The routine may be called by the names s17aqf or nagf_specfun_bessel_y0_real_vector.
s17aqf evaluates an approximation to the Bessel function of the second kind for an array of arguments , for .
Note: is undefined for and the routine will fail for such arguments.
The routine is based on four Chebyshev expansions:
For near zero, , where denotes Euler's constant. This approximation is used when is sufficiently small for the result to be correct to machine precision.
For very large , it becomes impossible to provide results with any reasonable accuracy (see Section 7), hence the routine fails. Such arguments contain insufficient information to determine the phase of oscillation of ; only the amplitude, , can be determined and this is returned on soft failure. The range for which this occurs is roughly related to machine precision; the routine will fail if
(see the Users' Note for your implementation for details).
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO
1: – IntegerInput
On entry: , the number of points.
2: – Real (Kind=nag_wp) arrayInput
On entry: the argument of the function, for .
, for .
3: – Real (Kind=nag_wp) arrayOutput
On exit: , the function values.
4: – Integer arrayOutput
On exit: contains the error code for , for .
On entry, is too large. contains the amplitude of the oscillation, .
On entry, , is undefined. contains .
5: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or 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).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, at least one value of x was invalid.
Check ivalid for more information.
On entry, .
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
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 representation error (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 the machine representation errors, then rounding errors could make slightly larger than the above relation predicts.
For very small , the errors are essentially independent of and the routine should provide relative accuracy bounded by the machine precision.
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 machine precision, it is impossible to calculate the phase of and the routine must fail.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s17aqf 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.