# Seader's Method for Real Roots of a Nonlinear Equation

Seader's Method for Real Roots of a Nonlinear Equation

Consider the two test functions:

1. Seader's function: , where =,

f(x)=sin(x)-+0.5

cosh(x)

5000

z

1

z

1

u

1

1-

u

1

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

f(x)=(x)

J

n

n

Seader's function admits multiple real roots (up to 14 roots for =0.99999) while Bessel's function has an infinite number of roots.

u

1

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.

H(t(s),x(s))=1-t(s)+f(x(s))=0

H(t,x)

t'+x'=1

2

(s)

2

(s)

f(x)

t(s)=1