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# NAG Toolbox: nag_specfun_bessel_i1_scaled (s18cf)

## Purpose

nag_specfun_bessel_i1_scaled (s18cf) returns a value of the scaled modified Bessel function ${e}^{-\left|x\right|}{I}_{1}\left(x\right)$ via the function name.

## Syntax

[result, ifail] = s18cf(x)
[result, ifail] = nag_specfun_bessel_i1_scaled(x)

## Description

nag_specfun_bessel_i1_scaled (s18cf) evaluates an approximation to ${e}^{-\left|x\right|}{I}_{1}\left(x\right)$, where ${I}_{1}$ is a modified Bessel function of the first kind. The scaling factor ${e}^{-\left|x\right|}$ removes most of the variation in ${I}_{1}\left(x\right)$.
The function uses the same Chebyshev expansions as nag_specfun_bessel_i1_real (s18af), which returns the unscaled value of ${I}_{1}\left(x\right)$.

## 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.

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

There are no actual failure exits from this function. ifail is always set to zero. This argument is included for compatibility with other functions in this chapter.

## 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 s18cf_example

fprintf('s18cf example results\n\n');

x = [0  0.5  1   3   6  10  1000  -1];
n = size(x,2);
result = x;

for j=1:n
[result(j), ifail] = s18cf(x(j));
end

disp('      x       e^-|x| I_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

```
```s18cf example results

x       e^-|x| I_1(x)
0.000e+00   0.000e+00
5.000e-01   1.564e-01
1.000e+00   2.079e-01
3.000e+00   1.968e-01
6.000e+00   1.521e-01
1.000e+01   1.213e-01
1.000e+03   1.261e-02
-1.000e+00  -2.079e-01
```

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Chapter Contents
Chapter Introduction
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