Seader's Method for Real Roots of a Nonlinear Equation

Consider the two test functions:

1. Seader's function: , where ,

2. Bessel function of the first kind: where is an integer.

Seader's function admits multiple real roots (up to 14 roots for ) while Bessel's function has an infinite number of roots.

This Demonstration finds all the roots using Seader's approach [1] and the arc length continuation technique.

The problem considered is described as follows: (the function was first proposed in [1]) and (i.e., the auxiliary equation). Using the built-in Mathematica function WhenEvent, all roots of are readily obtained when . A list of all roots is provided for both test functions. When you compare the present approach for the Bessel function of the first kind with the built-in Mathematica function BesselJZero perfect agreement is found.

[1] S. K. Rahimian, F. Jalali, J. D. Seader, and R. E. White, "A New Homotopy for Seeking all Real Roots of a Nonlinear Equation," Computers and Chemical Engineering, 35(3), 2011 pp. 403–411. doi:10.1016/j.compchemeng.2010.04.007.