naginterfaces.library.specfun.gamma_​incomplete

naginterfaces.library.specfun.gamma_incomplete(a, x, tol)

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

For full information please refer to the NAG Library document for s14ba

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s14baf.html

Parameters

afloat

The argument $a$ of the functions.

xfloat

The argument $x$ of the functions.

tolfloat

The relative accuracy required by you in the results. If gamma_incomplete is entered with $tol$ greater than $1.0$ or less than machine precision, then the value of machine precision is used instead.

Returns

pfloat

The value of $P(a,x)$

qfloat

The value of $P(a,x)$

Raises

NagValueError

(errno $1$)

On entry, $a=⟨value⟩$.

Constraint: $a>0.0$.

(errno $2$)

On entry, $x=⟨value⟩$.

Constraint: $x\ge 0.0$.

(errno $3$)

Algorithm fails to terminate in $⟨value⟩$ iterations.

Notes

gamma_incomplete evaluates the incomplete gamma functions in the normalized form

$$P(a,x)=\frac{1}{\Gamma \left(a\right)}{\int }_0^x{t}^{a-1}{e}^{-t}dt\text{,}$$

$$Q(a,x)=\frac{1}{\Gamma \left(a\right)}{\int }_x^{\infty }{t}^{a-1}{e}^{-t}dt\text{,}$$

with $x\ge 0$ 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 $a\simeq x$, 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 function is derived from the function GAM in Gautschi (1979b).

References

Gautschi, W, 1979, A computational procedure for incomplete gamma functions, ACM Trans. Math. Software (5), 466–481

Gautschi, W, 1979, 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