nag_bessel_i0 (s18aec) evaluates an approximation to the modified Bessel function of the first kind .
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 , the function must fail because of the danger of overflow in calculating .
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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 22.214.171.124 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 and the function returns the approximate value of at the nearest valid argument.
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 machine precision, then rounding errors could make slightly larger than the above relation predicts.
For small the amplification factor is approximately , which implies strong attenuation of the error, but in general can never be less than the machine precision.
For large , and we have strong amplification of errors. However the function must fail for quite moderate values of , because would overflow; hence in practice the loss of accuracy for large is not excessive. Note that for large the errors will be dominated by those of the standard function exp.
Parallelism and Performance
nag_bessel_i0 (s18aec) 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.