toldoubleInput
On entry: the relative accuracy required 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:     pdouble *Output
5:     qdouble *Output
On exit: the values of the functions $P\left(a,x\right)$ and $Q\left(a,x\right)$ respectively.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On error nag_incomplete_gamma (s14bac) returns with a value of 0.0 for p and q.
NE_ALG_NOT_CONV
The algorithm has failed to converge in $〈\mathit{\text{value}}〉$ iterations.
Convergence of the Taylor series or Legendre continued fraction has failed within the specified number of iterations. This error is extremely unlikely to occur; if it does, contact NAG.
NE_REAL_ARG_GT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\le 〈\mathit{\text{value}}〉$.
The argument is too large, the function returns the approximate value of $\Gamma \left(x\right)$ at the nearest valid argument.
NE_REAL_ARG_LE
On entry, a must not be less than or equal to 0.0: ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
NE_REAL_ARG_LT
On entry, x must not be less than 0.0: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.

## 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.

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.

## 9  Example

The following program reads values of the argument $a$ and $x$ from a file, evaluates the function and prints the results.

### 9.1  Program Text

Program Text (s14bace.c)

### 9.2  Program Data

Program Data (s14bace.d)

### 9.3  Program Results

Program Results (s14bace.r)