S20ARF (PDF version)
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NAG Library Manual

NAG Library Routine Document


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

S20ARF returns an array of values for the Fresnel integral Cx.

2  Specification

REAL (KIND=nag_wp)  X(N), F(N)

3  Description

S20ARF evaluates an approximation to the Fresnel integral
for an array of arguments xi, for i=1,2,,n.
Note:  Cx=-C-x, so the approximation need only consider x0.0.
The routine 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  Parameters

1:     N – INTEGERInput
On entry: n, the number of points.
Constraint: N0.
2:     X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the argument xi of the function, for i=1,2,,N.
3:     F(N) – REAL (KIND=nag_wp) arrayOutput
On exit: Cxi, the function values.
4:     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

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry, N=value.
Constraint: N0.

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


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 (s20arfe.f90)

9.2  Program Data

Program Data (s20arfe.d)

9.3  Program Results

Program Results (s20arfe.r)

S20ARF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

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