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Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

nag_specfun_bessel_y1_real (s17ad) returns the value of the Bessel function ${Y}_{1}\left(x\right)$, via the function name.

## Syntax

[result, ifail] = nag_specfun_bessel_y1_real(x)

## Description

nag_specfun_bessel_y1_real (s17ad) evaluates an approximation to the Bessel function of the second kind ${Y}_{1}\left(x\right)$.
Note:  ${Y}_{1}\left(x\right)$ is undefined for $x\le 0$ and the function will fail for such arguments.
The function is based on four Chebyshev expansions:
For $0,
 $Y1x=2π ln⁡xx8∑′r=0arTrt-2πx +x8∑′r=0brTrt, with ​t=2 x8 2-1.$
For $x>8$,
 $Y1x=2πx P1xsinx-3π4+Q1xcosx-3π4$
where ${P}_{1}\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{c}_{r}{T}_{r}\left(t\right)$,
and ${Q}_{1}\left(x\right)=\frac{8}{x}\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{d}_{r}{T}_{r}\left(t\right)$, with $t=2{\left(\frac{8}{x}\right)}^{2}-1$.
For $x$ near zero, ${Y}_{1}\left(x\right)\simeq -\frac{2}{\pi x}$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision. For extremely small $x$, there is a danger of overflow in calculating $-\frac{2}{\pi x}$ and for such arguments the function will fail.
For very large $x$, it becomes impossible to provide results with any reasonable accuracy (see Accuracy), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of ${Y}_{1}\left(x\right)$; only the amplitude, $\sqrt{\frac{2}{\pi x}}$, can be determined and this is returned on soft failure. The range for which this occurs is roughly related to machine precision; the function will fail if .

## References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

## 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:
${\mathbf{ifail}}=1$
x is too large. On soft failure the function returns the amplitude of the ${Y}_{1}$ oscillation, $\sqrt{\frac{2}{\pi x}}$.
${\mathbf{ifail}}=2$
${\mathbf{x}}\le 0.0$, ${Y}_{1}$ is undefined. On soft failure the function returns zero.
${\mathbf{ifail}}=3$
x is too close to zero, there is a danger of overflow. On soft failure, the function returns the value of ${Y}_{1}\left(x\right)$ at the smallest valid argument.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Let $\delta$ be the relative error in the argument and $E$ be the absolute error in the result. (Since ${Y}_{1}\left(x\right)$ oscillates about zero, absolute error and not relative error is significant, except for very small $x$.)
If $\delta$ is somewhat larger than the machine precision (e.g., if $\delta$ is due to data errors etc.), then $E$ and $\delta$ are approximately related by:
 $E ≃ x Y0 x - Y1 x δ$
(provided $E$ is also within machine bounds). Figure 1 displays the behaviour of the amplification factor $\left|x{Y}_{0}\left(x\right)-{Y}_{1}\left(x\right)\right|$.
However, if $\delta$ is of the same order as machine precision, then rounding errors could make $E$ slightly larger than the above relation predicts.
For very small $x$, absolute error becomes large, but the relative error in the result is of the same order as $\delta$.
For very large $x$, the above relation ceases to apply. In this region, ${Y}_{1}\left(x\right)\simeq \sqrt{\frac{2}{\pi x}}\mathrm{sin}\left(x-\frac{3\pi }{4}\right)$. The amplitude $\sqrt{\frac{2}{\pi x}}$ can be calculated with reasonable accuracy for all $x$, but $\mathrm{sin}\left(x-\frac{3\pi }{4}\right)$ cannot. If $x-\frac{3\pi }{4}$ is written as $2N\pi +\theta$ where $N$ is an integer and $0\le \theta <2\pi$, then $\mathrm{sin}\left(x-\frac{3\pi }{4}\right)$ is determined by $\theta$ only. If $x>{\delta }^{-1}$, $\theta$ cannot be determined with any accuracy at all. Thus if $x$ is greater than, or of the order of, the inverse of the machine precision, it is impossible to calculate the phase of ${Y}_{1}\left(x\right)$ and the function must fail. 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 s17ad_example

x = [0.5     1    3    6   8    10   100];
n = size(x,2);
result = x;

for j=1:n
end

disp('      x         Y_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

x = [0.1:0.1:0.4,0.5:0.5:50];
for j = 1:numel(x)
end

fig1 = figure;
plot(x,Y1,'-r');
xlabel('x');
ylabel('Y_1(x)');
title('Bessel Function Y_1(x)');
axis([0 50 -2 0.75]);

```
```s17ad example results

x         Y_1(x)
5.000e-01  -1.471e+00
1.000e+00  -7.812e-01
3.000e+00   3.247e-01
6.000e+00  -1.750e-01
8.000e+00  -1.581e-01
1.000e+01   2.490e-01
1.000e+02  -2.037e-02
``` 