NAG CL Interfaces14bnc (gamma_​incomplete_​vector)

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1Purpose

s14bnc computes an array of values for the incomplete gamma functions $P\left(a,x\right)$ and $Q\left(a,x\right)$.

2Specification

 #include
 void s14bnc (Integer n, const double a[], const double x[], double tol, double p[], double q[], Integer ivalid[], NagError *fail)
The function may be called by the names: s14bnc, nag_specfun_gamma_incomplete_vector or nag_incomplete_gamma_vector.

3Description

s14bnc evaluates the incomplete gamma functions in the normalized form, for an array of arguments ${a}_{\mathit{i}},{x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
 $P(a,x) = 1Γ(a) ∫0x ta-1 e-t dt ,$
 $Q(a,x) = 1Γ (a) ∫x∞ ta- 1 e-t dt ,$
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 function is derived from the function GAM in Gautschi (1979b).
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: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{a}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${a}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{a}}\left[\mathit{i}-1\right]>0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left[\mathit{i}-1\right]\ge 0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
4: $\mathbf{tol}$double Input
On entry: the relative accuracy required by you in the results. If s14bnc is entered with tol greater than $1.0$ or less than machine precision, then the value of machine precision is used instead.
5: $\mathbf{p}\left[{\mathbf{n}}\right]$double Output
On exit: $P\left({a}_{i},{x}_{i}\right)$, the function values.
6: $\mathbf{q}\left[{\mathbf{n}}\right]$double Output
On exit: $Q\left({a}_{i},{x}_{i}\right)$, the function values.
7: $\mathbf{ivalid}\left[{\mathbf{n}}\right]$Integer Output
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for ${a}_{\mathit{i}}$ and ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${a}_{i}\le 0$.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}<0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
Algorithm fails to terminate.
8: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of x was invalid.

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 function are given to this precision.

8Parallelism and Performance

s14bnc is not threaded in any implementation.

The time taken for a call of s14bnc depends on the precision requested through tol, and n.

10Example

This example reads values of a and x from a file, evaluates the functions at each value of ${a}_{i}$ and ${x}_{i}$ and prints the results.

10.1Program Text

Program Text (s14bnce.c)

10.2Program Data

Program Data (s14bnce.d)

10.3Program Results

Program Results (s14bnce.r)