The routine may be called by the names s18arf or nagf_specfun_bessel_k1_real_vector.
s18arf evaluates an approximation to the modified 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 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 routine must fail.
For large , where there is a danger of underflow due to the smallness of , the result is set exactly to zero.
is too small, there is a danger of overflow. contains zero. The threshold value is the same as for in s18adf
, as defined in the the Users' Note for your implementation.
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 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 routine will return zero. Note that for large the errors will be dominated by those of the standard function exp.
8Parallelism and Performance
s18arf 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.