The function may be called by the names: s20arc, nag_specfun_fresnel_c_vector or nag_fresnel_c_vector.
s20arc evaluates an approximation to the Fresnel integral
for an array of arguments
, for .
Note: , so the approximation need only consider .
The function 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 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.
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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.
8Parallelism and Performance
s20arc is not threaded in any implementation.
This example reads values of x from a file, evaluates the function at each value of and prints the results.