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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_bessel_k1_real (s18ad)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_bessel_k1_real (s18ad) returns the value of the modified Bessel function K1x, via the function name.


[result, ifail] = s18ad(x)
[result, ifail] = nag_specfun_bessel_k1_real(x)


nag_specfun_bessel_k1_real (s18ad) evaluates an approximation to the modified Bessel function of the second kind K1x.
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 on some machines, 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.


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


Compulsory Input Parameters

1:     x – double scalar
The argument x of the function.
Constraint: x>0.0.

Optional Input Parameters


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:

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

x0.0, K1 is undefined. On soft failure the function returns zero.
W  ifail=2
x is too small, there is a danger of overflow. On soft failure the function returns approximately the largest representable value.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
function s18ad_example

fprintf('s18ad example results\n\n');

x = [0.4   0.6   1.4   1.6    2.5    3.5    6    8    10   1000];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s18ad(x(j));

disp('      x         K_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);


function s18ad_plot
  x = [0.2:0.05:1,1.2:0.2:4];
  for j = 1:numel(x)
    [K(j), ifail] = s18ad(x(j));

  fig1 = figure;
  title('Bessel Function K_1(x)');
  axis([0 4 -0.1 4]);

s18ad example results

      x         K_1(x)
   4.000e-01   2.184e+00
   6.000e-01   1.303e+00
   1.400e+00   3.208e-01
   1.600e+00   2.406e-01
   2.500e+00   7.389e-02
   3.500e+00   2.224e-02
   6.000e+00   1.344e-03
   8.000e+00   1.554e-04
   1.000e+01   1.865e-05
   1.000e+03   0.000e+00

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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