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(x_n)/fʺ(x_n).

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.

EXAMPLE

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.

However,

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.