S20ACF evaluates an approximation to the Fresnel integral
Note: , so the approximation need only consider .
The routine is based on three Chebyshev expansions:
For small , . This approximation is used when is sufficiently small for the result to be correct to machine precision. For very small , this approximation would underflow; the result is then set exactly to zero.
For large , and . Therefore for moderately large , when is negligible compared with , the second term in the approximation for may be dropped. For very large , when becomes negligible, . However there will be considerable difficulties in calculating accurately before this final limiting value can be used. Since is periodic, its value is essentially determined by the fractional part of . If where is an integer and , then depends on and on modulo . By exploiting this fact, it is possible to retain significance in the calculation of either all the way to the very large limit, or at least until the integer part of is equal to the maximum integer allowed on the machine.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
1: – REAL (KIND=nag_wp)Input
On entry: the argument of the function.
2: – INTEGERInput/Output
On entry: IFAIL must be set to , . 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 is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: 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.
Let and be the relative errors in the argument and result respectively.
If is somewhat larger than the machine precision (i.e., if is due to data errors etc.), then and are approximately related by:
Figure 1 shows the behaviour of the error amplification factor
However if is of the same order as the machine precision, then rounding errors could make slightly larger than the above relation predicts.
For small , and hence there is only moderate amplification of relative error. Of course for very small where the correct result would underflow and exact zero is returned, relative error-control is lost.
For moderately large values of ,
and the result will be subject to increasingly large amplification of errors. However the above relation breaks down for large values of (i.e., when is of the order of the machine precision); in this region the relative error in the result is essentially bounded by .
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 Parallelism and Performance
9 Further Comments
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.