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