s14bn: FL CL CPP AD

NAG CL Interface
s14bnc (gamma_incomplete_vector)

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NAG Library Manual, Mark 27

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s14bn: 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

s14bnc computes an array of values for the incomplete gamma functions

P (a, x)

and

Q (a, x)

2 Specification

#include <nag.h>

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.

3 Description

s14bnc evaluates the incomplete gamma functions in the normalized form, for an array of arguments

a_{i}, x_{i}

, for

i = 1, 2, \dots, n

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 function is derived from the function 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: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $a [n]$ – const double Input

On entry: the argument

a_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

a [i - 1] > 0

, for

i = 1, 2, \dots, n

3: $x [n]$ – const double Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

x [i - 1] \geq 0

, for

i = 1, 2, \dots, n

4: $tol$ – 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: $p [n]$ – double Output

On exit:

P (a_{i}, x_{i})

, the function values.

6: $q [n]$ – double Output

On exit:

Q (a_{i}, x_{i})

, the function values.

7: $ivalid [n]$ – Integer Output

On exit:

ivalid [i - 1]

contains the error code for

a_{i}

and

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $a_{i} \leq 0$ .
$ivalid [i - 1] = 2$: $x_{i} < 0$ .
$ivalid [i - 1] = 3$: Algorithm fails to terminate.

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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.
Check ivalid for more 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 function are given to this precision.

8 Parallelism and Performance

s14bnc is not threaded in any implementation.

9 Further Comments

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

10 Example

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.

s14bn: FL CL CPP AD

NAG CL Interfaces14bnc (gamma_​incomplete_​vector)

▸▿ 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 CL Interface
s14bnc (gamma_incomplete_vector)