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# NAG Toolbox: nag_specfun_bessel_k1_scaled (s18cd)

## Purpose

nag_specfun_bessel_k1_scaled (s18cd) returns a value of the scaled modified Bessel function ${e}^{x}{K}_{1}\left(x\right)$ via the function name.

## Syntax

[result, ifail] = s18cd(x)
[result, ifail] = nag_specfun_bessel_k1_scaled(x)

## Description

nag_specfun_bessel_k1_scaled (s18cd) evaluates an approximation to ${e}^{x}{K}_{1}\left(x\right)$, where ${K}_{1}$ is a modified Bessel function of the second kind. The scaling factor ${e}^{x}$ removes most of the variation in ${K}_{1}\left(x\right)$.
The function uses the same Chebyshev expansions as nag_specfun_bessel_k1_real (s18ad), which returns the unscaled value of ${K}_{1}\left(x\right)$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}$ – 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:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}\le 0.0$: ${K}_{1}$ is undefined. On soft failure nag_specfun_bessel_k1_scaled (s18cd) returns zero.
${\mathbf{ifail}}=2$
On entry, x 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. On soft failure, nag_specfun_bessel_k1_scaled (s18cd) returns the reciprocal of the safe-range parameter.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## 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

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.
```function s18cd_example

fprintf('s18cd example results\n\n');

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

for j=1:n
[result(j), ifail] = s18cd(x(j));
end

disp('      x        e^xK_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

```
```s18cd example results

x        e^xK_1(x)
4.000e-01   3.259e+00
6.000e-01   2.374e+00
1.400e+00   1.301e+00
2.500e+00   9.002e-01
1.000e+01   4.108e-01
1.000e+03   3.965e-02
```

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