NAG FL Interface
s18acf (bessel_​k0_​real)

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1 Purpose

s18acf returns the value of the modified Bessel function K0(x), via the function name.

2 Specification

Fortran Interface
Function s18acf ( x, ifail)
Real (Kind=nag_wp) :: s18acf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s18acf_ (const double *x, Integer *ifail)
The routine may be called by the names s18acf or nagf_specfun_bessel_k0_real.

3 Description

s18acf evaluates an approximation to the modified Bessel function of the second kind K0(x).
Note:  K0(x) is undefined for x0 and the routine will fail for such arguments.
The routine is based on five Chebyshev expansions:
For 0<x1,
K0(x)=-lnxr=0arTr(t)+r=0brTr(t),   where ​t=2x2-1.  
For 1<x2,
K0(x)=e-xr=0crTr(t),   where ​t=2x-3.  
For 2<x4,
K0(x)=e-xr=0drTr(t),   where ​t=x-3.  
For x>4,
K0(x)=e-xx r=0erTr(t),where ​ t=9-x 1+x .  
For x near zero, K0(x)-γ-ln( x2) , where γ denotes Euler's constant. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For large x, where there is a danger of underflow due to the smallness of K0, 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 Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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=-99
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.
ifail=-399
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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 K1 (x) K0 (x) |δ.  
Figure 1 shows the behaviour of the error amplification factor
| x K1(x) K0 (x) |.  
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, the amplification factor is approximately | 1lnx |, which implies strong attenuation of the error, but in general ε can never be less than the machine precision.
For large x, εxδ and we have strong amplification of the relative error. Eventually K0, 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

s18acf 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 (s18acfe.f90)

10.2 Program Data

Program Data (s18acfe.d)

10.3 Program Results

Program Results (s18acfe.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 K0(x) x Example Program Returned Values for the Bessel Function K0(x) gnuplot_plot_1