Generalization of Newton’s Method
can improve the basin of attraction
for a zero of a function
Donald R. Burleson, Ph.D.
Copyright © 2014 Donald R. Burleson
This article may be reproduced in its entirety
provided original authorship is expressly acknowledged.
In the article at www.blackmesapress.com/NewtonGeneralization.htm
I demonstrated that since the traditional Newton’s Method
typically has a geometric derivation based on tangent lines
but can equivalently be derived using the n = 1 case of the
Taylor polynomial approximation to y = f(x), so long as
f is an analytic function, i.e. a function representable by a Taylor series,
it follows that a very natural generalization springs from using the
Taylor polynomial of degree n = 2, from which I derived
the faster-converging iterative scheme
where the sign must be chosen to correspond to the sign of fʹ(xn)/fʺ(xn).
I will refer to this as Burleson’s Method
for lack of anything else to call it.
As shown by computational examples in the original article,
this method typically converges significantly faster than does
the original Newton’s Method.
But another important advantage is that in some cases in which
Newton’s Method does not converge at all, or converges to some value
other than the desired zero, Burleson’s
Method does converge to the desired result.
Consider the function f(x) = sin(x) in the vicinity of the zero x = π .
I use this example since the zero is already known and the
result of iterated approximation can be checked.
Suppose we make what turns out to be an unhappy choice
of initial seed-value, x0 = 1.6. By Newton’s Method it would turn out
that this choice is too close to the value x = π/2 at which fʹ = 0.
What happens in seven iterations of Newton’s Method is this:
x1 = 35.83253273
x2 = 32.55084748
x3 = 30.40377975
x4 = 32.00360096
x5 = 31.33740823
x6 = 31.41608829
x7 = 31.41592654
where it becomes clear that what the process is converging to
is not the nearby zero x = π but rather the more distant zero x = 10π.
In geometric terms this of course is because the
tangent line to the curve y = sin(x) is so nearly horizontal
as to intersect the x-axis a considerable distance away
from the zero nearest the initial value 1.6.
Oddly enough, when we consider Newton’s Method to be
a dynamical system and when we use that terminology,
the initial value x = 1.6 is not in the basin of attraction of the
zero (the attractor) x = π but rather in the basin of attraction
of the zero (attractor) x = 10π.
when we use Burleson’s Method, three iterations
of the procedure starting at the same initial value x0 = 1.6
will produce the results
x1 = 2.985303253
x2 = 3.14096859
x3 = 3.14159265
where it is clear that the iterations are converging
to the attractor x = π.
The effect of this example is to demonstrate that at least
in some instances, considering the new method to be another
dynamical system, in those terms the basin of attraction
associated with a particular zero (attractor) is improved,
i.e. expanded to a richer field of workable initial values.
The particular effect of course depends upon the given function.