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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_bessel_k0_real_vector (s18aq)

## Purpose

nag_specfun_bessel_k0_real_vector (s18aq) returns an array of values of the modified Bessel function K0(x)${K}_{0}\left(x\right)$.

## 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 K0(xi)${K}_{0}\left({x}_{i}\right)$ for an array of arguments xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
Note:  K0(x)${K}_{0}\left(x\right)$ is undefined for x0$x\le 0$ and the function will fail for such arguments.
The function is based on five Chebyshev expansions:
For 0 < x1$0,
 K0(x) = − lnx ∑′ arTr(t) + ∑′ brTr(t),   where ​t = 2x2 − 1. r = 0 r = 0
$K0(x)=-ln⁡x∑′r=0arTr(t)+∑′r=0brTr(t), where ​t=2x2-1.$
For 1 < x2$1,
 K0(x) = e − x ∑′ crTr(t),   where ​t = 2x − 3. r = 0
$K0(x)=e-x∑′r=0crTr(t), where ​t=2x-3.$
For 2 < x4$2,
 K0(x) = e − x ∑′ drTr(t),   where ​t = x − 3. r = 0
$K0(x)=e-x∑′r=0drTr(t), where ​t=x-3.$
For x > 4$x>4$,
 K0(x) = (e − x)/(sqrt(x)) ∑′ erTr(t),where ​t = (9 − x)/(1 + x). r = 0
$K0(x)=e-xx ∑′r=0erTr(t),where ​ t=9-x 1+x .$
For x$x$ near zero, K0(x)γln(x/2) ${K}_{0}\left(x\right)\simeq -\gamma -\mathrm{ln}\left(\frac{x}{2}\right)$, where γ$\gamma$ denotes Euler's constant. This approximation is used when x$x$ is sufficiently small for the result to be correct to machine precision.
For large x$x$, where there is a danger of underflow due to the smallness of K0${K}_{0}$, 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:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
The argument xi${x}_{\mathit{i}}$ of the function, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: x(i) > 0.0${\mathbf{x}}\left(\mathit{i}\right)>0.0$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
n$n$, the number of points.
Constraint: n0${\mathbf{n}}\ge 0$.

None.

### Output Parameters

1:     f(n) – double array
K0(xi)${K}_{0}\left({x}_{i}\right)$, the function values.
2:     ivalid(n) – int64int32nag_int array
ivalid(i)${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
ivalid(i) = 0${\mathbf{ivalid}}\left(i\right)=0$
No error.
ivalid(i) = 1${\mathbf{ivalid}}\left(i\right)=1$
xi0.0${x}_{i}\le 0.0$, K0(xi)${K}_{0}\left({x}_{i}\right)$ is undefined. f(i)${\mathbf{f}}\left(\mathit{i}\right)$ contains 0.0$0.0$.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

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${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
ifail = 2${\mathbf{ifail}}=2$
Constraint: n0${\mathbf{n}}\ge 0$.

## Accuracy

Let δ$\delta$ and ε$\epsilon$ be the relative errors in the argument and result respectively.
If δ$\delta$ is somewhat larger than the machine precision (i.e., if δ$\delta$ is due to data errors etc.), then ε$\epsilon$ and δ$\delta$ are approximately related by:
 ε ≃ |( x K1 (x) )/( K0 (x) )|δ. $ε≃ | x K1 (x) K0 (x) |δ.$
Figure 1 shows the behaviour of the error amplification factor
 |( x K1(x) )/( K0 (x) )|. $| x K1(x) K0 (x) |.$
However, if δ$\delta$ is of the same order as machine precision, then rounding errors could make ε$\epsilon$ slightly larger than the above relation predicts.
For small x$x$, the amplification factor is approximately |1/(lnx)|$|\frac{1}{\mathrm{ln}x}|$, which implies strong attenuation of the error, but in general ε$\epsilon$ can never be less than the machine precision.
For large x$x$, εxδ$\epsilon \simeq x\delta$ and we have strong amplification of the relative error. Eventually K0${K}_{0}$, which is asymptotically given by (ex)/(sqrt(x)) $\frac{{e}^{-x}}{\sqrt{x}}$, becomes so small that it cannot be calculated without underflow and hence the function will return zero. Note that for large x$x$ the errors will be dominated by those of the standard function exp.
Figure 1

None.

## Example

```function nag_specfun_bessel_k0_real_vector_example
x = [0.4; 0.6; 1.4; 1.6; 2.5; 3.5; 6; 8; 10; 1000];
[f, ivalid, ifail] = nag_specfun_bessel_k0_real_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
```
```

X           Y
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

```
```function s18aq_example
x = [0.4; 0.6; 1.4; 1.6; 2.5; 3.5; 6; 8; 10; 1000];
[f, ivalid, ifail] = s18aq(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
```
```

X           Y
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

```