A Taylor-polynomial-based Faster-converging

Generalization of Newton’s Method for

Successive Approximation of Zeros

Donald R. Burleson, Ph.D.

Copyright © 2013, 2020 by Donald R. Burleson.

This article may be reproduced in its entirety

provided original authorship is expressly
acknowledged.

It is well known that Newton’s Method, a generally
reliable algorithm for successively

approximating a zero of a continuous and
differentiable function f(x), consists of

starting with an initial estimate x_{0} and
computing a sequence of improving

approximations x_{1}, x_{2}, x_{3},
etc. as needed, until the successive iterates no longer differ

significantly, where each “next” approximation x_{n+1}
is computed from the “present”

approximation x_{n}
by the formula

provided fʹ(x_{n})
is not too close to 0, in which case the successive computations may

fail to converge.

In calculus courses the usual method of deriving the
formula for Newton’s

Method is to write the equation of the tangent line to
the curve y = f(x) at the point

(x_{0}, f(x_{0})) and determine the
value x_{1} (which is then the first iterate) at which this

tangent line intersects the x-axis. One then repeats the process starting with x_{1}

to produce the next value x_{2} and so on.

But it is often not mentioned (mostly because Newton’s
Method and

Taylor series are commonly covered in different
courses) that if one

truncates the Taylor series

to approximate a curve y = f(x) roughly with a
degree-one Taylor polynomial

y = f_{1}(x) = f(x_{0}) + fʹ(x_{0})(x - x_{0})

then under the condition y = 0 one readily verifies
that this representation

is equivalent to the usual tangent line derivation of
Newton’s formula.

I propose that a natural generalization, then, is to
start by approximating

the curve y =
f(x) with a degree-two Taylor polynomial

y = f_{2}(x) = f(x_{0}) + fʹ(x_{0})(x-x_{0}) + (1/2)fʺ(x_{0})(x-x_{0})^{2}

from which, solving quadratically for x-x_{0}
under the condition y = 0

to determine the x-value at which this approximating
parabola intersects

the x-axis, we have

where x = x_{1} will represent the new
approximation of the zero in question.
This

approximation will generally be closer to the desired
zero of f(x) than x_{1}

would be by a one-pass Newton’s Method computation,
since the parabola

y = f_{2}(x) could be expected more closely to
approximate the curve y = f(x)

than does the linear y = f_{1}(x). Algebraically rearranging the above equation

and writing x_{n+1} for x and x_{n} for x_{0} gives our main result, an
iterative

algorithm that performs the same task as Newton’s
Method but

converges significantly faster:

where the sign is chosen to match the sign of fʹ(x_{n})/fʺ(x_{n}) since this
choice of sign

clearly minimizes the absolute difference between x_{n} and x_{n+1}.

(Under the right circumstances, with a negative
radicand and given a

workable starting value, this generalized method may
begin to converge

to a pair of conjugate complex numbers, in which case
of course

we employ both algebraic signs in front of the square
root.)

It is understood, of course, that similarly to what
happens in Newton’s Method

when fʹ is too close to 0, this parabolic
approach could fail to perform well

if fʺ were too close to 0, in particular if one
were to choose (x_{0}, f(x_{0}))

too close to an inflection point of the curve y =
f(x).

One naturally could devise further generalizations
using a degree-three

Taylor polynomial and so on, but the parabolic method
described here

often produces results correct to six or seven decimal
places

in only one step, with relatively simple computations,
whereas

higher-degree Taylor polynomial approaches would
entail computations

considerably more tedious with only moderately
faster-converging results.

The following examples show only one iteration of the
parabolic

method, with error less than 10^{ -6}. A second iteration would produce

values having error much smaller even than that.

EXAMPLE 1

To approximate the cube root of 7, i.e. to approximate
a zero of

the function
f(x) = x^{3}-7, using the initial estimate x_{0} = 1.9.

For f(x) = x^{3}-7,
with derivatives fʹ(x) = 3x^{2} and fʺ(x) = 6x,

the method described above gives the new estimate of
the zero as

= 1.9 - 0.95 + 0.962931380 =
1.912931380

As the actual cube root of 7 to nine decimal places is
1.912931183,

the absolute error of our one-pass computation is only
0.000000187.

By comparison, Newton’s Method for one pass gives

x_{1} = 1.913019391

for a larger absolute error of 0.000088208.

EXAMPLE 2

To approximate a zero of the function f(x) = sin(x) -
e^{ -x} close to x = 0.6.

The parabolic method with initial estimate x_{0}
= 0.6 gives the next iterate to be

= 0.6 + 1.234130118 - 1.245597323 =
0.588532795

with an absolute error of only 0.000000051

since the actual zero (correct to nine places as
determined by four

passes in Newton’s Method) is 0.588532744.

Newton’s Method for one pass gives x_{1} =
0.588479519

with larger absolute error 0.000053225.