Local Convergence Analysis Research Articles

Local convergence of a one parameter fourth-order Jarratt-type metho¨d in Banach spaces

We present a local convergence analysis of a one parameter Jarratt-type method. We use this method to approximate a solution of an equation in a Banach space setting. The semilocal convergence of this method was recently carried out in earlier studies under stronger hypotheses. Numerical examples are given where earlier results such as in [Ezquerro J.A., Hernandez M.A., {New iterations of $R$-order four with reduced computational cost}, BIT Numer. Math. {49} (2009), 325--342] cannot be used to solve equations but our results can be applied.

Synchronization and local convergence analysis of networks with dynamic diffusive coupling.

In this paper, we address the problem of achieving synchronization in networks of nonlinear units coupled by dynamic diffusive terms. We present two types of couplings consisting of a static linear term, corresponding to the diffusive coupling, and a dynamic term which can be either the integral or the derivative of the sum of the mismatches between the states of neighbouring agents. The resulting dynamic coupling strategy is a distributed proportional-integral (PI) or a proportional-derivative (PD) law that is shown to be effective in improving the network synchronization performance, for example, when the dynamics at nodes are nonidentical. We assess the stability of the network by extending the classical Master Stability Function approach to the case where the links are dynamic ones of PI/PD type. We validate our approach via a set of representative examples including networks of chaotic Lorenz and networks of nonlinear mechanical systems.

Convergence of Newton, Halley and Chebyshev iterative methods as methods for simultaneous determination of multiple polynomial zeros

In this paper, we provide a local convergence analysis of Newton, Halley and Chebyshev iterative methods considered as methods for simultaneous determination of all multiple zeros of a polynomial f over an arbitrary normed field K. Convergence theorems with a priori and a posteriori error estimates for each of the proposed methods are established. The obtained results for Newton and Chebyshev methods are new even in the case of simple zeros. Three numerical examples are given to compare the convergence properties of the considered methods and to confirm the theoretical results.

Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative

In this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Further, we also study the dynamic behaviour of that scheme. In an earlier study, Sharma and Arora (2015) did not discuss these properties. Furthermore, the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only the first order derivatives appear in the proposed scheme. To overcome this problem, we present the hypotheses for the proposed scheme maximum up to first order derivative. In this way, we not only expand the applicability of the methods but also suggest convergence domain. Finally, a variety of concrete numerical examples are proposed where earlier studies can not be applied to obtain the solutions of nonlinear equations on the other hand our study does not exhibit this type of problem/restriction.

An inexact proximal regularization method for unconstrained optimization

We present a regularization algorithm to solve a smooth unconstrained minimization problem.This algorithm is suitable to solve a degenerate problem, when the Hessian is singular at a local optimal solution. The main feature of our algorithm is that it uses an outer/inner iteration scheme. We show that the algorithm has a strong global convergence property under mild assumptions. A local convergence analysis shows that the algorithm is superlinearly convergent under a local error bound condition. Some numerical experiments are reported.

Convergence of Steffensen’s method for non-differentiable operators

We present a semilocal as well as a local convergence analysis of Steffensen's method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The novelty of this paper is twofold. On the one hand, we show convergence under general conditions including earlier ones as special cases. In particular, these conditions allow the important study when the operator involved is not differentiable. On the other hand, we use a combination of conditions that allow a more precise computation of the upper bounds on the norms of the inverses involved leading to a tighter convergence analysis. Finally, we provide numerical examples involving nonlinear integral equations on mixed Hammerstein type that appear in chemistry, vehicular traffic theory, biology, and queuing theory.

A stabilized SQP method: superlinear convergence

Stabilized sequential quadratic programming (sSQP) methods for nonlinear optimization generate a sequence of iterates with fast local convergence regardless of whether or not the active-constraint gradients are linearly dependent. This paper concerns the local convergence analysis of an sSQP method that uses a line search with a primal-dual augmented Lagrangian merit function to enforce global convergence. The method is provably well-defined and is based on solving a strictly convex quadratic programming subproblem at each iteration. It is shown that the method has superlinear local convergence under assumptions that are no stronger than those required by conventional stabilized SQP methods. The fast local convergence is obtained by allowing a small relaxation of the optimality conditions for the quadratic programming subproblem in the neighborhood of a solution. In the limit, the line search selects the unit step length, which implies that the method does not suffer from the Maratos effect. The analysis indicates that the method has the same strong first- and second-order global convergence properties that have been established for augmented Lagrangian methods, yet is able to transition seamlessly to sSQP with fast local convergence in the neighborhood of a solution. Numerical results on some degenerate problems are reported.

A Newton conditional gradient method for constrained nonlinear systems

In this paper, we consider the problem of solving constrained systems of nonlinear equations. We propose an algorithm based on a combination of Newton and conditional gradient methods, and establish its local convergence analysis. Our analysis is set up by using a majorant condition technique, allowing us to prove, in a unified way, convergence results for two large families of nonlinear functions. The first one includes functions whose derivative satisfies a Hölder-like condition, and the second one consists of a substantial subclass of analytic functions. Some preliminary numerical experiments are reported.

Local convergence of a parameter based iteration with Hölder continuous derivative in Banach spaces

The local convergence analysis of a parameter based iteration with Holder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives. An existence-uniqueness theorem along with the derivation of error bounds for the solution is established. Different numerical examples including nonlinear Hammerstein equation are solved. The radii of balls of convergence for them are obtained. Substantial improvements of these radii are found in comparison to some other existing methods under similar conditions for all examples considered.

On the Local Convergence of an Eighth-order Method for Solving Nonlinear Equations

Abstract We present a local convergence analysis of an eighth-order method for approximating a locally unique solution of a non-linear equation. Earlier studies such as have shown convergence of these methods under hypotheses up to the seventh derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

Abstract We present a local convergence analysis of inexact Gauss-Newton-like method (IGNLM) for solving nonlinear least-squares problems in a Euclidean space setting. The convergence analysis is based on our new idea of restricted convergence domains. Using this idea, we obtain a more precise information on the location of the iterates than in earlier studies leading to smaller majorizing functions. This way, our approach has the following advantages and under the same computational cost as in earlier studies: A large radius of convergence and more precise estimates on the distances involved to obtain a desired error tolerance. That is, we have a larger choice of initial points and fewer iterations are also needed to achieve the error tolerance. Special cases and numerical examples are also presented to show these advantages.

Ball Convergence for Two Optimal Eighth-Order Methods Using Only the First Derivative

We present a local convergence analysis of two optimal eighth-order methods in order to approximate a locally unique solution of a nonlinear equation defined on the real line or the complex plane. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative although only the first derivative appears in these methods. We only show convergence using hypotheses on the first derivative. We also provide computable: error bounds, radii of convergence as well as uniqueness of the solution with results based on Lipschitz constants not given in earlier studies. The computational order of convergence is also used to determine the order of convergence. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

Analysis of Stability, Local Convergence, and Transformation Sensitivity of a Variant of the Particle Swarm Optimization Algorithm

In this paper, we investigate three important properties (stability, local convergence, and transformation invariance) of a variant of particle swarm optimization (PSO) called standard PSO 2011 (SPSO2011). Through some experiments, we identify boundaries of coefficients for this algorithm that ensure particles converge to their equilibrium. Our experiments show that these convergence boundaries for this algorithm are: 1) dependent on the number of dimensions of the problem; 2) different from that of some other PSO variants; and 3) not affected by the stagnation assumption. We also determine boundaries for coefficients associated with different behaviors, e.g., nonoscillatory and zigzagging, of particles before convergence through analysis of particle positions in the frequency domain. In addition, we investigate the local convergence property of this algorithm and we prove that it is not locally convergent. We provide a sufficient condition and related proofs for local convergence for a formulation that represents updating rules of a large class of PSO variants. We modify the SPSO2011 in such a way that it satisfies that sufficient condition; hence, the modified algorithm is locally convergent. Also, we prove that the original standard PSO algorithm is not sensitive to rotation, scaling, and translation of the search space.

Local Convergence and the Dynamics of a Two-Step Newton-Like Method

We present the local convergence analysis and the study of the dynamics of a two-step Newton-like method in order to approximate a locally unique solution of multiplicity one of a nonlinear equation.

Ball convergence of some fourth and sixth-order iterative methods

We present a local convergence analysis for some families of fourth and sixth-order methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Earlier studies [V. Candela and A. Marquina, Recurrence relations for rational cubic methods II: The Chebyshev method, Computing 45 (1990) 355–367; C. Chun, P. Stanica and B. Neta, Third order family of methods in Banach spaces, Comput. Math. Appl. 61 (2011) 1665–1675; J. M. Gutiérrez and M. A. Hernández, Recurrence relations for the super-Halley method, Comput. Math. Appl. 36 (1998) 1–8; M. A. Hernández and M. A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math. 126 (2000) 131–143; M. A. Hernández, Chebyshev’s approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–455; M. A. Hernández, Second-derivative-free variant of the Chebyshev method for nonlinear equations, J. Optim. Theory Appl. 104(3) (2000) 501–515; J. L. Hueso, E. Martinez and C. Teruel, Convergence, efficiency and dynamics of new fourth and sixth-order families of iterative methods for nonlinear systems, J. Comput. Appl. Math. 275 (2015) 412–420; Á. A. Magre nán, Estudio de la dinámica del método de Newton amortiguado, Ph.D. thesis, Servicio de Publicaciones, Universidad de La Rioja (2013), http://dialnet.unirioja.es/servlet/tesis?codigo=38821 ; J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970); M. S. Petkovic, B. Neta, L. Petkovic and J. Džunič, Multi-Point Methods for Solving Nonlinear Equations (Elsevier, 2013); J. F. Traub, Iterative Methods for the Solution of Equations, Automatic Computation (Prentice-Hall, Englewood Cliffs, NJ, 1964); X. Wang and J. Kou, Semilocal convergence and [Formula: see text]-order for modified Chebyshev–Halley methods, Numer. Algorithms 64(1) (2013) 105–126] have used hypotheses on the fourth Fréchet derivative of the operator involved. We use hypotheses only on the first Fréchet derivative in our local convergence analysis. This way, the applicability of these methods is extended. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples illustrating the theoretical results are also presented in this study.

Repeated anticipatory network traffic control using iterative optimization accounting for model bias correction

Anticipatory signal control in traffic networks adapts the signal timings with the aim of controlling the resulting (equilibrium) flows and route choice patterns in the network. This study investigates a method to support control decisions for successful applications in real traffic systems that operate repeatedly, for instance from day to day, month to month, etc. The route choice response to signal control is usually predicted through models; however this leads to suboptimality because of unavoidable prediction errors between model and reality. This paper proposes an iterative optimizing control method to drive the traffic network towards the real optimal performance by observing modeling errors and correcting for them. Theoretical analysis of this Iterative Optimizing Control with Model Bias Correction (IOCMBC) on matching properties between the modeled optimal solution and the real optimum is presented, and the advantages over conventional iterative schemes are demonstrated. A local convergence analysis is also elaborated to investigate conditions required for a convergent scheme. The main innovation is the calculation of the sensitivity (Jacobian) information of the real route choice behavior with respect to signal control variables. To avoid performing additional perturbations, we introduce a measurement-based implementation method for estimating the operational Jacobian that is associated with the reality. Numerical tests confirm the effectiveness of the proposed IOCMBC method in tackling modeling errors, as well as the influence of the optimization step size on the reality-tracking convergence.

Ball convergence of a novel Newton-Traub composition for solving equations

We present a local convergence analysis of a Newton-Traub composition method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The method was shown to be of convergence order five, if defined on the m-dimensional Euclidean space using Taylor’s expansion and hypotheses reaching up to the fifth derivative (Hueso, Martinez, & Tervel, 2015; Sharma, 2014). We expand the applicability of this method using contractive techniques and hypotheses only on the first Frechet-derivative of the operator involved. Moreover, we provide computable radius of convergence and error estimates on the distances involved not given in the earlier studies (Hueso et al., 2015; Sharma, 2014). Numerical examples illustrating the theoretical results are also presented in this study.

A new highly efficient and optimal family of eighth-order methods for solving nonlinear equations

The principle aim of this manuscript is to present a new highly efficient and optimal eighth-order family of iterative methods to solve nonlinear equations in the case of simple roots. The derivation of this scheme is based on weight function and rational approximation approaches. The proposed family requires only four functional evaluations (viz. f(xn) f′(xn) f(yn) and f(zn)) per iteration. Therefore, the proposed family is optimal in the sense of Kung–Traub hypotheses. In addition, we given a theorem which describing the order of convergence of the proposed family. Moreover, we present a local convergence analysis using hypotheses only on the first-order derivative, since in our preceding theorem we used hypotheses on higher-order derivatives that do not appear in these methods. In this way, we expand the applicability of these methods even further. Furthermore, a variety of nonlinear equations is considered for the numerical experiments. It is observed from the numerical experiments that our proposed methods perform better than the existing optimal methods of same order, when the accuracy is checked in multi precision digits.

Local convergence for multipoint methods using only the first derivative

We present a local convergence analysis of two methods in order to approximate a locally unique solution of an equation in a Banach space setting. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative (or even higher order derivatives) although only the first derivative appears in these methods. We only show convergence using hypotheses on the first derivative. We also provide computable: error bounds, radii of convergence as well as uniqueness results for the solution based on Lipschitz constants not given in earlier studies. The computational order of convergence is also used to determine the order of convergence. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

Local convergence of deformed Jarratt-type methods in Banach space without inverses

We present a local convergence analysis for the Jarratt-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative. Hence, the applicability of these methods is expanded under weaker hypotheses and less computational cost for the constants involved in the convergence analysis. Numerical examples are also provided in this study.