# NAG Library Routine Document

## 1Purpose

s13adf returns the value of the sine integral
 $Six=∫0xsin⁡uudu,$
via the function name.

## 2Specification

Fortran Interface
 Real (Kind=nag_wp) :: s13adf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s13adf_ (const double *x, Integer *ifail)

## 3Description

s13adf calculates an approximate value for $\mathrm{Si}\left(x\right)$.
For $\left|x\right|\le 16.0$ it is based on the Chebyshev expansion
 $Six=x∑r=0′arTrt,t=2 x16 2-1.$
For $16<\left|x\right|<{x}_{\mathrm{hi}}$, where ${x}_{\mathrm{hi}}$ is an implementation-dependent number,
 $Six=signx π2-fxcos⁡xx-gxsin⁡xx2$
where $f\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{f}_{r}{T}_{r}\left(t\right)$ and $g\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{g}_{r}{T}_{r}\left(t\right)$, $t=2{\left(\frac{16}{x}\right)}^{2}-1$.
For $\left|x\right|\ge {x}_{\mathrm{hi}}$, $\mathrm{Si}\left(x\right)=\frac{1}{2}\pi \mathrm{sign}x$ to within machine precision.

## 4References

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

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

There are no failure exits from s13adf. The argument ifail has been included for consistency with other routines in this chapter.

## 7Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result, respectively, then in principle
 $ε≃ δ sin⁡x Six .$
The equality may hold if $\delta$ is greater than the machine precision ($\delta$ due to data errors etc.) but if $\delta$ is simply due to round-off in the machine representation, then since the factor relating $\delta$ to $\epsilon$ is always less than one, the accuracy will be limited by machine precision.

None.

## 10Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.