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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_bessel_k0_real (s18ac)

## Purpose

nag_specfun_bessel_k0_real (s18ac) returns the value of the modified Bessel function ${K}_{0}\left(x\right)$, 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 ${K}_{0}\left(x\right)$.
Note:  ${K}_{0}\left(x\right)$ is undefined for $x\le 0$ and the function will fail for such arguments.
The function is based on five Chebyshev expansions:
For $0,
 $K0x=-ln⁡x∑′r=0arTrt+∑′r=0brTrt, where ​t=2x2-1.$
For $1,
 $K0x=e-x∑′r=0crTrt, where ​t=2x-3.$
For $2,
 $K0x=e-x∑′r=0drTrt, where ​t=x-3.$
For $x>4$,
 $K0x=e-xx ∑′r=0erTrt,where ​ t=9-x 1+x .$
For $x$ near zero, ${K}_{0}\left(x\right)\simeq -\gamma -\mathrm{ln}\left(\frac{x}{2}\right)$, where $\gamma$ 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 ${K}_{0}$, 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:     $\mathrm{x}$ – double scalar
The argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
${\mathbf{x}}\le 0.0$, ${K}_{0}$ is undefined. On soft failure the function returns zero.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is somewhat larger than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ 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 $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.
For small $x$, the amplification factor is approximately $\left|\frac{1}{\mathrm{ln}x}\right|$, which implies strong attenuation of the error, but in general $\epsilon$ can never be less than the machine precision.
For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of the relative error. Eventually ${K}_{0}$, 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

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
```