nag_bessel_k1_vector (s18arc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_bessel_k1_vector (s18arc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_bessel_k1_vector (s18arc) returns an array of values of the modified Bessel function K1x.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_bessel_k1_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3  Description

nag_bessel_k1_vector (s18arc) evaluates an approximation to the modified Bessel function of the second kind K1xi for an array of arguments xi, for i=1,2,,n.
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 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:     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: K1xi, 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, K1xi 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 nag_bessel_k1 (s18adc), as defined in the Users' Note for your implementation.
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 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  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 (s18arce.c)

9.2  Program Data

Program Data (s18arce.d)

9.3  Program Results

Program Results (s18arce.r)


nag_bessel_k1_vector (s18arc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

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