Aurél Galántai

The local convergence of ABS methods for nonlinear algebraic equations

SummaryIn this paper we consider an extension to nonlinear algebraic systems of the class of algorithms recently proposed by Abaffy, Broyden and Spedicato for general linear systems. We analyze the convergence properties, showing that under the usual assumptions on the function and some mild assumptions on the free parameters available in the class, the algorithm is locally convergent and has a superlinear rate of convergence (per major iteration, which is computationally comparable to a single Newtons step). Some particular algorithms satisfying the conditions on the free parameters are considered.

Perturbation bounds for polynomials

Using two different elementary approaches we derive a global and a local perturbation theorem on polynomial zeros that significantly improve the results of Ostrowski (Acta Math 72:99–257, 1940), Elsner et al. (Linear Algebra Appl 142:195–209, 1990). A comparison of different perturbation bounds shows that our results are better in many cases than the similar local result of Beauzamy (Can Math Bull 42(1):3–12, 1999). Using the matrix theoretical approach we also improve the backward stability result of Edelman and Murakami (Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, SIAM, Philapdelphia, 1994; Math Comput 64:210–763, 1995).

Quasi-Newton ABS methods for solving nonlinear algebraic systems of equations

This paper is concerned with the solution of nonlinear algebraic systems of equations. For this problem, we suggest new methods, which are combinations of the nonlinear ABS methods and quasi-Newton methods. Extensive numerical experiments compare particular algorithms and show the efficiency of the proposed methods.

The global convergence of the ABS methods for a class of nonlinear problems

We prove the monotone convergence of a wide subclass of the nonlinear ABS methods. The convergence conditions are essentially those of the Newton-Baluev theorem [10,11]. Two members of the ABS class are shown to be at least as fast as Newtons method in the partial ordering.

The comparison of numerical methods for solving polynomial equations

In this paper we compare the Turainprocess [51-(61 with the Lehmer-Schur method [21. We prove that the latter is better. 1. The Algorithms. We first describe the Turan process [5] -[6] which can be considered as an improvement of Graeffes method. For the complex polynomial n (1.1) p0 ) a00o(z) --? ajozi = o (a E C, a =a0 0), j=0 the method can be formulated as follows. Let (1.2) Pi(Z) p11(/)p1E) akiz1 (j = 1, 2, . . . ) k=O be the jth Graeffe transformation and let [ k p0/lk (1.3) M[po(z), i] =l max 1 Ll ?k n n J where go = 2-m0, uoo = 0, k-1I (1.4) OJk= kakmo Ejajmok-i] /aomo (k = 1,*.., n) and mo > 1 is fixed. Let the constants o 1mo, I be defined by the inequalities 2.5 + a -1 0.5 7T arcc 2+2cs ma mI>2. Then with the notations (1.5) M(?)-M[p0(z), 0], s(0) -0, the dth step of the algorithm is the following: 1. Algorithm (T). (i) Let S(d+l ) = S(d) + 0.5(1 + a )exp(j Kwhere j-= 0, 1 . ., I and i T Received January 14, 1976; revised August 18, 1976. AMS (MOS) subject classifications (1970). Primary 65HOS. Copyright C) 1978, American Mathematical Society 391 This content downloaded from 157.55.39.147 on Wed, 18 May 2016 05:03:53 UTC All use subject to http://about.jstor.org/terms

Analysis of error propagation in the ABS class for linear systems

Broydens backward error analysis technique is applied to evaluate the numerical stability of the ABS class of methods for solving linear systems.

Projection Methods for Nonlinear Algebraic Equations

We investigate two types of projection methods for solving nonlinear algebraic equations of the form

Finite Projection Methods for Linear Systems

F\left( x \right) = 0\;\left( {F:{\mathbb{R}^m} \to {\mathbb{R}^m}} \right),