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Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_bessel_k1_real (s18ad)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_bessel_k1_real (s18ad) returns the value of the modified Bessel function

K_{1} (x)

, via the function name.

Syntax

[result, ifail] = s18ad(x)

[result, ifail] = nag_specfun_bessel_k1_real(x)

Description

nag_specfun_bessel_k1_real (s18ad) evaluates an approximation to the modified Bessel function of the second kind

K_{1} (x)

Note:

K_{1} (x)

is undefined for

x \leq 0

and the function will fail for such arguments.

The function is based on five Chebyshev expansions:

For

0 < x \leq 1

K_{1} (x) = \frac{1}{x} + x \ln x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t) - x \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t),   where ​ t = 2 x^{2} - 1 .

For

1 < x \leq 2

K_{1} (x) = e^{- x} \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t),   where ​ t = 2 x - 3 .

For

2 < x \leq 4

K_{1} (x) = e^{- x} \underset{r = 0}{\sum^{'}} d_{r} T_{r} (t),   where ​ t = x - 3 .

For

x > 4

K_{1} (x) = \frac{e^{- x}}{\sqrt{x}} \underset{r = 0}{\sum^{'}} e_{r} T_{r} (t),   where ​ t = \frac{9 - x}{1 + x} .

For

x

near zero,

K_{1} (x) ≃ \frac{1}{x}

. 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

\frac{1}{x}

without overflow and the function must fail.

For large

x

, where there is a danger of underflow due to the smallness of

K_{1}

, 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:

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

$ifail = 1$: $x \leq 0.0$ , $K_{1}$ 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.

$ifail = - 99$: 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.

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 ≃ |\frac{x K_{0} (x) - K_{1} (x)}{K_{1} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x K_{0} (x) - K_{1} (x)}{K_{1} (x)}| .

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

ε ≃ δ

and there is no amplification of errors.

For large

x

ε ≃ x δ

and we have strong amplification of the relative error. Eventually

K_{1}

, which is asymptotically given by

\frac{e^{- x}}{\sqrt{x}}

, 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

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.

Open in the MATLAB editor: s18ad_example

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));
end

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

s18ad_plot;



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));
  end

  fig1 = figure;
  plot(x,K,'-r');
  xlabel('x');
  ylabel('K_1(x)');
  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

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Chapter Contents

Chapter Introduction

NAG Toolbox