NAG FL Interfaces14bnf (gamma_​incomplete_​vector)

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1Purpose

s14bnf computes an array of values for the incomplete gamma functions $P\left(a,x\right)$ and $Q\left(a,x\right)$.

2Specification

Fortran Interface
 Subroutine s14bnf ( n, a, x, tol, p, q,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: a(n), x(n), tol Real (Kind=nag_wp), Intent (Out) :: p(n), q(n)
#include <nag.h>
 void s14bnf_ (const Integer *n, const double a[], const double x[], const double *tol, double p[], double q[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s14bnf or nagf_specfun_gamma_incomplete_vector.

3Description

s14bnf 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 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: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{a}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${a}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{a}}\left(\mathit{i}\right)>0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)\ge 0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
4: $\mathbf{tol}$Real (Kind=nag_wp) Input
On entry: the relative accuracy required by you in the results. If s14bnf is entered with tol greater than $1.0$ or less than machine precision, then the value of machine precision is used instead.
5: $\mathbf{p}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $P\left({a}_{i},{x}_{i}\right)$, the function values.
6: $\mathbf{q}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $Q\left({a}_{i},{x}_{i}\right)$, the function values.
7: $\mathbf{ivalid}\left({\mathbf{n}}\right)$Integer array Output
On exit: ${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${a}_{\mathit{i}}$ and ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${a}_{i}\le 0$.
${\mathbf{ivalid}}\left(i\right)=2$
${x}_{i}<0$.
${\mathbf{ivalid}}\left(i\right)=3$
Algorithm fails to terminate.
8: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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, at least one value of x was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

s14bnf is not threaded in any implementation.

The time taken for a call of s14bnf 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 (s14bnfe.f90)

10.2Program Data

Program Data (s14bnfe.d)

10.3Program Results

Program Results (s14bnfe.r)