Semilocal convergence and R-order for modified Chebyshev-Halley methods

In this paper, we study the semilocal convergence and R-order for a class of modified Chebyshev-Halley methods for solving non-linear equations in Banach spaces. To solve the problem that the third-order derivative of an operator is neither Lipschitz continuous nor Hölder continuous, the condition of Lipschitz continuity of third-order Fréchet derivative considered in Wang et al. (Numer Algor 56:497–516, 2011) is replaced by its general continuity condition, and the latter is weaker than the former. Furthermore, the R-order of these methods is also improved under the same condition. By using the recurrence relations, a convergence theorem is proved to show the existence-uniqueness of the solution and give a priori error bounds. We also analyze the R-order of these methods with the third-order Fréchet derivative of an operator under different continuity conditions. Especially, when the third-order Fréchet derivative is Lipschitz continuous, the R-order of the methods is at least six, which is higher than the one of the method considered in Wang et al. (Numer Algor 56:497–516, 2011) under the same condition.     

Semilocal convergence for the Super-Halley’s method

The semilocal convergence of Super-Halley’s method for solving nonlinear equations in Banach spaces is established under the assumption that the second Frëchet derivative satisfies the ω-continuity condition. This condition is milder than the well-known Lipschitz and Hölder continuity conditions. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where the Lipschitz and the Hölder continuity conditions fail. The difficult computation of second Frëchet derivative is also avoided by replacing it with the divided difference containing only the first Frëchet derivatives. A number of recurrence relations based on two parameters are derived. A convergence theorem is established to estimate a priori error bounds along with the domains of existence and uniqueness of the solutions. The R-order convergence of the method is shown to be at least three. Two numerical examples are worked out to demonstrate the efficacy of our method. It is observed that in both examples the existence and uniqueness regions of solution are improved when compared with those obtained in [7].     

Convergence of a third order method for fixed points in Banach spaces

In this paper, the semilocal convergence of a third order Stirling-like method used to find fixed points of nonlinear operator equations in Banach spaces is established under the assumption that the first Fréchet derivative of the involved operator satisfies ??-continuity condition. It turns out that this convergence condition is weaker than the Lipschitz and the H?lder continuity conditions on first Fréchet derivative of the involved operator. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where Lipschitz and H?lder continuity conditions on first Fréchet derivative fail. It also avoids the evaluation of second order Fréchet derivative which is difficult to compute at times. A priori error bounds along with the domains of existence and uniqueness of a fixed point are derived. The R-order of the method is shown to be equal to (2p?+?1) for p????(0,1]. Finally, two numerical examples involving nonlinear integral equations are worked out to show the efficacy of our approach.     

Semilocal convergence analysis for the modified Newton-HSS method under the Hölder condition

The present paper is concerned with theoretical properties of the modified Newton-HSS method for large sparse non-Hermitian positive definite systems of nonlinear equations. Assuming that the nonlinear operator satisfies the Hölder continuity condition, a new semilocal convergence theorem for the modified Newton-HSS method is established. The Hölder continuity condition is milder than the usual Lipschitz condition. The semilocal convergence theorem is established by using the majorizing principle, which is based on the concept of majorizing sequence given by Kantorovich. Two real valued functions and two real sequences are used to establish the convergence criterion. Furthermore, a numerical example is given to show application of our theorem.     

Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces

The aim of this paper is to establish the semilocal convergence of a multipoint third order Newton-like method for solving F(x)=0 in Banach spaces by using recurrence relations. The convergence of this method is studied under the assumption that the second Fréchet derivative of F satisfies Hölder continuity condition. This continuity condition is milder than the usual Lipschitz continuity condition. A new family of recurrence relations are defined based on the two new constants which depend on the operator F. These recurrence relations give a priori error bounds for the method. Two numerical examples are worked out to demonstrate the applicability of the method in cases where the Lipschitz continuity condition over second derivative of F fails but Hölder continuity condition holds.     

A Stirling-like method with Hölder continuous first derivative in Banach spaces

In this paper, the convergence of a Stirling-like method used for finding a solution for a nonlinear operator in a Banach space is examined under the relaxed assumption that the first Fréchet derivative of the involved operator satisfies the Hölder continuity condition. Many results exist already in the literature to cover the stronger case when the second Fréchet derivative of the involved operator satisfies the Lipschitz/Hölder continuity condition. Our convergence analysis is done by using recurrence relations. The error bounds and the existence and uniqueness regions for the solution are obtained. Finally, two numerical examples are worked out to show that our convergence analysis leads to better error bounds and existence and uniqueness regions for the fixed points.     

Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems

In this paper, we analyze the semilocal convergence of k-steps Newton’s method with frozen first derivative in Banach spaces. The method reaches order of convergence k + 1. By imposing only the assumption that the Fréchet derivative satisfies the Lipschitz continuity, we define appropriate recurrence relations for obtaining the domains of convergence and uniqueness. We also define the accessibility regions for this iterative process in order to guarantee the semilocal convergence and perform a complete study of their efficiency. Our final aim is to apply these theoretical results to solve a special kind of conservative systems.     

On Newton's method

In this note we discuss Newton's method in a setting somewhat more restrictive than customary. In this setting, however, we claim to have proved superlinear convergence of the Newton process without assuming twice differentiability or Lipschitz continuity of the first derivative of the operator. A further feature is that the iteration to be discussed is not initially but is eventually the Newton process. With this feature global rather than local convergence is achieved.This research was supported by the Boeing Scientific Research Laboratories and by Grant AF-AFOSR 937-65.     

Nonsmooth Steepest Descent Method by Proximal Subdifferentials in Hilbert Spaces

In this paper, we first study a nonsmooth steepest descent method for nonsmooth functions defined on a Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox-regularity and Lipschitz continuity. As an application, the established algorithm is used to search for the minimizer of a lower semicontinuous and convex function on a finite-dimensional space. A convergence theorem, as an extension and improvement of the existing converging result for twice continuously differentiable convex functions, is also presented therein.     

Weak sufficient convergence conditions and applications for newton methods

The famous Newton—Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton—Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton— Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.     

Convergence analysis for a family of improved super-Halley methods under general convergence condition

In this paper, we focus on the semilocal convergence for a family of improved super-Halley methods for solving non-linear equations in Banach spaces. Different from the results in Wang et al. (J Optim Theory Appl 153:779–793, 2012), the condition of Hölder continuity of third-order Fréchet derivative is replaced by its general continuity condition, and the latter is weaker than former. Moreover, the R-order of the methods is also improved. By using the recurrence relations, we prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is analyzed with the third-order Fréchet derivative of the operator satisfies general continuity condition and Hölder continuity condition.     

Convergence of a continuation method under Lipschitz continuous derivative in Banach spaces

The aim of this paper is to use recurrence relations instead of majorizing sequences to establish the semilocal convergence of a continuation method combining Chebyshev method and Convex acceleration of Newton??s method for solving nonlinear equations in Banach spaces under the assumption that the first Fréchet derivative satisfies the Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter ????[0,1]. Two numerical examples are worked out to demonstrate the efficacy of our convergence analysis. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences, it is found that our analysis gives better results in both the examples. Further, we observed that for particular values of ??, our analysis reduces to those for Chebyshev method (??=0) and Convex acceleration of Newton??s method (??=1) respectively with improved results.     

A product integration method for a class of singular first kind Volterra equations

Summary Convergence of a midpoint product integration method for singular first kind Volterra equations with kernels of the formk(t, s)(t–s) ^– , 0<<1, wherek(t, s) is continuous, is examined. It is shown that convergence of order one holds if the solution of the Volterra equation has a Lipschitz continuous first derivative andk(t, s) is suitably smooth. In addition, convergence is shown to hold when the solution has only Lipschitz continuity and the same conditions onk(t, s) apply. An existence theorem of Kowalewski is used to relate these conditions on the solution to conditions on the data andk(t, s).     

Convex composite non–Lipschitz programming

In this paper necessary, and sufficient optimality conditions are established without Lipschitz continuity for convex composite continuous optimization model problems subject to inequality constraints. Necessary conditions for the special case of the optimization model involving max-min constraints, which frequently arise in many engineering applications, are also given. Optimality conditions in the presence of Lipschitz continuity are routinely obtained using chain rule formulas of the Clarke generalized Jacobian which is a bounded set of matrices. However, the lack of derivative of a continuous map in the absence of Lipschitz continuity is often replaced by a locally unbounded generalized Jacobian map for which the standard form of the chain rule formulas fails to hold. In this paper we overcome this situation by constructing approximate Jacobians for the convex composite function involved in the model problem using ε-perturbations of the subdifferential of the convex function and the flexible generalized calculus of unbounded approximate Jacobians. Examples are discussed to illustrate the nature of the optimality conditions. Received: February 2001 / Accepted: September 2001?Published online February 14, 2002     

Convergence of a continuation method under the Hölder condition of the first derivative

The aim of this paper is to describe a continuation method combining the Chebyshev’s method and the convex acceleration of Newton’s method to solve nonlinear equations in Banach spaces. The semilocal convergence analysis of the method is established using recurrence relations under the assumption that the first Fréchet derivative satisfies the Hölder continuity condition. This condition is milder than the usual Lipschitz condition. The computation of second Fréchet derivative is also avoided. Two real valued functions and a real sequence are used to establish a convergence criterion of R-order (1+p), where p∈(0,1] is the order of the Hölder condition. An existence and uniqueness theorem along with the closed form of error bounds is derived in terms of a real parameter α∈[0,1]. Two numerical examples are worked out to demonstrate the efficacy of our convergence analysis. For both the examples, the convergence conditions hold for the Chebyshev’s method (α=0). However, for the convex acceleration of Newton’s method (α=1), these convergence conditions hold for the first example but fail for the second example. For particular values of α, our method reduces to the Chebyshev’s method (α=0) and the convex acceleration of Newton’s method (α=1).     

Supercalm Multifunctions For Convergence Analysis

Calmness of multifunctions is a well-studied concept of generalized continuity in which single-valued selections from the image sets of the multifunction exhibit a restricted type of local Lipschitz continuity where the base point is fixed as one point of comparison. Generalized continuity properties of multifunctions like calmness can be applied to convergence analysis when the multifunction appropriately represents the iterates generated by some algorithm. Since it involves an essentially linear relationship between input and output, calmness gives essentially linear convergence results when it is applied directly to convergence analysis. We introduce a new continuity concept called ‘supercalmness’ where arbitrarily small calmness constants can be obtained near the base point, which leads to essentially superlinear convergence results. We also explore partial supercalmness and use a well-known generalized derivative to characterize both when a multifunction is supercalm and when it is partially supercalm. To illustrate the value of such characterizations, we explore in detail a new example of a general primal sequential quadratic programming method for nonlinear programming and obtain verifiable conditions to ensure convergence at a superlinear rate.     

How to Solve Nonlinear Equations When a Third Order Method is Not Applicable

In this paper, we use a one-parametric family of second-order iterations to solve a nonlinear operator equation in a Banach space. A Kantorovich-type convergence theorem is proved, so that the first Fréchet derivative of the operator satisfies a Lipschitz condition. We also give an explicit error bound.Supported in part by the University of La Rioja (grants: API-98/A25 and API-98/B12)and DGES (grant: PB96-0120-C03-02).     

Local convergence analysis of inexact Newton-like methods under majorant condition

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.     

Complexity analysis of an interior-point algorithm for linear optimization based on a new proximity function

In this paper, we focus on a family of modified Chebyshev methods and study the semilocal convergence for these methods. Different from the results in reference (Hernández and Salanova, J. Comput. Appl. Math. 126:131–143, 2000), the Hölder continuity of the second derivative is replaced by its generalized continuity condition, and the latter is weaker than the former. Using the recurrence relations, we establish the semilocal convergence of these methods and prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is also analyzed. Especially, when the second derivative of the operator is Hölder continuous, the R-order of these methods is at least 3 + 2p, which is higher than the one of Chebyshev method considered in reference (Hernández and Salanova, J. Comput. Appl. Math. 126:131–143, 2000) under the same condition. Finally, we give some numerical results to show our approach.     

New Korovkin Type Theorem for Non-Tensor Meyer–König and Zeller Operators

In this paper, we introduce a certain class of linear positive operators via a generating function, which includes the non-tensor MKZ operators and their non-trivial extension. In investigating the approximation properties, we prove a new Korovkin type approximation theorem by using appropriate test functions. We compute the rate of convergence of these operators by means of the modulus of continuity and the elements of modified Lipschitz class functions. Furthermore, we give functional partial differential equations for this class. Using the corresponding equations, we calculate the first few moments of the non-tensor MKZ operators and investigate their approximation properties. Finally, we state the multivariate versions of the results and obtain the convergence properties of the multivariate Meyer–König and Zeller operators.