NAG Library Function Document
nag_bessel_i1 (s18afc)
1 Purpose
nag_bessel_i1 (s18afc) returns a value for the modified Bessel function .
2 Specification
#include <nag.h> |
#include <nags.h> |
double |
nag_bessel_i1 (double x,
NagError *fail) |
|
3 Description
nag_bessel_i1 (s18afc) 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
,
For
,
For
,
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 cannot be represented without overflow.
4 References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
5 Arguments
- 1:
x – doubleInput
On entry: the argument of the function.
- 2:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_INTERNAL_ERROR
-
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.
- NE_REAL_ARG_GT
-
On entry, .
Constraint: .
is too large and the function returns the approximate value of at the nearest valid argument.
7 Accuracy
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 , and there is no amplification of errors.
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 math library function exp.
8 Parallelism and Performance
Not applicable.
None.
10 Example
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
10.1 Program Text
Program Text (s18afce.c)
10.2 Program Data
Program Data (s18afce.d)
10.3 Program Results
Program Results (s18afce.r)