A Remarkable Chord Iterative Method for Roots of Uncertain Multiplicity

In this note we at first briefly review iterative methods for effectively approaching a root of an unknown multiplicity. We describe a first order, then a second order estimate for the multiplicity index m of the approached root. Next we present a second order, two-step method for iteratively nearing a root of an unknown multiplicity. Subsequently, we introduce a novel chord, or a twostep method, not requiring beforehand knowledge of the multiplicity index m of the sought root, nor requiring higher order derivatives of the equilibrium function, which is quadratically convergent for any m ≤ 4 , and then reverts to superlinear.


Introduction
The multiplicity index m of root x a = , ( ) 0 f a = of equilibrium function ( ) f x 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 4 m ≤ , and then reverts to superlinear.

Assumed Nature of the Equilibrium Function
We assume that near root ( ) , 0 a f a = , function ( ) f x has the power series representation where m is the multiplicity index of root a, and where , , , A B C etc. are, for 1 m = , the coefficients ( ) ( ) ( ) and so on.

Extrapolation to the Limit
nearly.Eliminating B A 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 3 x of Equation (8) we have For example, for ( )

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 [11]- [13].Using µ in Equation ( 14) for m in Equation ( 12) we obtain the method which is quadratic for any, provided, The method of Equation ( 15) is also obtained by applying Newton's method not to f, but rather to u f f ′ = .For ( ) ( ) + , we obtain by the method of Equation ( 15) that requires not only f ′ but also f ′′ , starting with 0 1 x = .
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

An Erroneous m
The method ( ) produces the superlinear and if 0 >  , convergence is alternating.

Estimation of the Leading Term
We readily have that For example, for

An Elementary Discrete Two-Step Newton Method for Roots of Any Multiplicity
If are already close to root a of multiplicity 1 m > , then according to Equation (5)   ( ) ( ) , and nearly, from which we extract the approximation Setting a back into Equation (26) yields and the two-step method where µ in Equation ( 28) is seen to be but the finite-difference approximation of we have from the method of Equation ( 29) that ( ) ( ) The repeated classical Newton's method, 1 See also [14] [15].

Derivation of the Chord Method
It is a rational two step method of the form ( ) ( ) ( ) the method is quadratic for 1, 2 m m = = and 3 m = .In fact; For 1 m = ( ) ( ) ( ) and for 2 k = the method is quadratic for 4 m = as well.According to Equation (36a), if 1, 9 7 m k = = − , then the method is higher than quadratic.

Lowering the Value of k
We leave k in 1 0 0 0 x x kf f ′ = + of Equation (34), free, and have by power series expansion, for multiplicity index 5 m = , for ( ) The linear term in the above expansion is annulled with 2 125 55 4 0, 1.9859043.
We do this for higher values of m and find that The general form of the linear part of 2 x a − in Equations ( 42) is of the form ( )( ) 0 c m x a − with a constant ( ) c m that is small if multiplicity index m is not much above 5.For instance, ( ) , meaning that at each iteration the error 2 x a − is reduced by this factor.Such convergence behavior we term superlinear.More concretely, for ( ) ( ) + , we obtain by the above method, using 1.95 k = , starting with 0 1 x = .

Conclusions
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, 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).