s14ba: FL CL CPP AD

NAG FL Interface
s14baf (gamma_incomplete)

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NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s14ba: FL CL CPP AD

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

s14baf computes values for the incomplete gamma functions

P (a, x)

and

Q (a, x)

2 Specification

Fortran Interface

Subroutine s14baf (

a, x, tol, p, q, ifail)

Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	a, x, tol
Real (Kind=nag_wp), Intent (Out)	::	p, q

C Header Interface

#include <nag.h>

void	s14baf_ (const double a, const double x, const double tol, double p, double q, Integer ifail)

The routine may be called by the names s14baf or nagf_specfun_gamma_incomplete.

3 Description

s14baf evaluates the incomplete gamma functions in the normalized form

P (a, x) = \frac{1}{Γ (a)} \int_{0}^{x} t^{a - 1} e^{- t} d t,

Q (a, x) = \frac{1}{Γ (a)} \int_{x}^{\infty} t^{a - 1} e^{- t} d t,

with

x \geq 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 ≃ x

, the uniform asymptotic expansion of Temme (1987) is employed for greater efficiency – specifically, this expansion is used when

a \geq 20

and

0.7 a \leq x \leq 1.4 a

Once either

P

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).

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

1: $a$ – Real (Kind=nag_wp) Input: On entry: the argument $a$ of the functions.

Constraint: $a > 0.0$ .
2: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the functions.

Constraint: $x \geq 0.0$ .
3: $tol$ – Real (Kind=nag_wp) Input: On entry: the relative accuracy required by you in the results. If s14baf is entered with tol greater than $1.0$ or less than machine precision, then the value of machine precision is used instead.
4: $p$ – Real (Kind=nag_wp) Output
5: $q$ – Real (Kind=nag_wp) Output: On exit: the values of the functions $P (a, x)$ and $Q (a, x)$ respectively.
6: $ifail$ – 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 $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $a = ⟨ value ⟩$ .
Constraint: $a > 0.0$ .

$ifail = 2$: On entry, $x = ⟨ value ⟩$ .
Constraint: $x \geq 0.0$ .

$ifail = 3$: Algorithm fails to terminate in $⟨ value ⟩$ iterations.

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 routine are given to this precision.

8 Parallelism and Performance

s14baf is not threaded in any implementation.

9 Further Comments

The time taken for a call of s14baf depends on the precision requested through tol, and also varies slightly with the input arguments

a

and

x

10 Example

This example reads values of the argument

a

and

x

from a file, evaluates the function and prints the results.

s14ba: FL CL CPP AD

NAG FL Interfaces14baf (gamma_​incomplete)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
s14baf (gamma_incomplete)