NAG CL Interface
s20adc (fresnel_c)
1
Purpose
s20adc returns a value for the Fresnel integral .
2
Specification
The function may be called by the names: s20adc, nag_specfun_fresnel_c or nag_fresnel_c.
3
Description
s20adc evaluates an approximation to the Fresnel integral
Note: , so the approximation need only consider
.
The function is based on three Chebyshev expansions:
For
,
For
,
where
,
and ,
with .
For small , . This approximation is used when is sufficiently small for the result to be correct to machine precision.
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 some 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.
4
References
5
Arguments
-
1:
– double
Input
-
On entry: the argument of the function.
6
Error Indicators and Warnings
None.
7
Accuracy
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 there is no amplification of relative error.
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
Background information to multithreading can be found in the
Multithreading documentation.
s20adc is not threaded in any implementation.
None.
10
Example
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
10.1
Program Text
10.2
Program Data
10.3
Program Results