Received 23 May 2016; accepted 8 July 2016; published 11 July 2016
The multiplicity index m of root, of equilibrium function may be a well latent property of the root, not cursorily revealed, nor readily available, yet this multiplicity can profoundly affect the behavior of the iterative approach  -  to the root. In this note, we briefly review the iterative methods  -  for approaching a root of an unknown multiplicity, and present a first oder  as well as a second order estimate for the multiplicity index m of the approached root. Then we present a novel chord, or a two-step method, not requiring beforehand knowledge of m, nor requiring the higher derivatives of the equilibrium function, which is quadratically convergent for any, and then reverts to superlinear.
2. Assumed Nature of the Equilibrium Function
We assume that near root, function has the power series representation
where m is the multiplicity index of root a, and where etc. are, for, the coefficients
and so on.
3. The Newton-Raphson Method
The Newton-Raphson method
if. However, if, the method declines to mere linear
See also  .
4. Extrapolation to the Limit
Let be already near root a. Then, if
nearly. Eliminating from the two equations we obtain
which we solve for an approximate a, as
The square root in Equation (8) may be approximated as
and for this extrapolated of Equation (8) we have
For example, for, and starting with, we compute,; and then from Equation (8),. Another such cycle starting with produces a next.
5. Always Quadratic Newton-Raphson Method
The modified Newton-Raphson method
converges quadratically to a root of any multiplicity m
But for this we need to know m.
By Equation (1) we readily deduce that, for any x
obtained at the price of a second derivative. For finite-difference approximations of the needed derivatives see  -  . Using in Equation (14) for m in Equation (12) we obtain the method
which is quadratic for any, provided, m
The method of Equation (15) is also obtained by applying Newton’s method not to f, but rather to. For, we obtain by the method of Equation (15) that requires not only but also, starting with.
Equation (15) may be written as
and it is of interest to know that
For the price of a third derivative we may have the quadratic approximation
6. An Erroneous m
produces the superlinear
and if, convergence is alternating.
7. Estimation of the Leading Term
We readily have that
For example, for, we compute using Equation (23) the approximations as depending on the chosen x
8. An Elementary Discrete Two-Step Newton Method for Roots of Any Multiplicity
are already close to root a of multiplicity, then according to Equation (5)
nearly, from which we extract the approximation
Setting a back into Equation (26) yields
and the two-step method
where in Equation (28) is seen to be but the finite-difference approximation of in Equation (14).
For example, for, and starting with, we compute by Equation (29), the successive approximations
Generally, starting with
we have from the method of Equation (29) that
The repeated classical Newton’s method, , we recall, is only linear if
See also   .
9. Derivation of the Chord Method
It is a rational two step method of the form
the method is quadratic for and. In fact;
For the method produces
and for the method is quadratic for as well.
According to Equation (36a), if, then the method is higher than quadratic.
10. The Method is Further Superlinear
For we have:
11. Lowering the Value of k
We leave k in of Equation (34), free, and have by power series expansion, for multiplicity index, for in Equation (1), that
The linear term in the above expansion is annulled with
We do this for higher values of m and find that
We try, and get
The general form of the linear part of in Equations (42) is of the form with a constant that is small if multiplicity index m is not much above 5. For instance, , meaning that at each iteration the error is reduced by this factor. Such convergence behavior we term superlinear. More concretely, for, we obtain by the above method, using, starting with.
The paper is predicated on the realistic assumption that the multiplicity index m of the iteratively targeted root is unknown. We conclude that if one prefers to shun second order derivatives, then the quadratic two-step method of Equation (29), that provides also ever better approximations for the multiplicity index m of the approached root, is a practically appealing alternative.
Otherwise, one may use the rational two-step method of Equation (34) with a constant k that is only slightly less than 2. Thus stating the method becomes superlinear, albeit, of a reduced speed of convergence for highly elevated root multiplicities. For the sake of brevity, the present paper does not explore the possibility of estimating the multiplicity index m of the sought root by the method of Equation (29), then applying this estimate to the choice of an optimal k in the method of Equations (34) and (35).
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