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.