NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s13ad: FL CL AD

NAG FL Interface
s13adf (integral_sin)

Keyword Search:

NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s13ad: FL CL AD

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2019

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

Fortran Interface

Function s13adf (

Real (Kind=nag_wp)	::	s13adf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include <nag.h>

double

s13adf_ (const double *x, Integer *ifail)

The routine may be called by the names s13adf or nagf_specfun_integral_sin.

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

NIST Digital Library of Mathematical Functions

5 Arguments

1: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.
2: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface 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 argument, 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 argument ifail has been included for consistency with other routines in this chapter.

7 Accuracy

If

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

to

ε

is always less than

1

, the accuracy will be limited by machine precision.

8 Parallelism and Performance

s13adf is not threaded in any implementation.

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.

10.1 Program Text

Program Text (s13adfe.f90)

10.2 Program Data

Program Data (s13adfe.d)

10.3 Program Results

Program Results (s13adfe.r)

NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s13ad: FL CL AD

© The Numerical Algorithms Group Ltd. 2019