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NAG C Library Manual

NAG Library Function Documentnag_dawson (s15afc)

1  Purpose

nag_dawson (s15afc) returns a value for Dawson's Integral, $F\left(x\right)$.

2  Specification

 #include #include
 double nag_dawson (double x)

3  Description

nag_dawson (s15afc) evaluates an approximation for Dawson's Integral
 $Fx=e-x2∫0xet2dt.$
The function 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  Arguments

1:     xdoubleInput
On entry: the argument $x$ of the function.

None.

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 (s15afce.c)

9.2  Program Data

Program Data (s15afce.d)

9.3  Program Results

Program Results (s15afce.r)