A Cubic Taylor Polynomial-based Refinement

of Newton’s Method

Donald R. Burleson, Ph.D.

Copyright © 2021 by Donald R. Burleson

This article may be reproduced in its entirety

provided original authorship is expressly acknowledged.

            In my previous article “A Taylor-polynomial-based Faster-converging Generalization of
Newton’s Method for Successive Approximation of Zeros” (see
www.blackmesapress.com/NewtonGeneralization.htm) I presented my initial refinement of
Newton’s Method derived from a truncated Taylor series in the form of a degree-2 Taylor
polynomial. I propose now to introduce a further such result that I have derived from a degree-3
Taylor polynomial


This requires a more elaborate computation that involves solving a cubic polynomial equation,
but the result turns out to be what is at least a moderately faster-converging algorithm.

            First, for the above polynomial written in order of descending powers, we divide through
by the leading coefficient (an operation which of course does not affect the zeros) to produce the
monic degree-3 polynomial function



so that in the functional format


we have


(For proper convergence the third derivative of the object function f should not be too close to 0 in value.)

It is expedient, when applying this method to a particular object function and its derivatives, to
compute these component values as one goes along, rather than writing out a single formula
explicitly. I.e., the computations are most readily done in stages, much in keeping with any
solution of a cubic polynomial equation.

We will later write ole4.gif in keeping with the present
purpose of producing a sequence of successive approximations to a zero of a given function,
though as we shall see, “sequence” is too inclusive a term since in my experience it is not at all
uncommon for this method to produce a sufficiently close approximation in only one step.

            By the traditional methods of computing zeros of a cubic polynomial, the substitution

ole5.gif    gives a reduced cubic ole6.gif ,                                                                     

Again, for a particular application, b and c and d should already have been computed at this point.

            Next, following the usual approach to solving a cubic, we let

and when that is computed, let ole9.gif
Then we may write

ole10.gif  and use this result to characterize

so that the template for the desired successive iterations is



            As a convenient illustration we will choose (using the initial value 0.7) the function

ole13.gif    since its zero is actually known to be ln(2) = 0.69314718056 and thus our computed result can be checked for proximity to this value.

            For this function,

so that

ole15.gif ,

the necessary Taylor polynomial then being


and following the above-described procedure we identify

Next we have


Then   ole19.gif

so that

Then taking ole21.gif

we have ole22.gif


with error less than

0.000000001 for only one step.

            By comparison, my earlier-described degree-2 Taylor polynomial refinement would give
(for one step) an absolute error of about 0.000000054, while the original linear Newton’s
Method for one step would give an error of about 0.000023427.