nag_bessel_k0_vector (s18aqc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_bessel_k0_vector (s18aqc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_bessel_k0_vector (s18aqc) returns an array of values of the modified Bessel function K0x.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_bessel_k0_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3  Description

nag_bessel_k0_vector (s18aqc) evaluates an approximation to the modified Bessel function of the second kind K0xi for an array of arguments xi, for i=1,2,,n.
Note:  K0x is undefined for x0 and the function will fail for such arguments.
The function is based on five Chebyshev expansions:
For 0<x1,
K0x=-lnxr=0arTrt+r=0brTrt,   where ​t=2x2-1.
For 1<x2,
K0x=e-xr=0crTrt,   where ​t=2x-3.
For 2<x4,
K0x=e-xr=0drTrt,   where ​t=x-3.
For x>4,
K0x=e-xx r=0erTrt,where ​ t=9-x 1+x .
For x near zero, K0x-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

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

5  Arguments

1:     nIntegerInput
On entry: n, the number of points.
Constraint: n0.
2:     x[n]const doubleInput
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]doubleOutput
On exit: K0xi, the function values.
4:     ivalid[n]IntegerOutput
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, K0xi is undefined. f[i-1] contains 0.0.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
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 K1x 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  Further Comments

None.

9  Example

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

9.1  Program Text

Program Text (s18aqce.c)

9.2  Program Data

Program Data (s18aqce.d)

9.3  Program Results

Program Results (s18aqce.r)


nag_bessel_k0_vector (s18aqc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012