s13ad: FL CL CPP AD PY MB

NAG CL Interface
s13adc (integral_sin)

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NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s13ad: FL CL CPP AD PY MB

▸▿ 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

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CL Name Style:
Short (a00aac)
Long (impl_details)
Full (nag_info_impl_details)

1 Purpose

s13adc returns the value of the sine integral

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

2 Specification

#include <nag.h>

double

s13adc (double x)

The function may be called by the names: s13adc, nag_specfun_integral_sin or nag_sin_integral.

3 Description

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

NIST Digital Library of Mathematical Functions

5 Arguments

1: $x$ – double Input: 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

1

, the accuracy will be limited by machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

s13adc 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.

s13ad: FL CL CPP AD PY MB

NAG CL Interfaces13adc (integral_​sin)

▸▿ 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
s13adc (integral_sin)