NAG Library Routine Document

s18aqf  (bessel_k0_real_vector)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

s18aqf returns an array of values of the modified Bessel function K0x.

2
Specification

Fortran Interface
Subroutine s18aqf ( n, x, f, ivalid, ifail)
Integer, Intent (In):: n
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: ivalid(n)
Real (Kind=nag_wp), Intent (In):: x(n)
Real (Kind=nag_wp), Intent (Out):: f(n)
C Header Interface
#include nagmk26.h
void  s18aqf_ ( const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)

3
Description

s18aqf 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 routine will fail for such arguments.
The routine 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:     n – IntegerInput
On entry: n, the number of points.
Constraint: n0.
2:     xn – Real (Kind=nag_wp) arrayInput
On entry: the argument xi of the function, for i=1,2,,n.
Constraint: xi>0.0, for i=1,2,,n.
3:     fn – Real (Kind=nag_wp) arrayOutput
On exit: K0xi, the function values.
4:     ivalidn – Integer arrayOutput
On exit: ivalidi contains the error code for xi, for i=1,2,,n.
ivalidi=0
No error.
ivalidi=1
xi0.0, K0xi is undefined. fi contains 0.0.
5:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6
Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
ifail=2
On entry, n=value.
Constraint: n0.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further 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 routine 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

s18aqf 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 (s18aqfe.f90)

10.2
Program Data

Program Data (s18aqfe.d)

10.3
Program Results

Program Results (s18aqfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017