In this paper we consider the error propagation of the Root and Bell's disk iteration methods enhanced by incorporating a correction term and a choice of a disk inversion formula in the methods, for the simultaneous computation of the zeros of a polynomial. The asymptotic error propagation is proved to be the same in both methods. This result is important considering the fact that these methods are in popular usage in the simultaneous computation of the zeros of a polynomial. The proof of the results herein follows the ideas of [M.S. Petkovic, C. Carstensen, Some improved inclusion methods for polynomial roots with Weierstrass corrections, Comput. Math. Appl. 25 (3) (1993) 59-67]. When the refinement process of correction is efficient, it is this mode of correction we have desired to propose.

The Root and Bell's disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial, Part I: Correction methods

In this paper, we present iterative methods of Weierstress’ type for the simultaneous inclusion of all simple zeros of a polynomial. The main advantage of the proposed methods is the increase of the convergence rate attained by applying suitable correction terms. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis for the total-step and the single-step methods. Numerical examples are given.

Weierstrass-like Methods with Corrections for the Inclusion of Polynomial Zeros

In Part I (Ikhile, 2008) [4], it was established that the root and Bell's disk/point iteration methods with or without correction term are of the same asymptotic error propagation characteristics in the simultaneous determination of the zeros of a polynomial. This concluding part of the investigation is a study in round-offs, its propagation and its effects on convergence employing interval arithmetic means. The purpose is to consequently draw attention on the effects of round-off errors introduced from the point arithmetic part, on the rate of convergence of the generalized root and Bell's simultaneous interval iteration algorithms and its enhanced modifications introduced in Part I for the numerical inclusion of all the zeros of a polynomial simultaneously. The motivation for studying the effects of round-off error propagation comes from the fact that the readily available computing devices at the moment are limited in precision, more so that accuracy expected from some programming or computing environments or from these numerical methods are or can be machine dependent. In fact, a part of the finding is that round-off propagation effects beyond a certain controllable order induces overwhelmingly delayed or even a severely retarded convergence speed which manifest glaringly as poor accuracy of these interval iteration methods in the computation of the zeros of a polynomial simultaneously. However, in this present consideration and even in the presence of overwhelming influence of round-offs, we give conditions under which convergence is still possible and derive the error/round-off relations along with the order/R-order of convergence of these methods with the results extended to similar interval iteration methods for computing the zeros of a polynomial simultaneously, especially to Bell's interval methods for refinement of zeros that form a cluster. Our findings are instructive and quite revealing and supported by evidence from numerical experiments. The analysis is preferred in circular interval arithmetic.

The root and Bell's disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial, Part II: Round-off error analysis by use of interval arithmetic

The aim of this paper is to develop improved methods for finding the multiple and simple roots of polynomial equations. The improved methods are based on Durand-Kerner method (a modification of Weierstrass method) and Laguerre method. The new simultaneous method has higher order of convergence and high computational efficiency since the accelerated convergence is attained with only negligible number of additional numerical operations. We have studied two classes of algorithms for solving polynomial equations: those with a known order of multiplicity and others with no information on multiplicity. For the second class we have proposed a suitable algorithm that computes simultaneously the distinct roots of polynomials and their respective multiplicities.

Accelerating simultaneous methods for the determination of polynomial multiple roots

In this paper we present iterative methods of Weierstrass's type for the simultaneous inclusion of all multiple zeros of a polynomial. The order of convergence of the proposed interval method is 1 + √2 ≈ 2.414 or 3, depending on the type of the applied disk inversion. The criterion for the choice of a proper circular root-set is given. This criterion uses the already calculated entries which increases the computational efficiency of the presented algorithms. Numerical results are given to demonstrate the convergence behavior.

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Inclusion Weierstrass-like root-finders with corrections

The third order interval Gargantini method for the simultaneous inclusion of polynomial zeros was improved to the fourth order method by Carstensen and Petkovic [An improvement of Gargantini’s simultaneous inclusion method for polynomial roots by Schroeder’s correction, Appl. Numer. Math. 13 (1994) 453-468]. We investigate the numerical stability of this improved method in the presence of rounding errors. The dependence of the convergence rate of the considered method on the magnitude of rounding errors is studied.

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M. Dušan Milošević

Papers

Inclusion Weierstrass-like root-finders with corrections

The numerical stability of the improved Gargantini method