NAG FL Interface
s18adf (bessel_​k1_​real)

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1 Purpose

s18adf returns the value of the modified Bessel function K1(x), 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 <nag.h>
double  s18adf_ (const double *x, Integer *ifail)
The routine may be called by the names s18adf or nagf_specfun_bessel_k1_real.

3 Description

s18adf evaluates an approximation to the modified Bessel function of the second kind K1(x).
Note:  K1(x) is undefined for x0 and the routine will fail for such arguments.
The routine is based on five Chebyshev expansions:
For 0<x1,
K1(x)=1x+xlnxr=0arTr(t)-xr=0brTr(t),   where ​ t=2x2-1.  
For 1<x2,
K1(x)=e-xr=0crTr(t),   where ​t=2x-3.  
For 2<x4,
K1(x)=e-xr=0drTr(t),   where ​t=x-3.  
For x>4,
K1(x)=e-xx r=0erTr(t),   where ​t=9-x 1+x .  
For x near zero, K1(x) 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 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=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 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 K0(x)- K1(x) K1(x) |δ.  
Figure 1 shows the behaviour of the error amplification factor
| xK0(x) - K1 (x) K1(x) |.  
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