Dusan M. Milosevic

A higher order family for the simultaneous inclusion of multiple zeros of polynomials

Abstract Starting from a suitable fixed point relation, a new family of iterative methods for the simultaneous inclusion of multiple complex zeros in circular complex arithmetic is constructed. The order of convergence of the basic family is four. Using Newton’s and Halley’s corrections, we obtain families with improved convergence. Faster convergence of accelerated methods is attained with only few additional numerical operations, which provides a high computational efficiency of these methods. Convergence analysis of the presented methods and numerical results are given.

Ostrowski-Like Method with Corrections for the Inclusion of Polynomial Zeros

In this paper we construct iterative methods of Ostrowskis type for the simultaneous inclusion of all zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods with Newton and Halleys corrections. The case of multiple zeros is also considered. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. Numerical examples and an analysis of computational efficiency are given.

Improved Halley-like methods for the inclusion of polynomial zeros

Improved iterative methods of Halleys type for the simultaneous inclusion of all simple complex zeros of a polynomial are proposed. The presented convergence analysis, which uses the concept of the R-order of convergence of mutually dependent sequences, shows that the convergence rate of the basic fourth order method is increased to 5 and 6 using Newtons and Halleys corrections, respectively. The proposed inclusion methods possess a high computational efficiency since the increase of the convergence is attained without additional calculations. This advantage, together with very fast convergence, make that the presented methods are ranking among the most powerful inclusion methods for polynomial zeros. In order to demonstrate convergence properties of the proposed methods, two numerical examples are given.

On a new family of simultaneous methods with corrections for the inclusion of polynomial zeros

A high-order one-parameter family of inclusion methods for the simultaneous inclusion of all simple complex zeros of a polynomial is presented. For specific values of the parameter, some known interval methods are obtained. The convergence rate of the basic fourth-order family is increased to 5 and 6 using Newtons and Halleys corrections, respectively. Using the concept of the R-order of convergence of mutually dependent sequences, we present a convergence analysis of the accelerated total-step and single-step methods with corrections. The suggested inclusion methods have great computational efficiency since an increase of the convergence rate is attained with only a few additional calculations. Two numerical examples are included to demonstrate the convergence properties of the proposed methods.

Efficient methods for the inclusion of polynomial zeros

Abstract Using a suitable zero-relation and the inclusion isotonicity property, new interval iterative methods for the simultaneous inclusion of simple complex zeros of a polynomial are derived. These methods produce disks in the complex plane that contain the polynomial zeros in each iteration, providing in this manner an information about upper error bounds of approximations. Starting from the basic method of the fourth order, two accelerated methods with Newton’s and Halley’s corrections, having the order of convergence five and six respectively, are constructed. This increase of the convergence rate is obtained without any additional operations, which means that the methods with corrections are very efficient. The convergence analysis of the basic method and the methods with corrections is performed under computationally verifiable initial conditions, which is of practical importance. Two numerical examples are presented to demonstrate the convergence behavior of the proposed interval methods.

New higher-order methods for the simultaneous inclusion of polynomial zeros

Higher-order methods for the simultaneous inclusion of complex zeros of algebraic polynomials are presented in parallel (total-step) and serial (single-step) versions. If the multiplicities of each zeros are given in advance, the proposed methods can be extended for multiple zeros using appropriate corrections. These methods are constructed on the basis of the zero-relation of Gargantini’s type, the inclusion isotonicity property and suitable corrections that appear in two-point methods of the fourth order for solving nonlinear equations. It is proved that the order of convergence of the proposed methods is at least six. The computational efficiency of the new methods is very high since the acceleration of convergence order from 3 (basic methods) to 6 (new methods) is attained using only n polynomial evaluations per iteration. Computational efficiency of the considered methods is studied in detail and two numerical examples are given to demonstrate the convergence behavior of the proposed methods.

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newtons type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratts corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.

High order Euler-like method for the inclusion of polynomial zeros

Abstract Improved iterative method of Euler’s type for the simultaneous inclusion of polynomial zeros is considered. To accelerate the convergence of the basic method of the fourth order we applied Borsch–Supan’s correction. It is proved that the R -order of convergence of the improved Euler-like method is six. The convergence analysis is derived under computationally verifiable initial conditions. The proposed algorithm possesses great computational efficiency since the increase of the convergence rate from 4 to 6 is obtained with negligible number of additional calculations. In order to demonstrate convergence properties of the suggested method, two numerical examples are given.

On an efficient inclusion method for finding polynomial zeros

New efficient iterative method of Halleys type for the simultaneous inclusion of all simple complex zeros of a polynomial is proposed. The presented convergence analysis, which uses the concept of the R -order of convergence of mutually dependent sequences, shows that the convergence rate of the basic fourth order method is increased from 4 to 9 using a two-point correction. The proposed inclusion method possesses high computational efficiency since the increase of convergence is attained with only one additional function evaluation per sought zero. Further acceleration of the proposed method is carried out using the Gauss-Seidel procedure. Some computational aspects and three numerical examples are given in order to demonstrate high computational efficiency and the convergence properties of the proposed methods.

Improved methods for the simultaneous inclusion of multiple polynomial zeros

Using a new fixed point relation, the interval methods for the simultaneous inclusion of complex multiple zeros in circular complex arithmetic are constructed. 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 with Schroders and Halley-like corrections under computationally verifiable initial conditions. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. Two numerical examples are given to demonstrate convergence characteristics of the proposed method.