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].     

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.     

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.     

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.     

Forcing sequences and inexact Newton iterates in Banach space

We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately and in some unspecified manner. In the elegant paper [1], natural assumptions under which the forcing sequence is uniformly less than one were given based on the first Fréchet-derivative of the operator involved. Here, we use assumption on the second Fréchet-derivative. This way, we essentially reproduce all results found earlier. However, our upper error bounds on the distances involved are smaller.     

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.     

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.     

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)     

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.     

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.     

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.     

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.     

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 local convergence of inexact Newton-type methods under residual control-type conditions

A local convergence analysis of inexact Newton-type methods using a new type of residual control was recently presented by C. Li and W. Shen. Here, we introduce the center-Hölder condition on the operator involved, and use it in combination with the Hölder condition to provide a new local convergence analysis with the following advantages: larger radius of convergence, and tighter error bounds on the distances involved. These results are obtained under the same hypotheses and computational cost. Numerical examples further validating the theoretical results are also provided in this study.     

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].     

Local convergence theorems of Newton's method for nonlinear equations using outer or generalized inverses

We provide local 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 convergence balls differ from earlier ones. In fact we show that with a simple numerical example that our convergence ball contains earlier ones. This way we have a wider choice of initial guesses than before. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations.     

The inexact, inexact perturbed, and quasi-Newton methods are equivalent models

A classical model of Newton iterations which takes into account some error terms is given by the quasi-Newton method, which assumes perturbed Jacobians at each step. Its high convergence orders were characterized by Dennis and Moré [Math. Comp. 28 (1974), 549-560]. The inexact Newton method constitutes another such model, since it assumes that at each step the linear systems are only approximately solved; the high convergence orders of these iterations were characterized by Dembo, Eisenstat and Steihaug [SIAM J. Numer. Anal. 19 (1982), 400-408]. We have recently considered the inexact perturbed Newton method [J. Optim. Theory Appl. 108 (2001), 543-570] which assumes that at each step the linear systems are perturbed and then they are only approximately solved; we have characterized the high convergence orders of these iterates in terms of the perturbations and residuals.

In the present paper we show that these three models are in fact equivalent, in the sense that each one may be used to characterize the high convergence orders of the other two. We also study the relationship in the case of linear convergence and we deduce a new convergence result.

     

Automatic continuity of n-homomorphisms between FrÉchet algebras

We impose a condition on a commutative regular Fréchet algebra (A, (p_m )) to ensure that A/kerp_m is a Fréchet Q-algebra. This implies that if θ is an n-homomorphism on certain Fréchet algebras (A, (p_m )) into semisimple commutative Fréchet algebras (B,(q_m)) such that θ(kerp_m) ? kerq_m, for large enough m, then θ is continuous. We also show that if A is a Fréchet Q-algebra, B is a semisimple Fréchet algebra, θ: A → B is a dense range n-homomorphism such that θ(A) is factorizable, and the spectral radius v_B is continuous on the separating space (θ), then θ is automatically continuous.