A study of accelerated Newton methods for multiple polynomial roots

Abstract

We analyze and compare several accelerated Newton methods with built in multiplicity estimates. We also introduce the concept of indicator functions and discuss the Crouse-Putt method. It is shown that many of the accelerated Newton methods not only derive from Schröder's classic approach but are equivalent. The related computational experiments show that the built in multiplicity estimates can significantly decrease the number of Newton iterations, while the error of these estimates may significantly increase.

The rate of multiplicity of the roots of nonlinear equations and its application to iterative methods

“…In this case, no additional information is needed in order to solve the equations, unlike other traditional methods which use either the exact order of multiplicity or other kind of properties (if this multiplicity is integer, in order to approximate it via extrapolation, or the sign of the order multiplicity, to know if there is a discontinuous pole or a continuous root). Some of these methods are found in [4,6].…”

“…In many equations, m is, or can be, accurately known. If m > 0 is an integer, even if it is not known, can be estimated by extrapolation methods ( [4,11]) (it is sufficient to approximate it with an error less than 0.5, which can often be reached with a small increase of computational cost).…”

“…Similar processes are introduced in [7,11,12,16], among others. Good surveys, including comparison of the performances of the algorithms when solving polynomial equations appear in [4,11].…”

“…This rate does not only analyze the theoretical properties of the function but it can also be used as an accelerator of the iteration. Though indicators, as defined in [4] based on [3] are closely related to first order rates of multiplicity, our definition is more general and can be used for accelerating higher order methods (such as Halley's and similar third order ones, or even higher than third order). Besides, they are useful for noninteger orders of multiplicity.…”

Accurate fourteenth-order methods for solving nonlinear equations

“…zero) on the interval D. That is, f (α) = 0 and f (α) = 0 in this neighborhood. Approximating the multiple roots of the equation (1) is out of the scope of this paper, thus kindly see [4] and the references therein for more information on this subject matter. The famous Newton's method converges locally quadratically and it consists of one evaluation of the function and one evaluation of the first derivative per iteration [13].…”