NAG CL Interface
s18aqc (bessel_​k0_​real_​vector)

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1 Purpose

s18aqc returns an array of values of the modified Bessel function K0(x).

2 Specification

#include <nag.h>
void  s18aqc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s18aqc, nag_specfun_bessel_k0_real_vector or nag_bessel_k0_vector.

3 Description

s18aqc evaluates an approximation to the modified Bessel function of the second kind K0(xi) for an array of arguments xi, for i=1,2,,n.
Note:  K0(x) is undefined for x0 and the function will fail for such arguments.
The function 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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x[n] const double Input
On entry: the argument xi of the function, for i=1,2,,n.
Constraint: x[i-1]>0.0, for i=1,2,,n.
3: f[n] double Output
On exit: K0(xi), the function values.
4: ivalid[n] Integer Output
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
ivalid[i-1]=0
No error.
ivalid[i-1]=1
xi0.0, K0(xi) is undefined. f[i-1] contains 0.0.
5: 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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
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.
NW_IVALID
On entry, at least one value of x was invalid.
Check ivalid for more 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 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

s18aqc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.

10.1 Program Text

Program Text (s18aqce.c)

10.2 Program Data

Program Data (s18aqce.d)

10.3 Program Results

Program Results (s18aqce.r)