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 .
- 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:
x = jroot(1,4);
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];
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.