On theR-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 Newton's and Halley's 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.     

A family of simultaneous methods for the determination of polynomial complex zeros

Using the development of a rational function by elementary fractions, a family of methods for the simultaneous determination of polynomial complex zeros is derived. All the methods of the family are cubically convergent for simple zeros. The known simultaneous procedures of the third order are included. The presented class of iteration functions is suitable for the parallel inclusion of polynomial complex zeros by circular regions. The family of methods, defined in complex circular arithmetic, gives a new interval method with cubic convergence. Numerical example is given.     

Derivative Free Inclusion Methods for Polynomial Zeros

Two iterative methods for the simultaneous inclusion of complex zeros of a polynomial are presented. Both methods are realized in circular interval arithmetic and do not use polynomial derivatives. The first method of the fourth order is composed as a combination of interval methods with the order of convergence two and three. The second method is constructed using double application of the inclusion method of Weierstrass’ type in serial mode. It is shown that its R-order of convergence is bounded below by the spectral radius of the corresponding matrix. Numerical examples illustrate the convergence rate of the presented methods     

A family of root-finding methods with accelerated convergence

A parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given.     

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

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.     

On the convergence condition of generalized root iterations for the inclusion of polynomial zeros

Using a fixed point relation based on the logarithmic derivative of the k-th order of an algebraic polynomial and the definition of the k-th root of a disk, a family of interval methods for the simultaneous inclusion of complex zeros in circular complex arithmetic was established by Petković [M.S. Petković, On a generalization of the root iterations for polynomial complex zeros in circular interval arithmetic, Computing 27 (1981) 37–55]. In this paper we give computationally verifiable initial conditions that guarantee the convergence of this parallel family of inclusion methods. These conditions are significantly relaxed compared to the previously stated initial conditions presented in literature.     

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

The aim of this paper is to present some modifications of Newton's 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 Jarratt's 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.     

The convergence of Euler-like method for the simultaneous inclusion of polynomial zeros

In this paper, we give the convergence analysis of the Euler-like iterative method for the simultaneous inclusion of all simple real or complex zeros of a polynomial. The established initial conditions provide the safe convergence of the considered method and the fourth order of convergence. These conditions are computationally verifiable, which is of practical importance. A procedure for the choice of initial inclusion disks is also given.     

On the convergence order of a modified method for simultaneous finding polynomial zeros

Using Newton's corrections and Gauss-Seidel approach, a modification of single-step method [1] for the simultaneous finding all zeros of ann-th degree polynomial is formulated in this paper. It is shown thatR-order of convergence of the presented method is at least 2(1+τ _n ) where τ _n ∈(1,2) is the unique positive zero of the polynomial \(\tilde f_n (\tau ) = \tau ^n - \tau - 1\) . Faster convergence of the modified method in reference to the similar methods is attained without additional calculations. Comparison is performed in the example of an algebraic equation.     

An iterative method for algebraic equation by Padé approximation

In this paper, we consider iterative formulae with high order of convergence to solve a polynomial equation,f(z)=0. First, we derive the numerator of the Padé approximant forf(z)/f′(z) by combining Viscovatov's and Euclidean algorithms, and then calculate the zeros of the numerator so as to apply one of the zeros for the next approximation. Regardless of whether the root is simple or multiple, the convergence order of this iterative formula is always attained for arbitrary positive integerm with the Taylor polynomial of degreem for a given polynomialf(z). Since it is easy to systematically obtain formulae of different order, we can choose formulae of suitable order according to the required accuracy.     

On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic

Consider a polynomialP (z) of degreen whose zeros are known to lie inn closed disjoint discs, each disc containing one and only one zero. Starting from the known simultaneous interval processes of the third and fourth order, based on Laguerre iterations, two generalised iterative methods in terms of circular regions are derived in this paper. These interval methods make use of the definition of thek-th root of a disc. The order of convergence of the proposed interval methods isk+2 (k≧1). Both procedures are suitable for simultaneous determination of interval approximations containing real or complex zeros of the considered polynomialP. A criterion for the choice of the appropriatek-th root set is also given. For one of the suggested methods a procedure for accelerating the convergence is proposed. Starting from the expression for interval center, the generalised iterative method of the (k+2)-th order in standard arithmetic is derived.     

The simultaneous determination of all zeros of a polynomial

A class of adaptive iterative methods of higher order for the simultaneous determination of all zeros of a polynomial is constructed. These methods preserve their order of convergence also in the case of multiple roots. Numerical examples are included.     

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

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's 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.     

Some modifications of the parallel Halley iteration method and their convergence

In this paper we derive five kinds of algorithms for simultaneously finding the zeros of a complex polynomial. The convergence and the convergence rate with higher order are obtained. The algorithms are numerically illustrated by an example of degree 10, and the numerical results are satisfactory.     

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.     

An improvement on two iteration methods for simultaneous determination of the zeros of a polynomial

The Durand-Kerner and the Ehrlich? Methods of respective quadratic and cubic convergence are two of the current methods for determining simultaneously all zeros of a polynomial. By respectively including a Durand-Kerner and a Newton correction term in the above formulae, we establish two new methods-the Improved Durand-Kerner and the Improved Ehrlich.

We show that the improvement is reflected by an increase of unity in the order of convergence of each of the two methods.     

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

In this paper we construct iterative methods of Ostrowski's 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 Halley's 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.     

On The Convergence Of Some Interval Methods For Simultaneous Computation Of Polynomial Zeros

In this paper we consider the convergence of a certain interval method for simultaneous computation of polynomial zeros. Under the legitimacy of suitable isolation of the roots in a restrictive respective circular disks it is established in a limiting sense a finite positive constant in existence for which convergence is certain. This positive constant which is the limiting convergence factor is dependent on the minimum distance between the zeros of the polynomial. This provides a qualitative information that may be found useful on the occasion the roots are clustered. The climax however, is an introduction of a new interval method and an improved modification of an existing one considered.     

The zeros of complex polynomials,matrix inequalities and non-linear programming†

The complex zeros of a general complex polynomial are localized by constructing the intersection of areas in the complex plane defined by various inequality bounds on the eigenvalues of the companion matrix and also, possibly, by other inequalities on the zeros of polynomials. This localization then provides an efficient starting point for determining the zeros by applying a non-linear optimizer, such as the Fletcher-Powell method, to the square of the modulus of the polynomial, |p(x+iy)|², in order to determine its minimums. The minimums of | p |² are zero and occur at the zeros of p(z). Experimentation indicates that Gershgorin's discs and similar results for Cassini's ovals supply rather sharp bounds for this purpose     

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

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.