nag_fresnel_s (s20acc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_fresnel_s (s20acc)

+ Contents

    1  Purpose
    7  Accuracy

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 = 0 x sin π 2 t 2 dt .
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:     xdoubleInput
On entry: the argument x  of the function.

6  Error Indicators and Warnings


7  Accuracy

Let δ  and ε  be the relative errors in the argument and result respectively.
If 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


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.

9.1  Program Text

Program Text (s20acce.c)

9.2  Program Data

Program Data (s20acce.d)

9.3  Program Results

Program Results (s20acce.r)

nag_fresnel_s (s20acc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012