nag_bessel_y1 (s17adc) evaluates an approximation to the Bessel function of the second kind .
Note: is undefined for and the function will fail for such arguments.
The function is based on four Chebyshev expansions:
and , with .
For near zero, . This approximation is used when is sufficiently small for the result to be correct to machine precision. For extremely small , there is a danger of overflow in calculating and for such arguments the function will fail.
For very large , it becomes impossible to provide results with any reasonable accuracy (see Section 7), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of ; only the amplitude, , can be determined and this is returned on failure. The range for which this occurs is roughly related to machine precision; the function will fail if (see the Users' Note for your implementation for details).
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO
On entry: the argument of the function.
– NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 18.104.22.168 in How to Use the NAG Library and its Documentation for further information.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, . Constraint: . is too large, the function returns the amplitude of the oscillation, .
On entry, . Constraint: . is undefined, the function returns zero.
x is too close to zero and there is danger of overflow, . Constraint: . The function returns the value of at the smallest valid argument.
Let be the relative error in the argument and be the absolute error in the result. (Since oscillates about zero, absolute error and not relative error is significant, except for very small .)
If is somewhat larger than the machine precision (e.g., if is due to data errors etc.), then and are approximately related by:
(provided is also within machine bounds). Figure 1 displays the behaviour of the amplification factor .
However, if is of the same order as machine precision, then rounding errors could make slightly larger than the above relation predicts.
For very small , absolute error becomes large, but the relative error in the result is of the same order as .
For very large , the above relation ceases to apply. In this region, . The amplitude can be calculated with reasonable accuracy for all , but cannot. If is written as where is an integer and , then is determined by only. If , cannot be determined with any accuracy at all. Thus if is greater than, or of the order of, the inverse of the machine precision, it is impossible to calculate the phase of and the function must fail.
Parallelism and Performance
nag_bessel_y1 (s17adc) is not threaded in any implementation.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.