## 1Purpose

s17adf returns the value of the Bessel function ${Y}_{1}\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Real (Kind=nag_wp) :: s17adf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s17adf_ (const double *x, Integer *ifail)
The routine may be called by the names s17adf or nagf_specfun_bessel_y1_real.

## 3Description

s17adf 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 routine will fail for such arguments.
The routine 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 routine will fail.
For very large $x$, it becomes impossible to provide results with any reasonable accuracy (see Section 7), hence the routine 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 routine will fail if (see the Users' Note for your implementation for details).

## 4References

NIST Digital Library of Mathematical Functions
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\le 〈\mathit{\text{value}}〉$.
$x$ is too large, the function returns the amplitude of the ${Y}_{1}$ oscillation, $\sqrt{2/\left(\pi x\right)}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>0.0$.
${Y}_{1}$ is undefined, the function returns zero.
${\mathbf{ifail}}=3$
x is too close to zero and there is danger of overflow, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>〈\mathit{\text{value}}〉$.
The function returns the value of ${Y}_{1}\left(x\right)$ at the smallest valid argument.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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 routine must fail.

None.

## 10Example

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