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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_bessel_k1_scaled_vector (s18cr)

## Purpose

nag_specfun_bessel_k1_scaled_vector (s18cr) returns an array of values of the scaled modified Bessel function exK1(x)${e}^{x}{K}_{1}\left(x\right)$.

## Syntax

[f, ivalid, ifail] = s18cr(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_k1_scaled_vector(x, 'n', n)

## Description

nag_specfun_bessel_k1_scaled_vector (s18cr) evaluates an approximation to exiK1(xi)${e}^{{x}_{i}}{K}_{1}\left({x}_{i}\right)$, where K1${K}_{1}$ is a modified Bessel function of the second kind for an array of arguments xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$. The scaling factor ex${e}^{x}$ removes most of the variation in K1(x)${K}_{1}\left(x\right)$.
The function uses the same Chebyshev expansions as nag_specfun_bessel_k1_real_vector (s18ar), which returns an array of the unscaled values of K1(x)${K}_{1}\left(x\right)$.

## 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
exiK1(xi)${e}^{{x}_{i}}{K}_{1}\left({x}_{i}\right)$, 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${\mathbf{ivalid}}\left(i\right)=1$
 On entry, xi ≤ 0.0${x}_{i}\le 0.0$, K1(xi)${K}_{1}\left({x}_{i}\right)$ is undefined. f(i)${\mathbf{f}}\left(\mathit{i}\right)$ contains 0.0$0.0$.
ivalid(i) = 2${\mathbf{ivalid}}\left(i\right)=2$
xi${x}_{i}$ is too close to zero, as determined by the value of the safe-range parameter nag_machine_real_safe (x02am): there is a danger of causing overflow. f(i)${\mathbf{f}}\left(\mathit{i}\right)$ contains the reciprocal of the safe-range parameter.
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

Relative errors in the argument are attenuated when propagated into the function value. When the accuracy of the argument is essentially limited by the machine precision, the accuracy of the function value will be similarly limited by at most a small multiple of the machine precision.

None.

## Example

```function nag_specfun_bessel_k1_scaled_vector_example
x = [0.4; 0.6; 1.4; 2.5; 10; 1000];
[f, ivalid, ifail] = nag_specfun_bessel_k1_scaled_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   3.259e+00    0
6.000e-01   2.374e+00    0
1.400e+00   1.301e+00    0
2.500e+00   9.002e-01    0
1.000e+01   4.108e-01    0
1.000e+03   3.965e-02    0

```
```function s18cr_example
x = [0.4; 0.6; 1.4; 2.5; 10; 1000];
[f, ivalid, ifail] = s18cr(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   3.259e+00    0
6.000e-01   2.374e+00    0
1.400e+00   1.301e+00    0
2.500e+00   9.002e-01    0
1.000e+01   4.108e-01    0
1.000e+03   3.965e-02    0

```