# NAG Library Routine Document

## 1Purpose

s15aff returns a value for Dawson's Integral, $F\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s15aff ( x,
 Real (Kind=nag_wp) :: s15aff Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s15aff_ (const double *x, Integer *ifail)

## 3Description

s15aff evaluates an approximation for Dawson's Integral
 $Fx=e-x2∫0xet2dt.$
The routine is based on two Chebyshev expansions:
For $0<\left|x\right|\le 4$,
 $Fx=x∑r=0′arTrt, where t=2 x4 2-1.$
For $\left|x\right|>4$,
 $Fx=1x∑r=0′brTrt, where t=2 4x 2-1.$
For $\left|x\right|$ near zero, $F\left(x\right)\simeq x$, and for $\left|x\right|$ large, $F\left(x\right)\simeq \frac{1}{2x}$. These approximations are used for those values of $x$ for which the result is correct to 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 this routine.

## 7Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is considerably greater than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
 $ε≃ x 1-2xFx Fx δ.$
The following graph shows the behaviour of the error amplification factor $\left|\frac{x\left(1-2xF\left(x\right)\right)}{F\left(x\right)}\right|$:
Figure 1
However if $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ somewhat larger than the above relation indicates. In fact $\epsilon$ will be largely independent of $x$ or $\delta$, but will be of the order of a few times the machine precision.

## 8Parallelism and Performance

s15aff is not threaded in any implementation.

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.

### 10.1Program Text

Program Text (s15affe.f90)

### 10.2Program Data

Program Data (s15affe.d)

### 10.3Program Results

Program Results (s15affe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017