NAG Library Routine Document

S13ADF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S13ADF returns the value of the sine integral

Si (x) = \int_{0}^{x} \frac{\sin u}{u} d u,

via the function name.

2 Specification

FUNCTION S13ADF (

X, IFAIL)

REAL (KIND=nag_wp) S13ADF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X

3 Description

S13ADF calculates an approximate value for

Si (x)

For

|x| \leq 16.0

it is based on the Chebyshev expansion

Si (x) = x \sum_{r = 0}^{'} a_{r} T_{r} (t), t = 2 {(\frac{x}{16})}^{2} - 1 .

For

16 < |x| < x_{hi}

, where

x_{hi}

is an implementation-dependent number,

Si (x) = sign (x) \{\frac{π}{2} - \frac{f (x) \cos x}{x} - \frac{g (x) \sin x}{x^{2}}\}

where

f (x) = \underset{r = 0}{\sum^{'}} f_{r} T_{r} (t)

and

g (x) = \underset{r = 0}{\sum^{'}} g_{r} T_{r} (t)

t = 2 {(\frac{16}{x})}^{2} - 1

For

|x| \geq x_{hi}

Si (x) = \frac{1}{2} π sign x

to within machine precision.

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5 Parameters

1: X – REAL (KIND=nag_wp)Input: On entry: the argument $x$ of the function.
2: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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

There are no failure exits from S13ADF. The parameter IFAIL has been included for consistency with other routines in this chapter.

7 Accuracy

δ

and

ε

are the relative errors in the argument and result, respectively, then in principle

|ε| ≃ |\frac{δ \sin x}{Si (x)}| .

The equality may hold if

δ

is greater than the machine precision (

δ

due to data errors etc.) but if

δ

is simply due to round-off in the machine representation, then since the factor relating

δ

ε

is always less than one, the accuracy will be limited by machine precision.

8 Further Comments

None.

9 Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Routine DocumentS13ADF