Lidija V. Stefanović

On some improvements of square root iteration for polynomial complex zeros

Abstract Using Newtons and Halleys corrections, some modifications of the simultaneous method for finding polynomial complex zeros, based on square root iteration, are obtained. The convergence order of the proposed methods is five and six respectively. Further improvements of these methods are performed by applying the Gauss—Seidel approach. The lower bounds of the R-order of convergence and the convergence conditions for the accelerated (single-step) methods are given. Faster convergence is attained without additional calculations. The considered iterative procedures are illustrated numerically in the example of an algebraic equation.

On some iteration functions for the simultaneous computation of multiple complex polynomial zeros

Second order methods for simultaneous approximation of multiple complex zeros of a polynomial are presented. Convergence analysis of new iteration formulas and an efficient criterion for the choice of the appropriate value of a root are discussed. A numerical example is given which demonstrates the effectiveness of the presented methods.

On a second order method for the simultaneous inclusion of polynomial complex zeros in rectangular arithmetic

Starting from separated rectangles in the complex plane which contain polynomial complex zeros, an iterative method of second order for the simultaneous inclusion of these zeros is formulated in rectangular arithmetic. The convergence and a condition for convergence are considered. Applying Gauss-Seidel approach to the proposed method, two accelerated interval methods are formulated. TheR-order of convergence of these methods is determined. An analysis of the convergence order is given in the presence of rounding errors. The presented methods are illustrated numerically in examples of polynomial equations.ZusammenfassungAusgehend von disjukten Rechtecken in der komplexen Ebene, die komplexe Polynomnullstellen enthalten, wird ein iteratives Verfahren 2. Ordnung zur gleichzeitigen Einschließung dieser Nullstellen in der Rechteckarithmetik formuliert und die Konvergenz des Verfahrens betrachtet. Es werden zwei Einzelschritt-Varianten des vorgestellten Verfahrens formuliert, die zu beschleunigter Konvergenz führen, und dieR-Konvergenz beider Varianten bestimmt. Anschließend wird die Konvergenzordnung unter Berücksichtigung von Rundungsfehler untersucht und es werden die vorgestellten Verfahren an Beispielen illustriert.

Some higher-order methods for the simultaneous approximation of multiple polynomial zeros

Abstract Applying Newtons and Halleys corrections, some modified methods of higher order for the simultaneous approximation of multiple zeros of a polynomial are derived. Further acceleration of convergence of these methods is performed by approximating all zeros in a serial fashion using the new approximations as they become available. Faster convergence is attained without additional calculations. Lower bounds of the R -order of convergence for the serial (single-step) methods are given.

On the convergence order of accelerated root iterations

SummaryA Gauss-Seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the (k+2)-th order (k∈N) for finding polynomial complex zeros, given in [7], is considered in this paper. It is shown that theR-order of convergence of the accelerated method is at leastk+1+σn(k), where σn(k)>1 is the unique positive root of the equation σn-σ-k-1 = 0 andn is the degree of the polynomial. The examples of algebraic equations in ordinary and circular arithmetic are given.

The numerical stability of the generalised root iterations for polynomial zeros

Abstract The generalised root iterations for simultaneous finding polynomial complex zeros, with the convergence order k + 2(k ϵN), was formulated in [6] in terms of circular regions. The method produces the disks which contain the exact zeros, giving error bounds automatically for each approximation. This paper investigates the numerical stability of this algorithm in the presence of rounding errors. The conditions for preserving the convergence rate are established. The presented analysis contains the results given in [3] and [4].

On the R-order of some accelerated methods for the simultaneous finding of polynomial zeros

A Gauss-Seidel procedure is applied to increase the convergence of a basic fourth order method for finding polynomial complex zeros. Further acceleration of convergence is performed by using Newtons and Halleys corrections. It is proved that the lower bounds of theR-order of convergence for the proposed serial (single-step) methods lie between 4 and 7. Computational efficiency and numerical examples are also given.ZusammenfassungEin Gauss-Seidel Verfahren wird verwendet zur Beschleunigung der Konvergenz eines Verfahrens vierter Ordnung zur Bestimmung komplexer Polynomwurzeln. Eine weitere Beschleunigung der Konvergenz wird erreicht durch Anwendung von Newton und Halley-Korrekturformeln. Es wird bewiesen, daß die unteren Schranken für dieR-Ordnung der vorgeschlagenen seriellen (Einzelschritt-) Verfahren zwischen 4 und 7 liegen. Die Effizienzzahl und ein numerisches Beispiel wird angegeben.

A family of simultaneous zero finding methods

A class of iterative methods with arbitrary high order of convergence for the simultaneous approximation of multiple complex zeros is considered in this paper. A special attention is paid to the fourth order method and its modifications because of their good computational efficiency. The order of convergence of the presented methods is determined. Numerical examples are given.

Forward-backward serial iteration methods for simultaneously approximating polynomial zeros

Two approaches in realization of iteration methods for simultaneously improving approximations to the polynomial zeros in a serial manner are considered. The new procedure alleviates some unwanted effects which occur in the implementation of iteration methods on sequential computers due to the finite mantissa of floating-point arithmetic and improves R-order of convergence. The accelerated convergence is demonstrated in the examples of most frequently used simultaneous methods.