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## 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( , 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:
is a scalar denoting the order of the Bessel function, with .

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
The second column of this matrix contains the derivatives of the Bessel function at each of the roots xi. The expression is a solution to the differential equation
One of the expressions for such a function is given by the series
where 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 , 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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