NAG FL Interface
s18aef (bessel_i0_real)
1
Purpose
s18aef returns the value of the modified Bessel function , via the function name.
2
Specification
Fortran Interface
| Real (Kind=nag_wp) |
:: |
s18aef |
| Integer, Intent (Inout) |
:: |
ifail |
| Real (Kind=nag_wp), Intent (In) |
:: |
x |
|
C Header Interface
|
#include <nag.h>
| double |
s18aef_ (const double *x, Integer *ifail) |
|
C++ Header Interface
|
#include <nag.h> extern "C" {
| double |
s18aef_ (const double &x, Integer &ifail) |
}
|
The routine may be called by the names s18aef or nagf_specfun_bessel_i0_real.
3
Description
s18aef evaluates an approximation to the modified Bessel function of the first kind .
Note: , so the approximation need only consider .
The routine 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 routine must fail because of the danger of overflow in calculating .
4
References
5
Arguments
-
1:
– Real (Kind=nag_wp)
Input
-
On entry: the argument of the function.
-
2:
– Integer
Input/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).
6
Error 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, .
Constraint: .
is too large and the function returns the approximate value of at the nearest valid argument.
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
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 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 routine 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.
8
Parallelism and Performance
s18aef is not threaded in any implementation.
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
10.2
Program Data
10.3
Program Results
