NAG Library Function Document

nag_sin_integral (s13adc)

+− 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

nag_sin_integral (s13adc) returns the value of the sine integral

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

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_sin_integral (double x)

3 Description

nag_sin_integral (s13adc) 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 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.

6 Error Indicators and Warnings

None.

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 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 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 Function Documentnag_sin_integral (s13adc)