A Cubic Taylor Polynomial-based Refinement

of Newton’s Method

Donald R. Burleson, Ph.D.

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 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 gives a reduced cubic ,
where 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 Then we may write and use this result to characterize so that the template for the desired successive iterations is EXAMPLE

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

the necessary Taylor polynomial then being and following the above-described procedure we identify Next we have Then so that Then taking we have  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.