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.     

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.     

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.     

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 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 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.     

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 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.     

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.     

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.     

Sorting-based localization and stable computation of zeros of a polynomial. II

The stability of a sorting-based scheme for identifying polynomial zeros under coefficient perturbation is discussed. A method is proposed for simultaneous reconstruction (with a logarithmic estimate of time complexity) of the coefficients of an arbitrary polynomial from the values of its zeros. The method of identification of polynomial zeros is based on the operator of localization of extremal elements of a numerical sequence after its preliminary sorting. The method is extended to pattern recognition. Part I of this article is published in No. 1 (2007). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 161–174, March–April 2007.     

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.     

On initial conditions for the convergence of simultaneous root finding methods

Conditions for the convergence of iteration methods for the simultaneous approximation of polynomial complex roots, treated in the literature, are most frequently based on unattainable data. In this paper we give simple initial conditions involving only initial approximations to the roots of a polynomial and the polynomial degree. Convergence theorems are stated for the simultaneous methods in ordinary complex arithmetic and complex interval arithmetic.     

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

In this paper we present iteration methods of Halley'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 for the total-step and the single-step methods with Newton's corrections. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. A numerical example is given. Received: June 23, 1998     

On the type of a polynomial f, when f(0)f−f(0)f is a monomial

A polynomial is said to be of type (p₁, p₂, p₃) relative to a directed line in the complex plane if, counting multiplicities, it has p₁ zeros to the left of, p₂ zeros on, and p₃ zeros to the right of the line. In this paper we determine explicitly the types of all polynomials belonging to a very restricted (but infinite) family of polynomials. A polynomial ƒ belongs to this family if and only if its coefficients are such that the polynomial ƒ^*(0)ƒ(z)−ƒ(0) ƒ^*(z) is a monomial; here ƒ^* denotes the reflection of ƒ in the directed line.

A special case of the present result appeared in an earlier publication.     

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     

An algebraic algorithm to isolate complex polynomial zeros using Sturm sequences

In this paper, algorithms to enumerate and isolate complex polynomial roots are developed, analyzed, and implemented. We modified an algorithm due to Wilf, in which Sturm sequences and the principle of argument are used, by employing algebraic methods, aiming to enumerate zeros inside a rectangle in an exact way. Several improvements are introduced, such as dealing with zeros on the boundary of the rectangle. The performance of this new algorithm is evaluated in a theoretical as well as from a practical point of view, by means of experimental tests. The robustness of the algorithm is verified by using tests with ill-conditioned polynomials. We also compare the performance of this algorithm with the results of a recent paper, using different polynomial classes.     

一个同时决定多项式零点的迭代法的有效加速技巧

§1.引言 文献[1-7]讨论高次代数方程的迭代解法,目的在于提高迭代法的收敛速度并改善迭代效率,迭代法中常用Gauss-Seidel技巧以提高收敛速度,实际上就是一个迭代法与自身的结合应用,事实上,不同的迭代法用类似于Gauss-Seidel技巧的思想进行结合,有可能在提     

On the simultaneous determination of zeros of analytic or sectionally analytic functions

It is shown how the total-step and single-step iterative methods, as well as their improvements, for the simultaneous determination of simple zeros of polynomials can be used (with one slight modification) for the determination of simple zeros of analytic functions (inside or outside a simple smooth closed contour in the complex plane) or sectionally analytic functions (outside their arcs of discontinuity). Numerical results, obtained by the single-step method, are also presented.