Iterative methods    for locating roots of nonlinear equations are of further appeal if converging oppositely  or alternatingly   so as to establish bounds, or to bracket, the targeted root.
In this note we are mainly concerned with the creation of oppositely and alternatingly converging iterative methods for ever tighter bounds on the targeted root. By a slight parametric perturbation of Newton’s method we create an alternating super-linear method approaching the targeted root in turns from above and from below. Further extension of Newton’s method  creates an oppositely converging quadratic counterpart to it. This New method requires a second derivative, but for it, the average of the two opposite methods rises to become a cubic method.
This note examines also the creation of high order iterative methods by a repeated evaluation of undetermined coefficients  .
2. The Function
At the heart of this note lies the seeking of simple root a of function , , a function which we assume throughout the paper to have the expanded form
and so on.
The condition guarantees that root a of is simple, or, otherwise said, of multiplicity one.
The one-step iterative method is of the general, and expanded, form
and, evidently, if , namely, if a is a fixed-point of iteration function in Equation (3), and if, further, , then iterative method (3) converges quadratically to a. Higher order derivatives of being zero at point a moves iterative method (3) to still higher orders of convergence to fixed- point .
3. Newton’s Method
With comprehensiveness in mind we actually derive this classical, mainstay iterative method of numerical analysis. We start by generally stating it as
for any value of free parameter P. By the fact that , point a is a fixed- point of iteration function of method (4), that is to say, .
Power series expansion of yields
Equation (5) suggests that the choice should result in a quadratic method. However, since , and since a is unknown, we replace it by the known approximation , actually take to have
which happens to be still quadratic; convergence of Newton’s method to a simple, namely of multiplicity one ( ), root is verified to be of second order
where are as in Equations ((1) and (2)).
For example, for , we generate by Equation (6), starting with , the converging sequence
4. Extrapolation to the Limit
Let be already near root a, then, by Equation (7)
nearly. Eliminating from the two equations we are left with
which we solve for an approximate a, as
The square root in Equation (11) may be approximated as
For example, for , and starting with , we compute , ; and then from Equation (11), . Another such cycle starting with produces a next .
5. Estimating the Leading Term B/A
From Equation (9) we have that, approximately
if is already close to a.
We pick from the list in Equation (8) the values
and have from Equation (15) that .
6. Hopping to the Other Side of the Root
If instead of of Equation (6) we vault over by the double step
then we land at
implying that asymptotically, as
and this is good to know.
For example, seeking a root of we start with , which is above the root , and using we obtain which is, indeed, under the root .
7. A Chord Method
Each step of Newton’s method provides us with an and values, which can be used to polynomially extrapolate to zero. A linear extrapolation through the pair of points results in the line
We set and obtain the extrapolated value
which may now be repeated to form a chord iterative method.
For example, from the two values , taken from the list in Equation (8) for , we compute from Equation (21) .
Starting with , we repeatedly compute
with no need for any further derivative function evaluation.
Theoretically, by power series expansion
According to Equation (23) and are ultimately of the same sign.
8. A Rational Higher Order Method
To have this we start with
for open parameter Q. Power series expansion yields
To have a cubic method we take
with A and B as in Equation (2), but evaluated at rather than at a, to have Halley’s method
which is cubic
9. A Polynomial Higher Order Method
Here we start with the quadratic in
for undetermined parameter Q. Power series expansion of yields
To have a cubic method we take
with A and B evaluated at rather than at a as in Equation (2), and
which is verified to be still cubic
10. An Alternating Super-Linear Method
We take Newton’s method of Equation (6) and parametrically perturb it into
for some parameter , to have
which is super-liner if . Moreover, for a positive convergence here is ultimately alternating: and are of opposite signs.
For example, for , , , we compute from Equation (34)
11. An Opposite Quadratic Method
To create a method oppositely converging to Newton’s method of Equations ((6) and (7)) we start with the perturbed Newton’s method
or in power series form
To have a quadratic method opposite to Newton’s we set, in view of Equation (7)
which is, indeed, opposite to Newton’s
Compare Equation (41) with Equation (7).
12. The Average of the Opposites Is a Cubic Method
Method (40) requires and above the mere of Newton’s method, but for this, the average of the two opposite methods rises to become the cubic method
of Equation (32).
13. A Quartic Method
Hereby we advance another undetermined coefficients strategy for constructing high order iterative methods      to locate root a of , .
We start with the polynomial iteration function
of undetermined coefficients , and expand as
The coefficients of up to are made zero with
to have a quartic method.
However, as root a is unavailable we replace it in by to have the variable
But, with this replacement of a by method (43) falls back to a mere quadratic
To repair this retreat in the order of convergence we propose to further correct of Equations ((43) and (46)) into
for new parameters . Power series expansion reveals that method (48) is restored to fourth order with , to have
14. The Rational Method Revisited
We start here with
and expand it into
To have a cubic method we set
Inserting these last P and Q values into Equation (51) we have the disappointing
We retry for a cubic method by writing, still for P and Q of Equation (54)
for which we get by power series expansion
We take , and have the method
which is now restored to third order of convergence
15. Undetermined Variable Factors
Here we start with
and expand to have
in which . To have a cubic method, and considering that we impose the two conditions:
and solve system (62) for P as
Since root a is unknown we evaluate P instead at and have iterative method (60) in the form
which we recognize as being the cubic method of Halley
In Section 6, we have demonstrated that the double-step Newton’s method places the next point on the other side of the root. Passing a line through two such points we create a chord method as that of Section 7.
A slight perturbation of Newton’s method, as in Equation (34) creates a super-linear method of alternating convergence as in Section 10. By a further modification of Newton’s method we have created a quadratic method opposite to Newton’s. The average of these two opposite methods is a cubic method, as shown in Section 12.
In Section 13, a quartic method is successfully created by repeated undetermined coefficients.
 Fried, I. (2009) Oppositely Converging Newton-Raphson Method for Nonlinear Equilibrium Problems. International Journal for Numerical Methods in Engineering, 79, 375-378.
 Fried, I. (2014) Effective High-Order Iterative Methods via the Asymptotic Form of the Taylor-Lagrange Remainder. Journal of Applied Mathematics, 2014, Article ID: 108976.
 Chun, C. and Neta, B. (2008) Some Modification of Newton’s Method by the Method of Undetermined Coefficients. Computers and Mathematics with Applications, 56, 2528-2538.