JROOT Function

Language Reference

JROOT Function

computes the first nonzero roots of a Bessel function of the first kind and the derivative of the Bessel function at each root

JROOT( $\nu$ , n)

The JROOT function returns an n ×2 matrix with the calculated roots in the first column and the derivatives in the second column.

The inputs to the JROOT function are as follows:

$\nu$: is a scalar denoting the order of the Bessel function, with $\nu \gt -1$ .
n: is a positive integer denoting the number of roots.

The JROOT function returns a matrix in which the first column contains the first n roots of the Bessel function; these roots are the solutions to the equation

$J_{\nu}(x_i) = 0 ,\hspace*{0.20in} i = 1, ... , n$

The second column of this matrix contains the derivatives $J^'_\nu(x_i)$ of the Bessel function at each of the roots x_i. The expression $J_\nu(x)$ is a solution to the differential equation

$x^2 \frac{d^2 J_{\nu}}{dx^2} + x \frac{dJ_{\nu}}{dx} + (x^2 - \nu^2)J_{\nu} = 0$

One of the expressions for such a function is given by the series

$J_{\nu}(x) = ( \frac{1}2 z )^{\nu} \sum_{k=0}^{\infty} \frac{ ( -\frac{1}4 z^2 )^k} {k! \Gamma(\nu + k + 1)}$

where $\Gamma(\cdot)$ is the gamma function. Refer to Abramowitz and Stegun (1973) for more details concerning the Bessel and gamma functions. The algorithm is a Newton method coupled with a reasonable initial guess. For large values of n or $\nu$ , the algorithm could fail due to machine limitations. In this case, JROOT returns a matrix with zero rows and zero columns. The values that cause the algorithm to fail are machine dependent.

The following code provides an example:

proc iml;
  x = jroot(1,4);
  print x;

To obtain only the roots, you can use the following statement, which extracts the first column of the returned matrix:

  x = jroot(1,4)[,1];

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