The function may be called by the names: s18arc, nag_specfun_bessel_k1_real_vector or nag_bessel_k1_vector.
s18arc evaluates an approximation to the modified Bessel function of the second kind for an array of arguments , for .
Note: is undefined for and the function will fail for such arguments.
The function is based on five Chebyshev expansions:
For near zero, . This approximation is used when is sufficiently small for the result to be correct to machine precision. For very small it is impossible to calculate without overflow and the function must fail.
For large , where there is a danger of underflow due to the smallness of , the result is set exactly to zero.
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.
On entry, at least one value of x was invalid.
Check ivalid for more 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 errors.
For large , and we have strong amplification of the relative error. Eventually , which is asymptotically given by , becomes so small that it cannot be calculated without underflow and hence the function will return zero. Note that for large the errors will be dominated by those of the standard function exp.
8Parallelism and Performance
s18arc 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.