Lidija Rancic

On the new fourth-order methods for the simultaneous approximation of polynomial zeros

A new iterative method of the fourth-order for the simultaneous determination of polynomial zeros is proposed. This method is based on a suitable zero-relation derived from the fourth-order method for a single zero belonging to the Schroder basic sequence. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is studied in detail for the proposed method. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. The construction of improved methods in ordinary complex arithmetic and complex circular arithmetic is discussed. Finally, numerical examples and the comparison with existing fourth-order methods are given.

A new fourth-order family of simultaneous methods for finding polynomial zeros

Using a suitable fixed point relation with a complex parameter, a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic. Convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations. Further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented. Some numerical results for various values of the parameter are given.

Higher-order simultaneous methods for the determination of polynomial multiple zeros

Starting from Laguerres method and using Newtons and Halleys corrections for a multiple zero, new simultaneous methods of Laguerres type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.

The improved Farmer–Loizou method for finding polynomial zeros

An improvement of the Farmer–Loizou method for the simultaneous determination of simple roots of algebraic polynomials is proposed. Using suitable corrections of Newtons type, the convergence of the basic method is increased from 4 to 5 without any additional calculations. In this manner, a higher computational efficiency of the improved method is achieved. We prove a local convergence of the presented method under initial conditions which depend on a geometry of zeros and their initial approximations. Numerical examples are given to demonstrate the convergence behaviour of the proposed method and related methods.

On the guaranteed convergence of a cubically convergent Weierstrass-like root-finding method

Initial conditions that provide guaranteed and fast convergence of the Weierstrass-like cubically convergent iterative method for the simultaneous determination of all simple zeros of a polynomial are considered. It is proved that this method is convergent under suitable conditions stated in the spirit of Smales point estimation theory. The proposed convergence conditions are computationally verifiable since they depend only on initial approximations and the degree of a given polynomial, which is of practical importance.

On the guaranteed convergence of new two-point root-finding methods for polynomial zeros

We presented new two-point methods for the simultaneous approximation of all n simple (real or complex) zeros of a polynomial of degree n. We proved that the R-order of convergence of the total-step version is three. Moreover, computationally verifiable initial conditions that guarantee the convergence of one of the proposed methods are stated. These conditions are stated in the spirit of Smale’s point estimation theory; they depend only on available data, the polynomial coefficients, polynomial degree n and initial approximations x1(0),…,xn(0)

New simultaneous root-finding methods with accelerated convergence for analytic functions

x_{1}^{(0)},\ldots ,x_{n}^{(0)}

On a modification of the Ehrlich–Aberth method for simultaneous approximation of polynomial zeros

, which is of practical importance. Using the Gauss-Seidel approach we state the corresponding single-step version and consequently its prove that the lower bound of its R-order of convergence is at least 2 + yn > 3, where yn ∈ (1, 2) is the unique positive root of the equation yn − y − 2 = 0. Two numerical examples are given to demonstrate the convergence behavior of the considered methods, including global convergence.

Point estimation of simultaneous methods for solving polynomial equations: A survey (II)

A new iterative method of the fourth order for the simultaneous determination of zeros of a class of analytic functions, is proposed. Further improvements of the basic method are attained by using Newtons and Halleys corrections giving the orders of convergence five and six, respectively. The improved convergence is achieved with negligible number of additional calculations, which significantly increases the computational efficiency of the accelerated methods. Numerical examples demonstrate a good convergence properties, fitting very well theoretical results.

Point estimation of simultaneous methods for solving polynomial equations

A modified method of the fourth order for the simultaneous determination of simple complex zeros of a polynomial, which may be regarded as an extension of the Ehrlich–Aberth method, is given. This method is derived using a very simple procedure which is also applicable for the construction of a whole class of simultaneous methods. The convergence analysis of the presented method is performed under computationally verifiable initial conditions, which is of significant practical importance. Numerical results obtained by several iterative methods of the fourth order are also given.