naginterfaces.library.specfun.gamma_​incomplete_​vector

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

gamma_incomplete_vector computes an array of values for the incomplete gamma functions $P(a,x)$ and $Q(a,x)$.

For full information please refer to the NAG Library document for s14bn

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s14bnf.html

Parameters

afloat, array-like, shape $\left(n\right)$

The argument $a_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

xfloat, array-like, shape $\left(n\right)$

The argument $x_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

tolfloat

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

Returns

pfloat, ndarray, shape $\left(n\right)$

$P(a_i,x_i)$, the function values.

qfloat, ndarray, shape $\left(n\right)$

$Q(a_i,x_i)$, the function values.

ivalidint, ndarray, shape $\left(n\right)$

$ivalid[\text{i}-1]$ contains the error code for $a_{\text{i}}$ and $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

$a_i\le 0$.

$ivalid[i-1]=2$

$x_i<0$.

$ivalid[i-1]=3$

Algorithm fails to terminate.

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $x$ was invalid.

Check $ivalid$ for more information.

Notes

gamma_incomplete_vector evaluates the incomplete gamma functions in the normalized form, for an array of arguments $a_{\text{i}},x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

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

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

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