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NAG Library Manual

# NAG Library Routine DocumentS15AFF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

S15AFF returns a value for Dawson's Integral, $F\left(x\right)$, via the function name.

## 2  Specification

 FUNCTION S15AFF ( X, IFAIL)
 REAL (KIND=nag_wp) S15AFF
 INTEGER IFAIL REAL (KIND=nag_wp) X

## 3  Description

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. For very large $x$ on some machines, $F\left(x\right)$ may underflow and then the result is set exactly to zero (see the Users' Note for your implementation for details).

## 4  References

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

## 5  Parameters

1:     X – REAL (KIND=nag_wp)Input
On entry: the argument $x$ of the function.
2:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

## 6  Error Indicators and Warnings

There are no failure exits from this routine.

## 7  Accuracy

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.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s15affe.f90)

### 9.2  Program Data

Program Data (s15affe.d)

### 9.3  Program Results

Program Results (s15affe.r)