NAG Library Function Document

nag_dawson (s15afc)

+− 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

1 Purpose

nag_dawson (s15afc) returns a value for Dawson's Integral,

F (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_dawson (double x)

3 Description

nag_dawson (s15afc) evaluates an approximation for Dawson's Integral

F (x) = e^{- x^{2}} \int_{0}^{x} e^{t^{2}} d t .

The function is based on two Chebyshev expansions:

For

0 < |x| \leq 4

F (x) = x \sum_{r = 0}^{'} a_{r} T_{r} (t),   where t = 2 {(\frac{x}{4})}^{2} - 1 .

For

|x| > 4

F (x) = \frac{1}{x} \sum_{r = 0}^{'} b_{r} T_{r} (t),   where t = 2 {(\frac{4}{x})}^{2} - 1 .

For

|x|

near zero,

F (x) ≃ x

, and for

|x|

large,

F (x) ≃ \frac{1}{2 x}

. These approximations are used for those values of

x

for which the result is correct to machine precision.

4 References

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

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.

6 Error Indicators and Warnings

None.

7 Accuracy

Let

δ

and

ε

be the relative errors in the argument and result respectively.

δ

is considerably greater than the machine precision (i.e., if

δ

is due to data errors etc.), then

ε

and

δ

are approximately related by:

ε ≃ |\frac{x (1 - 2 x F (x))}{F (x)}| δ .

The following graph shows the behaviour of the error amplification factor

|\frac{x (1 - 2 x F (x))}{F (x)}|

Figure 1

However if

δ

is of the same order as machine precision, then rounding errors could make

ε

somewhat larger than the above relation indicates. In fact

ε

will be largely independent of

x

δ

, but will be of the order of a few times the machine precision.

8 Parallelism and Performance

Not applicable.

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.

NAG Library Function Documentnag_dawson (s15afc)