NAG Library Function Document

nag_fresnel_s (s20acc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_fresnel_s (s20acc) returns a value for the Fresnel Integral

S (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_fresnel_s (double x)

3 Description

nag_fresnel_s (s20acc) evaluates an approximation to the Fresnel Integral

S (x) = \int_{0}^{x} \sin (\frac{π}{2} t^{2}) d t .

The function is based on Chebyshev expansions.

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 somewhat larger than the machine precision (i.e., if

δ

is due to data errors etc.), then

ε

and

δ

are approximately related by

ε ≃ |x \sin (π x^{2} / 2) / S (x)| δ

However, if

δ

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

ε

slightly larger than the above relation predicts.

For small

x

ε ≃ 3 δ

and hence there is only moderate amplification of relative error. Of course for very small

x

where the correct result would underflow and exact zero is returned, relative error-control is lost.

For moderately large values of

x

|ε| ≃ |2 x \sin (π x^{2} / 2)| |δ|

and the result will be subject to increasingly large amplification of errors. However, the above relation breaks down for large values of

x

(i.e., when

1 / x^{2}

is of the order of the machine precision); in this region the relative error in the result is essentially bounded by

2 / π x

Hence the effects of error amplification are limited and at worst the relative error loss should not exceed half the possible number of significant figures.

8 Further Comments

None.

9 Example

The following program 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_fresnel_s (s20acc)