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NAG Toolbox: nag_specfun_bessel_k0_real (s18ac)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_bessel_k0_real (s18ac) returns the value of the modified Bessel function K0x, via the function name.

Syntax

[result, ifail] = s18ac(x)
[result, ifail] = nag_specfun_bessel_k0_real(x)

Description

nag_specfun_bessel_k0_real (s18ac) evaluates an approximation to the modified Bessel function of the second kind K0x.
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:     x – double scalar
The argument x of the function.
Constraint: x>0.0.

Optional Input Parameters

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     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:
   ifail=1
x0.0, K0 is undefined. On soft failure the function returns zero.
   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 the argument x from a file, evaluates the function at each value of x and prints the results.
function s18ac_example


fprintf('s18ac example results\n\n');

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

for j=1:numel(x)
  [result(j), ifail] = s18ac(x(j));
end

disp('      x          K_0(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s18ac_plot;



function s18ac_plot
  x = [0.01:0.01:0.1,0.12:0.02:4];
  for j = 1:numel(x)
    [K(j), ifail] = s18ac(x(j));
  end

  fig1 = figure;
  plot(x,K,'-r');
  xlabel('x');
  ylabel('K_0(x)');
  title('Bessel Function K_0(x)');

s18ac example results

      x          K_0(x)
   4.000e-01   1.115e+00
   6.000e-01   7.775e-01
   1.400e+00   2.437e-01
   1.600e+00   1.880e-01
   2.500e+00   6.235e-02
   3.500e+00   1.960e-02
   6.000e+00   1.244e-03
   8.000e+00   1.465e-04
   1.000e+01   1.778e-05
   1.000e+03   0.000e+00
s18ac_fig1.png

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