Enlarging the domain of starting points for Newton’s method under center conditions on the first Fréchet-derivative

We analyze the semilocal convergence of Newton’s method under center conditions on the first Fréchet-derivative of the operator involved. We see that we can extend the known results so far, since we provide different starting points from the point where the first Fréchet-derivative is centered (that is the situation usually considered by other authors), so that the domain of starting points is enlarged for Newton’s method. We also illustrate the theoretical results obtained with some mildly nonlinear elliptic equations.     

The effect of rounding errors on Newton methods

In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newtonlike methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitztype hypotheses on the second Fréchet-derivative instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.     

Affine invariant local convergence theorems for inexact Newton-like methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on the second. Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the second Fréchet-derivative our radius of convergence is larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

Local convergence theorems for Newton methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on themth (m ≥ 2 an integer). Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the mth Fréchet-derivative our radius of convergence can sometimes be larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

Chebysheff-Halley-like methods in Banach spaces

Chebysheff-Halley methods are probably the best known cubically convergent iterative procedures for solving nonlinear equations. These methods however require an evaluation of the second Fréchet-derivative at each step which means a number of function evaluations proportional to the cube of the dimension of the space. To reduce the computational cost we replace the second Fréchet derivative with a fixed bounded bilinear operator. Using the majorant method and Newton-Kantorovich type hypotheses we provide sufficient conditions for the convergence of our method to a locally unique solution of a nonlinear equation in Banach space. Our method is shown to be faster than Newton’s method under the same computational cost. Finally we apply our results to solve nonlinear integral equations appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field.     

The secant method and fixed points of nonlinear operators

Our results answer the following question: given that a mapping has a fixed point, when is it true that the Secant iterates produce a sequence of nearby points which converge to the fixed point? We assume only that the nonlinear operator has a Hölder continuous Fréchet-derivative at the fixed point.     

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.     

Newton–Kantorovich approximations under weak continuity conditions

The paper is concerned with the application of Kantorovich-type majorants for the convergence of Newton’s method to a locally unique solution of a nonlinear equation in a Banach space setting. The Fréchet-derivative of the operator involved satisfies only a rather weak continuity condition. Using our new idea of recurrent functions, we obtain sufficient convergence conditions, as well as error estimates. The results compare favorably to earlier ones (Ezquerro, Hernández in IMA J. Numer. Anal. 22:187–205, 2002 and Proinov in J. Complex. 26:3–42, 2010).     

On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.     

On the Convergence of Newton’s Method for Polynomial Equations and Applications in Radiative Transfer

 Newton’s method is used to approximate a locally unique zero of a polynomial operator F of degree in Banach space. So far, convergence conditions have been found for Newton’s method based on the Newton-Kantorovich hypothesis that uses Lipschitz-type conditions and information only on the first Fréchet-derivative of F. Here we provide a new semilocal convergence theorem for Newton’s method that uses information on all Fréchet-derivatives of F except the first. This way, we obtain sufficient convergence conditions different from the Newton-Kantorovich hypothesis. Our results are extended to include the case when F is a nonlinear operator whose kth Fréchet-derivative satisfies a H?lder continuity condition. An example is provided to show that our conditions hold where all previous ones fail. Moreover, some applications of our results to the solution of polynomial systems and differential equations are suggested. Furthermore, our results apply to solve a nonlinear integral equation appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field. Received 9 December 1997 in revised form 30 March 1998     

Semilocal convergence theorems for a certain class of iterative procedures

We provide semilocal convergence theorems for Newton-like methods in Banach space using outer and generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least square problems and ill-posed nonlinear operator equations.     

Concerning the radius of convergence of Newton’s method and applications

We present local and semilocal convergence results for Newton’s method in a Banach space setting. In particular, using Lipschitz-type assumptions on the second Fréchet-derivative we find results concerning the radius of convergence of Newton’s method. Such results are useful in the context of predictor-corrector continuation procedures. Finally, we provide numerical examples to show that our results can apply where earlier ones using Lipschitz assumption on the first Fréchet-derivative fail.     

Semilocal Convergence Theorems for Newton’s Method Using Outer Inverses and Hypotheses on the Second Fréchet-Derivative

 We provide semilocal convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations. We complete our study with some very simple examples to show that our results apply, where others fail.     

Semilocal Convergence Theorems for Newton’s Method Using Outer Inverses and Hypotheses on the Second Fréchet-Derivative

 We provide semilocal convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations. We complete our study with some very simple examples to show that our results apply, where others fail. (Received 26 April 2000; in final form 17 November 2000)     

Regularized continuous analog of the newton method for monotone equations in a Hilbert space

In a Hilbert space we construct a regularized continuous analog of the Newton method for nonlinear equation with a Fréchet differentiable and monotone operator. We obtain sufficient conditions of its strong convergence to the normal solution of the given equation under approximate assignment of the operator and the right-hand of the equation.     

An error analysis for a certain class of iterative methods

We provide local convergence results in affine form for in-exact Newton-like as well as quasi-Newton iterative methods in a Banach space setting. We use hypotheses on the second or on the first andmth Fréchet-derivative (m ≥ 2 an integer) of the operator involved. Our results allow a wider choice of starting points since our radius of convergence can be larger than the corresponding one given in earlier results using hypotheses on the first-Fréchet-derivative only. A numerical example is provided to illustrate this fact. Our results apply when the method is, for example, a difference Newton-like or update-Newton method. Furthermore, our results have direct applications to the solution of autonomous differential equations.     

Local path-following property of inexact interior methods in nonlinear programming

We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of the original system. In case of an exact solution of the Newton system, it has been shown that the sequence of iterates is asymptotically tangent to the central path (Armand and Benoist in Math. Program. 115:199?C222, 2008). The purpose of the present paper is to extend this result to an inexact solution of the Newton system. We give quite general conditions on the different parameters of the algorithm, so that this asymptotic property is satisfied. Some numerical tests are reported to illustrate our theoretical results.     

New Version of the Newton Method for Nonsmooth Equations

In this paper, an inexact Newton scheme is presented which produces a sequence of iterates in which the problem functions are differentiable. It is shown that the use of the inexact Newton scheme does not reduce the convergence rate significantly. To improve the algorithm further, we use a classical finite-difference approximation technique in this context. Locally superlinear convergence results are obtained under reasonable assumptions. To globalize the algorithm, we incorporate features designed to improve convergence from an arbitrary starting point. Convergence results are presented under the condition that the generalized Jacobian of the problem function is nonsingular. Finally, implementations are discussed and numerical results are presented.     

A Hybrid Smoothing Method for Mixed Nonlinear Complementarity Problems

In this paper, we describe a new, integral-based smoothing method for solving the mixed nonlinear complementarity problem (MNCP). This approach is based on recasting MNCP as finding the zero of a nonsmooth system and then generating iterates via two types of smooth approximations to this system. Under weak regularity conditions, we establish that the sequence of iterates converges to a solution if the limit point of this sequence is regular. In addition, we show that the rate is Q-linear, Q-superlinear, or Q-quadratic depending on the level of inexactness in the subproblem calculations and we make use of the inexact Newton theory of Dembo, Eisenstat, and Steihaug. Lastly, we demonstrate the viability of the proposed method by presenting the results of numerical tests on a variety of complementarity problems.     

A stable primal–dual approach for linear programming under nondegeneracy assumptions

This paper studies a primal–dual interior/exterior-point path-following approach for linear programming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primal–dual optimality equations. Under nondegeneracy assumptions, this nonlinear system is well-posed, i.e. it has a nonsingular Jacobian at optimality and is not necessarily ill-conditioned as the iterates approach optimality. Assuming that a basis matrix (easily factorizable and well-conditioned) can be found, we apply a simple preprocessing step to eliminate both the primal and dual feasibility equations. This results in a single bilinear equation that maintains the well-posedness property. Sparsity is maintained. We then apply either a direct solution method or an iterative solver (within an inexact Newton framework) to solve this equation. Since the linearization is well posed, we use affine scaling and do not maintain nonnegativity once we are close enough to the optimum, i.e. we apply a change to a pure Newton step technique. In addition, we correctly identify some of the primal and dual variables that converge to 0 and delete them (purify step). We test our method with random nondegenerate problems and problems from the Netlib set, and we compare it with the standard Normal Equations NEQ approach. We use a heuristic to find the basis matrix. We show that our method is efficient for large, well-conditioned problems. It is slower than NEQ on ill-conditioned problems, but it yields higher accuracy solutions.