NAG Library Routine Document

s18aff (bessel_i1_real)

1
Purpose

s18aff returns a value for the modified Bessel function I1x, via the function name.

2
Specification

Fortran Interface
Function s18aff ( x, ifail)
Real (Kind=nag_wp):: s18aff
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include <nagmk26.h>
double  s18aff_ (const double *x, Integer *ifail)

3
Description

s18aff evaluates an approximation to the modified Bessel function of the first kind I1x.
Note:  I1-x=-I1x, so the approximation need only consider x0.
The routine is based on three Chebyshev expansions:
For 0<x4,
I1x=xr=0arTrt,   where ​t=2 x4 2-1;  
For 4<x12,
I1x=exr=0brTrt,   where ​t=x-84;  
For x>12,
I1x=exx r=0crTrt,   where ​t=2 12x -1.  
For small x, I1xx. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For large x, the routine must fail because I1x cannot be represented without overflow.

4
References

NIST Digital Library of Mathematical Functions

5
Arguments

1:     x – Real (Kind=nag_wp)Input
On entry: the argument x of the function.
2:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6
Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, x=value.
Constraint: xvalue.
x is too large and the function returns the approximate value of I 1 x  at the nearest valid argument.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

Let δ and ε be the relative errors in the argument and result respectively.
If δ is somewhat larger than the machine precision (i.e., if δ is due to data errors etc.), then ε and δ are approximately related by:
ε xI0x- I1x I1 x δ.  
Figure 1 shows the behaviour of the error amplification factor
xI0x - I1x I1x .  
Figure 1
Figure 1
However, if δ is of the same order as machine precision, then rounding errors could make ε slightly larger than the above relation predicts.
For small x, εδ and there is no amplification of errors.
For large x, εxδ and we have strong amplification of errors. However, the routine must fail for quite moderate values of x because I1x would overflow; hence in practice the loss of accuracy for large x is not excessive. Note that for large x, the errors will be dominated by those of the standard function exp.

8
Parallelism and Performance

s18aff is not threaded in any implementation.

9
Further Comments

None.

10
Example

This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

10.1
Program Text

Program Text (s18affe.f90)

10.2
Program Data

Program Data (s18affe.d)

10.3
Program Results

Program Results (s18affe.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 I1(x) x Example Program Returned Values for the Bessel Function I1(x) gnuplot_plot_1