naginterfaces.library.specfun.bessel_​zeros

naginterfaces.library.specfun.bessel_zeros(a, n, mode, rel=1.0536712127723509e-08)

bessel_zeros determines the leading $n$ zeros of one of the Bessel functions $J_{\alpha }\left(x\right)$, $Y_{\alpha }\left(x\right)$, $J_{\alpha }^{\prime }\left(x\right)$ or $Y_{\alpha }^{\prime }\left(x\right)$ for real $x$ and non-negative $\alpha$.

For full information please refer to the NAG Library document for s17al

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s17alf.html

Parameters

afloat

The order $\alpha$ of the function.

nint

The number $N$ of zeros required.

modeint

Specifies the form of the function whose zeros are required.

$mode=1$

The zeros of $J_{\alpha }\left(x\right)$ are required.

$mode=2$

The zeros of $Y_{\alpha }\left(x\right)$ are required;

$mode=3$

The zeros of $J_{\alpha }^{\prime }\left(x\right)$ are required;

$mode=4$

The zeros of $Y_{\alpha }^{\prime }\left(x\right)$ are required.

relfloat, optional

The relative accuracy to which the zeros are required.

Returns

xfloat, ndarray, shape $\left(n\right)$

The $N$ required zeros of the function specified by $mode$.

Raises

NagValueError

(errno $1$)

On entry, $rel=⟨value⟩$.

Constraint: $rel>0.0$.

(errno $1$)

On entry, $mode=⟨value⟩$.

Constraint: $mode\le 4$.

(errno $1$)

On entry, $mode=⟨value⟩$.

Constraint: $mode\ge 1$.

(errno $1$)

On entry, $a=⟨value⟩$.

Constraint: $a\le 100000.0$.

(errno $1$)

On entry, $a=⟨value⟩$.

Constraint: $a\ge 0.0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Notes

bessel_zeros attempts to find the leading $N$ zeros of one of the Bessel functions $J_{\alpha }\left(x\right)$, $Y_{\alpha }\left(x\right)$, $J_{\alpha }^{\prime }\left(x\right)$ or $Y_{\alpha }^{\prime }\left(x\right)$, where $x$ is real. When $\alpha$ is real, these functions each have an infinite number of real zeros, all of which are simple with the possible exception of $x=0$. If $\alpha \ge 0$, the $\text{n}$th positive zero is denoted by $j_{\alpha ,\text{n}},j_{\alpha ,\text{n}}^{\prime },y_{\alpha ,\text{n}}$ and $y_{\alpha ,\text{n}}^{\prime }$, respectively, for $\text{n}=1,2,\dots ,N$, except that $x=0$ is counted as the first zero of $J_{\alpha }^{\prime }\left(x\right)$ when $\alpha =0$. Since $J_0^{\prime }\left(x\right)=-J_1\left(x\right)$, it, therefore, follows that $j_{0,1}^{\prime }=0$ and $j_{0,n}^{\prime }=-j_{1,n-1}$ for $n=2,3,\dots ,N-1$. Further details can be found in Section 9.5 of Abramowitz and Stegun (1972).

bessel_zeros is based on Algol 60 procedures given by Temme (1979). Initial approximations to the zeros are computed from asymptotic expansions. These are then improved by higher-order Newton iteration making use of the differential equation for the Bessel functions.

References

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

Temme, N M, 1976, On the numerical evaluation of the ordinary Bessel function of the second kind, J. Comput. Phys. (21), 343–350

Temme, N M, 1979, An algorithm with Algol 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives, J. Comput. Phys. (32), 270–279