public static double s17ae(
	double x,
	out int ifail
)

Public Shared Function s17ae ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s17ae(
	double x, 
	[OutAttribute] int% ifail
)

static member s17ae : 
        x : float * 
        ifail : int byref -> float 

s17ae returns the value of the Bessel function $J_0\left(x\right)$.

s17ae evaluates an approximation to the Bessel function of the first kind $J_0\left(x\right)$.

Note:  $J_0\left(-x\right)=J_0\left(x\right)$, so the approximation need only consider $x\ge 0$.

The method is based on three Chebyshev expansions:

For $0<x\le 8$,

For $x>8$,

where $P_0\left(x\right)={\displaystyle \underset{r=0}{{\sum }^{\prime }}}{b}_r{T}_r\left(t\right)$,

and $Q_0\left(x\right)=\frac{8}{x}{\displaystyle \underset{r=0}{{\sum }^{\prime }}}{c}_r{T}_r\left(t\right)$,

with $t=2{\left(\frac{8}{x}\right)}^2-1$.

For $x$ near zero, $J_0\left(x\right)\simeq 1$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.

For very large $x$, it becomes impossible to provide results with any reasonable accuracy (see [Accuracy]), hence the method fails. Such arguments contain insufficient information to determine the phase of oscillation of $J_0\left(x\right)$; only the amplitude, $\sqrt{\frac{2}{\pi \left|x\right|}}$, can be determined and this is returned on failure. The range for which this occurs is roughly related to machine precision; the method will fail if $\left|x\right|\gtrsim 1/\mathit{machine precision}$ (see the Users' Note for your implementation for details).

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

Errors or warnings detected by the method:

Let $\delta$ be the relative error in the argument and $E$ be the absolute error in the result. (Since $J_0\left(x\right)$ oscillates about zero, absolute error and not relative error is significant.)

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: (provided $E$ is also within machine bounds). Figure 1 displays the behaviour of the amplification factor $\left|x{J}_1\left(x\right)\right|$.

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 large $x$, the above relation ceases to apply. In this region, $J_0\left(x\right)\simeq \sqrt{\frac{2}{\pi \left|x\right|}}\cos \left(x-\frac{\pi }{4}\right)$. The amplitude $\sqrt{\frac{2}{\pi \left|x\right|}}$ can be calculated with reasonable accuracy for all $x$, but $\cos \left(x-\frac{\pi }{4}\right)$ cannot. If $x-\frac{\pi }{4}$ is written as $2N\pi +\theta$ where $N$ is an integer and $0\le \theta <2\pi$, then $\cos \left(x-\frac{\pi }{4}\right)$ is determined by $\theta$ only. If $x\gtrsim {\delta }^{-1}$, $\theta$ 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 $J_0\left(x\right)$ and the method must fail.

None.

None.

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