Extending the applicability of Secant methods and nondiscrete induction

We use nondiscrete mathematical induction to extend the applicability of the Secant methods for solving equations in a Banach setting. Our approach has the following advantages over earlier works under the same information: weaker sufficient convergence conditions; tighter error bounds on the distances involved and more information on the location of the solution. Numerical examples where our results apply but earlier ones fail to solve nonlinear equation as well as tighter error bounds are also provided in this study.     

Convergence conditions for secant-type methods

We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.     

Mysovskii-type theorem for the Secant method under Hölder continuous Fréchet derivative

The Mysovskii-type condition is considered in this study for the Secant method in Banach spaces to solve a nonlinear operator equation. We suppose the inverse of divided difference of order one is bounded and the Fréchet derivative of the nonlinear operator is Hölder continuous. By use of Fibonacci generalized sequence, a semilocal convergence theorem is established which matches with the convergence order of the method. Finally, two simple examples are provided to show that our results apply, where earlier ones fail.     

On Newton’s method for solving equations containing Fréchet-differentiable operators of order at least two

We provide sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear operator equation containing operators that are Fréchet-differentiable of order at least two, in a Banach space setting. Numerical examples are also provided to show that our results apply to solve nonlinear equations in cases earlier ones cannot [J.M. Gutiérrez, A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79(1997) 131-145; Z. Huang, A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217; F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Mathematica 5 (1985) 71-84].     

The Secant method for nondifferentiable operators

In this paper, we use the Secant method to find a solution of a nonlinear operator equation in Banach spaces. A semilocal convergence result is obtained. For that, we consider a condition for divided differences which generalizes the usual ones, i.e., Lipschitz continuous or Hölder continuous conditions. Besides, we apply our results to approximate the solution of a nonlinear equation.     

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.     

On the convergence of Newton-like methods using restricted domains

We present a new semi-local convergence analysis for Newton-like methods in order to approximate a locally unique solution of a nonlinear equation containing a non-differentiable term in a Banach space setting. The new idea uses more precise convergence domains. This way the new sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, are also provided in this study.     

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)     

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.     

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.     

Numerical simulation of the nonlinear generalized time‐fractional Klein–Gordon equation using cubic trigonometric B‐spline functions

In this paper, an efficient numerical procedure for the generalized nonlinear time‐fractional Klein–Gordon equation is presented. We make use of the typical finite difference schemes to approximate the Caputo time‐fractional derivative, while the spatial derivatives are discretized by means of the cubic trigonometric B‐splines. Stability and convergence analysis for the numerical scheme are discussed. We apply our scheme to some typical examples and compare the obtained results with the ones found by other numerical methods. The comparison shows that our scheme is quite accurate and can be applied successfully to a variety of problems of applied nature.     

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.     

Convergence analysis for the Secant method based on new recurrence relations

A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.     

A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F(x)+G(x)=0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton-Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton-Kantorovich theorem.     

Kantorovich-type semilocal convergence analysis for inexact Newton methods

We provide a new semilocal convergence analysis for generating an inexact Newton method converging to a solution of a nonlinear equation in a Banach space setting. Our analysis is based on our idea of recurrent functions. Our results are compared favorably to earlier ones by others and us (Argyros (2007, 2009) [5] and [6], Argyros and Hilout (2009) [7], Guo (2007) [15], Shen and Li (2008) [18], Li and Shen (2008) [19], Shen and Li (2009) [20]). Numerical examples are provided to show that our results apply, but not earlier ones [15], [18], [19] and [20].     

A uniparametric family of iterative processes for solving nondifferentiable equations

In this work we study a class of secant-like iterations for solving nonlinear equations in Banach spaces. We consider a condition for divided differences which generalizes the usual ones, i.e., Lipschitz and Hölder continuous conditions. A semilocal convergence result is obtained for nondifferentiable operators. For that, we use a technique based on a new system of recurrence relations to obtain domains of existence and uniqueness of the solution. Finally, we apply our results to the numerical solution of several examples.     

Directional Secant-Type Methods for Solving Equations

A semilocal convergence analysis for directional Secant-type methods in multidimensional space is provided. Using weaker hypotheses than the ones exploited by An and Bai, we provide a semilocal convergence analysis with the following advantages: weaker convergence conditions, larger convergence domain, finer error estimates on the distances involved, and more precise information on the location of the solution. A numerical example, where our results apply to solve an equation but not the ones of An and Bai, is also provided. In a second example, we show how to implement the method.     

Multilevel augmentation methods for solving the sine-Gordon equation

In this paper we develop the multilevel augmentation method for solving nonlinear operator equations of the second kind and apply it to solving the one-dimensional sine-Gordon equation. We first give a general setting of the multilevel augmentation method for solving the second kind nonlinear operator equations and prove that the multilevel augmentation method preserves the optimal convergence order of the projection method while reducing computational cost significantly. Then we describe the semi-discrete scheme and the fully-discrete scheme based on multiscale methods for solving the sine-Gordon equation, and apply the multilevel augmentation method to solving the discrete equation. A complete analysis for convergence order is proposed. Finally numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the method.     

Improved convergence analysis for Newton-like methods

We present a new semilocal convergence analysis for Newton-like methods in order to approximate a locally unique solution of an equation in a Banach space setting. This way, we expand the applicability of these methods in cases not covered in earlier studies. The advantages of our approach include a more precise convergence analysis under the same computational cost on the Lipschitz constants involved. Applications are also given in this study to show that our estimates on the distances involved are tighter than the older ones.     

Weak convergence conditions for Inexact Newton-type methods

We establish a new semilocal convergence results for Inexact Newton-type methods for approximating a locally unique solution of a nonlinear equation in a Banach spaces setting. We show that our sufficient convergence conditions are weaker and the estimates of error bounds are tighter in some cases than in earlier works [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] and [31]. Special cases and numerical examples are also provided in this study.