S20ACF (PDF version)
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NAG Library Manual

NAG Library Routine Document

S20ACF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

S20ACF returns a value for the Fresnel integral Sx, via the function name.

2  Specification

FUNCTION S20ACF ( X, IFAIL)
REAL (KIND=nag_wp) S20ACF
INTEGER  IFAIL
REAL (KIND=nag_wp)  X

3  Description

S20ACF evaluates an approximation to the Fresnel integral
Sx=0xsinπ2t2dt.
Note:  Sx=-S-x, so the approximation need only consider x0.0.
The routine is based on three Chebyshev expansions:
For 0<x3,
Sx=x3r=0arTrt,   with ​ t=2 x3 4-1.
For x>3,
Sx=12-fxxcosπ2x2-gxx3sinπ2x2 ,
where fx=r=0brTrt,
and gx=r=0crTrt,
with t=2 3x 4-1.
For small x, Sx π6x3. This approximation is used when x is sufficiently small for the result to be correct to machine precision. For very small x, this approximation would underflow; the result is then set exactly to zero.
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, Sx12 . However there will be considerable difficulties in calculating cos π2x2 accurately before this final limiting value can be used. Since cos 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 cos π2x2 depends on θ and on N modulo 4. By exploiting this fact, it is possible to retain significance in the calculation of cos 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  Parameters

1:     X – REAL (KIND=nag_wp)Input
On entry: the argument x of the function.
2:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

There are no failure exits from S20ACF. The parameter IFAIL has been included for consistency with other routines in this chapter.

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 π2 x2 Sx δ.
Figure 1 shows the behaviour of the error amplification factor x sin π2 x2 Sx .
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, ε3δ 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 2x sin 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 the argument x from a file, evaluates the function at each value of x and prints the results.

9.1  Program Text

Program Text (s20acfe.f90)

9.2  Program Data

Program Data (s20acfe.d)

9.3  Program Results

Program Results (s20acfe.r)


S20ACF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

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