NAG Library Function Document

nag_bessel_k1 (s18adc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_bessel_k1 (s18adc) returns the value of the modified Bessel function K1x.

2
Specification

#include <nag.h>
#include <nags.h>
double  nag_bessel_k1 (double x, NagError *fail)

3
Description

nag_bessel_k1 (s18adc) evaluates an approximation to the modified Bessel function of the second kind K1x.
Note:  K1x is undefined for x0 and the function will fail for such arguments.
The function 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 function 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

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5
Arguments

1:     x doubleInput
On entry: the argument x of the function.
Constraint: x>0.0.
2:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_LE
On entry, x=value.
Constraint: x>0.0.
K0 is undefined and the function returns zero.
NE_REAL_ARG_TOO_SMALL
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.

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 function 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

nag_bessel_k1 (s18adc) 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 (s18adce.c)

10.2
Program Data

Program Data (s18adce.d)

10.3
Program Results

Program Results (s18adce.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
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017