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 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.     

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     

Modified Kovarik Algorithm For Approximate Orthogonalization Of Arbitrary Matrices

In a previous paper the author presented an extension of an iterative approximate orthogonalization algorithm, due to Z. Kovarik, for arbitrary rectangular matrices. In this algorithm, as Kovarik already observed in his paper, at each iteration an inversion of a symmetric and positive definite matrix is made. The dimension of this matrix equals the number of rows of the initial one, thus the inverse computation can be very expensive. In the present paper we describe an algorithm in which the above matrix inversion step is replaced by an arbitrary odd degree polynomial matrix expression. We prove that this new algorithm converges to the same matrix as the original Kovarik's method. Some numerical experiments described in the last section of the paper show us that, even for small degree polynomial expressions the convergence properties of the new algorithm are comparable with those of the original one.     

A modification of Muller’s method

Abstract It is well-known that Muller’s method for the computation of the zeros of continuous functions has order ≈ 1.84 [10], and does not have the character of global convergence. Muller’s method is based on the interpolating polynomial built on the last three points of the iterative sequence. In this paper the authors take as nodes of the interpolating polynomial the last two points of the sequence and the middle point between them. The resulting method has order p=2 for regular functions. This method leads to a globally convergent algorithm because it uses dichotomic techniques. Many numerical examples are given to show how the proposed code improves on Muller’s method.     

On optimal parameter of Laguerre's family of zero-finding methods

ABSTRACT

A one parameter Laguerre's family of iterative methods for solving nonlinear equations is considered. This family includes the Halley, Ostrowski and Euler methods, most frequently used one-point third-order methods for finding zeros. Investigation of convergence quality of these methods and their ranking is reduced to searching optimal parameter of Laguerre's family, which is the main goal of this paper. Although methods from Laguerre's family have been extensively studied in the literature for more decades, their proper ranking was primarily discussed according to numerical experiments. Regarding that such ranking is not trustworthy even for algebraic polynomials, more reliable comparison study is presented by combining the comparison by numerical examples and the comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. This combined approach has shown that Ostrowski's method possesses the best convergence behaviour for most polynomial equations.     

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.     

Exact Computation of Polynomial Zeros Expressible by Square Roots

In this paper we give an efficient algorithm to find symbolically correct zeros of a polynomial f ∈ R[X] which can be represented by square roots. R can be any domain if a factorization algorithm over R[X] is given, including finite rings or fields, integers, rational numbers, and finite algebraic or transcendental extensions of those. Asymptotically, the algorithm needs O(T_f(d²)) operations in R, where T_f(d) are the operations for the factorization algorithm over R[X] for a polynomial of degree d. Thus, the algorithm has polynomial running time for instance for polynomials over finite fields or the rationals. We also present a quick test for deciding whether a given polynomial has zeros expressible by square roots and describe some additional methods for special cases.     

Bounds for an interval polynomial

We discuss the evaluation of the range of values of an interval polynomial over an interval. Several algorithms are proposed and tested on numerical examples. The algorithms are based on ideas by Cargo and Shiska [2] and Rivlin [4]. The one basic algorithm uses Bernstein polynomials. It is shown to converge to the exact bounds and it has furthermore the property that if the maximum respectively the minimum of the polynomials occurs at an endpoint of the interval then the bound is exact. This is a useful property in routines for polynomials zeros. The other basic method is based on the meanvalue theorem and it has the advantage that the degree of approximation required for a certain apriori tolerance is smaller than the degree required in the Bernstein polynomial case. The mean value method is shown to be at least quadratically convergent and the Bernstein polynomial method is shown to be at least linearly convergent.     

An Adaptive Weighted Bisection Method For Finding Roots Of Non-Linear Equations

In this paper, an algorithm for finding the roots of non-linear equations is developed by introducing a weight in the formula of the Bisection Method (BM). Initially, we use a fixed weight to solve an equation in the least possible number of iterations. In a second stage, we develop a method termed as the Adaptive Weighted Bisection Method (AWBM) in order to update the weight at each iteration. The adaptation is achieved by minimizing the function values of the iterates with respect to the weight. Our numerical experiments show that, the AWBM, based on minimizing a function value with respect to the weight, achieves quadratic convergence. However, the method differs from the classic second order Newton-Raphson by guaranteeing convergence through bracketing.     

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.     

Unified convergence analysis for Picard iteration in <Emphasis Type="Italic">n</Emphasis>-dimensional vector spaces

In this paper, we provide three types of general convergence theorems for Picard iteration in n-dimensional vector spaces over a valued field. These theorems can be used as tools to study the convergence of some particular Picard-type iterative methods. As an application, we present a new semilocal convergence theorem for the one-dimensional Newton method for approximating all the zeros of a polynomial simultaneously. This result improves in several directions the previous one given by Batra (BIT Numer Math 42:467–476, 2002).     

Safe bounds for the solutions of nonlinear problems using a parallel multisplitting method

For some systems of nonlinear equationsF(x)=0 we derive an algorithm which iteratively constructs tight lower and upper bounds for the zeros ofF. The algorithm is based on a multisplitting of certain matrices thus showing a natural parallelism. We prove criteria for the convergence of the bounds towards the zeros and we investigate the speed of convergence.     

多项式方程区间内重根的快速判定和裁剪

目的多项式求实根问题有着广泛的应用。改进传统的裁剪方法,在多项式重根的情形下,保持计算稳定性的同时显著地提高相应的收敛阶。方法提出了基于R~3空间内的3次裁剪方法。该方法继承了传统裁剪求根方法的优点,充分利用了Bernstein基函数较好的计算稳定性,同时给出简单方法判别重根的存在性,从而使得重根的情形可以转化为单根的情形。结果与已有的基于R~1和R~2空间的3次裁剪方法相比,本文方法可以具有更好的逼近效果。单根情形下,本文方法与基于R~2空间的3次裁剪方法同时具有5次收敛阶,略高于基于R~1空间3次裁剪方法的4次收敛阶;m(≥2)重根情形下,本文方法理论上可具有5次收敛阶,明显优于已有的基于R~1和R~2空间的3次裁剪方法的4/m或5/m收敛阶。基于R~1,R~2和R~3空间的3次裁剪方法的计算时间复杂度大致相当,均为O(n~2)。结论本文方法可以快速判定重根的情形,同时具有更高的收敛阶和更好的逼近效果。     

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.     

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.     

关于同时求解多项式所有零点的改进的Newton法

讨论了同时求解n次多项式所有零点的牛顿法及其改进；给出了保证它们收敛的初值应满足的一个充分条件,并证明了收敛性．数值实例的计算结果是满意的．     

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 Smale's 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 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 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.