NAG Library Routine Document

s18adf (bessel_k1_real)

1
Purpose

s18adf returns the value of the modified Bessel function K1x, via the function name.

2
Specification

Fortran Interface
Function s18adf ( x, ifail)
Real (Kind=nag_wp):: s18adf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include <nagmk26.h>
double  s18adf_ (const double *x, Integer *ifail)

3
Description

s18adf evaluates an approximation to the modified Bessel function of the second kind K1x.
Note:  K1x is undefined for x0 and the routine will fail for such arguments.
The routine is based on five Chebyshev expansions:
For 0<x1,
K1x=1x+xlnxr=0arTrt-xr=0brTrt,   where ​ t=2x2-1.  
For 1<x2,
K1x=e-xr=0crTrt,   where ​t=2x-3.  
For 2<x4,
K1x=e-xr=0drTrt,   where ​t=x-3.  
For x>4,
K1x=e-xx r=0erTrt,   where ​t=9-x 1+x .  
For x near zero, K1x 1x . This approximation is used when x is sufficiently small for the result to be correct to machine precision. For very small x on some machines, it is impossible to calculate 1x  without overflow and the routine must fail.
For large x, where there is a danger of underflow due to the smallness of K1, the result is set exactly to zero.

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.
Constraint: x>0.0.
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: x>0.0.
K0 is undefined and the function returns zero.
ifail=2
On entry, x=value.
Constraint: x>value.
x is too small, there is a danger of overflow and the function returns approximately the largest representable value.
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 x K0x- K1x K1x δ.  
Figure 1 shows the behaviour of the error amplification factor
xK0x - K1 x K1x .  
However, if δ is of the same order as the 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 the relative error. Eventually K1, which is asymptotically given by e-xx , becomes so small that it cannot be calculated without underflow and hence the routine will return zero. Note that for large x the errors will be dominated by those of the standard function exp.
Figure 1
Figure 1

8
Parallelism and Performance

s18adf 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 (s18adfe.f90)

10.2
Program Data

Program Data (s18adfe.d)

10.3
Program Results

Program Results (s18adfe.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 K1(x) x Example Program Returned Values for the Bessel Function K1(x) gnuplot_plot_1