nag_fresnel_c_vector (s20arc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_fresnel_c_vector (s20arc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_fresnel_c_vector (s20arc) returns an array of values for the Fresnel integral Cx.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_fresnel_c_vector (Integer n, const double x[], double f[], NagError *fail)

3  Description

nag_fresnel_c_vector (s20arc) evaluates an approximation to the Fresnel integral
Cxi=0xicosπ2t2dt
for an array of arguments xi, for i=1,2,,n.
Note:  Cx=-C-x, so the approximation need only consider x0.0.
The function is based on three Chebyshev expansions:
For 0<x3,
Cx=xr=0arTrt,   with ​ t=2 x3 4-1.
For x>3,
Cx=12+fxxsinπ2x2-gxx3cosπ2x2 ,
where fx=r=0brTrt,
and gx=r=0crTrt,
with t=2 3x 4-1.
For small x, Cxx. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For large x, fx 1π  and gx 1π2 . Therefore for moderately large x, when 1π2x3  is negligible compared with 12 , the second term in the approximation for x>3 may be dropped. For very large x, when 1πx  becomes negligible, Cx 12 . However there will be considerable difficulties in calculating sin π2x2 accurately before this final limiting value can be used. Since sin π2x2 is periodic, its value is essentially determined by the fractional part of x2. If x2=N+θ, where N is an integer and 0θ<1, then sin π2x2 depends on θ and on N modulo 4. By exploiting this fact, it is possible to retain some significance in the calculation of sin π2x2 either all the way to the very large x limit, or at least until the integer part of x2  is equal to the maximum integer allowed on the machine.

4  References

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

5  Arguments

1:     nIntegerInput
On entry: n, the number of points.
Constraint: n0.
2:     x[n]const doubleInput
On entry: the argument xi of the function, for i=1,2,,n.
3:     f[n]doubleOutput
On exit: Cxi, the function values.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

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 cos π2 x2 Cx δ.
Figure 1 shows the behaviour of the error amplification factor x cos π2 x2 Cx .
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, εδ and there is no amplification of relative error.
For moderately large values of x,
ε 2x cos π2 x2 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 1x2  is of the order of the machine precision); in this region the relative error in the result is essentially bounded by 2πx .
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.
Figure 1
Figure 1

8  Further Comments

None.

9  Example

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

9.1  Program Text

Program Text (s20arce.c)

9.2  Program Data

Program Data (s20arce.d)

9.3  Program Results

Program Results (s20arce.r)


nag_fresnel_c_vector (s20arc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

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