NAG CL Interface
s18adc (bessel_​k1_​real)

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1 Purpose

s18adc returns the value of the modified Bessel function K1(x).

2 Specification

#include <nag.h>
double  s18adc (double x, NagError *fail)
The function may be called by the names: s18adc, nag_specfun_bessel_k1_real or nag_bessel_k1.

3 Description

s18adc evaluates an approximation to the modified Bessel function of the second kind K1(x).
Note:  K1(x) is undefined for x0 and the function will fail for such arguments.
The function 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 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

NIST Digital Library of Mathematical Functions

5 Arguments

1: x double Input
On entry: the argument x of the function.
Constraint: x>0.0.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, x=value.
Constraint: x>0.0.
K0 is undefined and the function returns zero.
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 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 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

Background information to multithreading can be found in the Multithreading documentation.
s18adc is not threaded in any implementation.

9 Further Comments


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 5.4 patchlevel 6 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 "s18adfe.r" Example Program Returned Values for the Bessel Function K1(x)