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  and the arc length continuation technique.
The problem considered is described as follows: (the function was first proposed in ) 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.
 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.