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 [1] - [3] to the root. In this note, we briefly review the iterative methods [4] - [8] for approaching a root of an unknown multiplicity, and present a first oder [9] 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.

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.

The Newton-Raphson method

is quadratic

if. However, if, the method declines to mere linear

See also [10] .

Let be already near root a. Then, if

nearly. Eliminating from the two equations we obtain

which we solve for an approximate a, as

where

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.

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 [11] - [13] . 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.

For

For

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

The method

produces the superlinear

and if, convergence is alternating.

We readily have that

For example, for, we compute using Equation (23) the approximations as depending on the chosen x

If

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 [14] [15] .

It is a rational two step method of the form

With

the method is quadratic for and. In fact;

For

For

For

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.

For we have:

For

For

For

For

For

For

For

For

For

For

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

For

For

For

For

For

For

For

For

For

For

For

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.

For

For

For

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).

I. Fried, (2016) A Remarkable Chord Iterative Method for Roots of Uncertain Multiplicity. Applied Mathematics,07,1207-1214. doi: 10.4236/am.2016.711106