Burleson’s Refinement: Convergence to Conjugate
Complex Zeros of a Polynomial
Donald R. Burleson, Ph.D.
Copyright © 2020 by Donald R. Burleson.
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provided original authorship is expressly acknowledged.
In an earlier research article (see www.blackmesapress.com/NewtonGeneralization.htm) I presented my refinement/generalization of Newton’s Method for successive approximation of real zeros of a function, based on Taylor polynomials:
where the algebraic sign in front of the radical is chosen to match the sign of the quantity
when the process is to converge to a real-number zero.
It is possible for this generalized approach to converge to a conjugate pair of complex-number zeros in the case of a polynomial function with all real-number coefficients. For the circumstance of a real-number initial value producing a negative real-number radicand, so that the resulting new value involves non-real complex numbers, one thinks of both algebraic signs in front of the radical as being operative, though one does not always necessarily use them both at the same time in subsequent computations. Consider the following example.
Let where for purposes of straightforward illustration I have chosen a function whose zeros are actually 0 and the conjugate complex pair
For successive approximation we will choose the initial value so that for the given function and its derivatives we have, plugging f(1) = 2 and f’(1) = 4 and f”(1) = 6 into the refinement formula, the value
Let us first choose to pursue the process by taking in particular
from which we compute and
Plugging these into f and its derivatives we find that
and in turn plugging these into the refinement formula we have the next approximation
which after some computations (and using only the + sign, more about this below) reduces to
and the final radical expression can be evaluated in principal complex value using De Moivre’s Formula as
which in approximate decimal form works out to
2.317818225 + 0.610148608i which when multiplied by the givcs
0.446064325 + 0.117423154i, where we then add the term to produce
A similar computation starting with the conjugate value (and again using only the + sign in the refinement formula) will produce the complex conjugate of the value we just computed. If we were to use the minus sign both times instead of plus, the result would again be a next pair of conjugate complex numbers but with the sign reversed for the real-number component. Either way, when one draws a graph representing these complex numbers as vectors in the complex plane, the result is a sequence of vectors starting to converge in a kind of arc to the actual conjugate complex zeros 0 + 1i and 0 - 1i . Clearly the next stage of approximation will be tedious. (In my experience this convergence is slower than for real-number zeros.)
In any case the point is that this refinement of Newton’s Method, given a prudent starting value, can concern itself with a pair of conjugate complex-number zeros, whereas the original method xn+1 = xn - f(xn)/f '(xn) (applied to a real-valued function) produces non-real complex values only if one chooses such a number as an initial value.