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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_bessel_k0_real_vector (s18aq)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_bessel_k0_real_vector (s18aq) returns an array of values of the modified Bessel function K0x.

Syntax

[f, ivalid, ifail] = s18aq(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_k0_real_vector(x, 'n', n)

Description

nag_specfun_bessel_k0_real_vector (s18aq) 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.

References

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

Parameters

Compulsory Input Parameters

1:     xn – double array
The argument xi of the function, for i=1,2,,n.
Constraint: xi>0.0, for i=1,2,,n.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the number of points.
Constraint: n0.

Output Parameters

1:     fn – double array
K0xi, the function values.
2:     ivalidn int64int32nag_int array
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.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
   ifail=2
Constraint: n0.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

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

Further Comments

None.

Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.
function s18aq_example


fprintf('s18aq example results\n\n');

x = [0.4; 0.6; 1.4; 1.6; 2.5; 3.5; 6; 8; 10; 1000];

[f, ivalid, ifail] = s18aq(x);

fprintf('     x           K_0(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end


s18aq example results

     x           K_0(x)   ivalid
   4.000e-01   1.115e+00    0
   6.000e-01   7.775e-01    0
   1.400e+00   2.437e-01    0
   1.600e+00   1.880e-01    0
   2.500e+00   6.235e-02    0
   3.500e+00   1.960e-02    0
   6.000e+00   1.244e-03    0
   8.000e+00   1.465e-04    0
   1.000e+01   1.778e-05    0
   1.000e+03   0.000e+00    0

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