NAG CL Interface
s18arc (bessel_​k1_​real_​vector)

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1 Purpose

s18arc returns an array of values of the modified Bessel function K1(x).

2 Specification

#include <nag.h>
void  s18arc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s18arc, nag_specfun_bessel_k1_real_vector or nag_bessel_k1_vector.

3 Description

s18arc evaluates an approximation to the modified Bessel function of the second kind K1(xi) for an array of arguments xi, for i=1,2,,n.
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 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: 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: K1(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, K1(xi) is undefined. f[i-1] contains 0.0.
ivalid[i-1]=2
xi is too small, there is a danger of overflow. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_TOO_SMALL in s18adc , as defined in the the Users' Note for your implementation.
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 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

s18arc 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 (s18arce.c)

10.2 Program Data

Program Data (s18arce.d)

10.3 Program Results

Program Results (s18arce.r)