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].     

New sufficient convergence conditions for the secant method

We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses 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 the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

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.     

On the Secant method for solving nonsmooth equations

In this study we are concerned with the problem of approximating a locally unique solution of an operator equation in Banach space using the Secant method. The differentiability of the operator involved is not assumed. Using a flexible point-based approximation, we provide a local as well as a semilocal convergence analysis for the Secant method. Our results are justified by numerical examples that cannot be handled with 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.     

A new eighth-order iterative method for solving nonlinear equations

In this paper we present an improvement of the fourth-order Newton-type method for solving a nonlinear equation. The new Newton-type method is shown to converge of the order eight. Per iteration the new method requires three evaluations of the function and one evaluation of its first derivative and therefore the new method has the efficiency index of , which is better than the well known Newton-type methods of lower order. We shall examine the effectiveness of the new eighth-order Newton-type method by approximating the simple root of a given nonlinear equation. Numerical comparisons are made with several other existing methods to show the performance of the presented method.     

On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004)  [2], Bosarge and Falb (1969)  [9], Dennis (1971)  [10], Ezquerro et al. (2010)  [11], Hernández et al. (2005, 2000)   and , Kantorovich and Akilov (1982)  [14], Laasonen (1969)  [15], Ortega and Rheinboldt (1970)  [16], Parida and Gupta (2007)  [17], Potra (1982, 1984–1985, 1985)  ,  and , Proinov (2009, 2010)   and , Schmidt (1978) [23], Wolfe (1978)  [24] and Yamamoto (1987)  [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.     

On convergence of a new secant-like method for solving nonlinear equations

In this paper, we prove that the order of a new secant-like method presented recently with the so-called order of 2.618 is only 2.414. Some mistakes in the derivation leading to such a conclusion are pointed out. Meanwhile, under the assumption that the second derivative of the involved function is bounded, the convergence radius of the secant-like method is given, and error estimates matching its convergence order are also provided by using a generalized Fibonacci sequence.     

A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations

We revisit a fast iterative method studied by us in [I.K. Argyros, On a two-point Newton-like method of convergent order two, Int. J. Comput. Math. 88 (2) (2005) 219-234] to approximate solutions of nonlinear operator equations. The method uses only divided differences of order one and two function evaluations per step. This time we use a simpler Kantorovich-type analysis to establish the quadratic convergence of the method in the local as well as the semilocal case. Moreover we show that in some cases our method compares favorably, and can be used in cases where other methods using similar information cannot [S. Amat, S. Busquier, V.F. Candela, A class of quasi-Newton generalized Steffensen's methods on Banach spaces, J. Comput. Appl. Math. 149 (2) (2002) 397-406; D. Chen, On the convergence of a class of generalized Steffensen's iterative procedures and error analysis, Int. J. Comput. Math. 31 (1989) 195-203]. Numerical examples are provided to justify the theoretical results.     

On a quadratically convergent method using divided differences of order one under the gamma condition

We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.      

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.     

An iterative method for solving nonlinear functional equations

An iterative method for solving nonlinear functional equations, viz. nonlinear Volterra integral equations, algebraic equations and systems of ordinary differential equation, nonlinear algebraic equations and fractional differential equations has been discussed.     

On a method of Bownds for solving Volterra integral equations

An initial-value method of Bownds for solving Volterra integral equations is reexamined using a variable-step integrator to solve the differential equations. It is shown that such equations may be easily solved to an accuracy ofO(10^–8), the error depending essentially on that incurred in truncating expansions of the kernel to a degenerate one.This work was sponsored by a University of Nevada at Las Vegas Research Grant.     

On separable Banach subspaces

We show that any infinite-dimensional Banach (or more generally, Fréchet) space contains linear subspaces of arbitrarily high Borel complexity which admit separable complete norms giving rise to the inherited Borel structure.     

Two-step Newton methods

We present sufficient convergence conditions for two-step Newton methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The advantages of our approach over other studies such as Argyros et al. (2010) [5], Chen et al. (2010) [11], Ezquerro et al. (2000) [16], Ezquerro et al. (2009) [15], Hernández and Romero (2005) [18], Kantorovich and Akilov (1982) [19], Parida and Gupta (2007) [21], Potra (1982) [23], Proinov (2010) [25], Traub (1964) [26] for the semilocal convergence case are: weaker sufficient convergence conditions, more precise error bounds on the distances involved and at least as precise information on the location of the solution. In the local convergence case more precise error estimates are presented. These advantages are obtained under the same computational cost as in the earlier stated studies. Numerical examples involving Hammerstein nonlinear integral equations where the older convergence conditions are not satisfied but the new conditions are satisfied are also presented in this study for the semilocal convergence case. In the local case, numerical examples and a larger convergence ball are obtained.     

On an initial-value method for quickly solving Volterra integral equations: A review

A method of converting nonlinear Volterra equations to systems of ordinary differential equations is compared with a standard technique, themethod of moments, for linear Fredholm equations. The method amounts to constructing a Galerkin approximation when the kernel is either finitely decomposable or approximated by a certain Fourier sum. Numerical experiments from recent work by Bownds and Wood serve to compare several standard approximation methods as they apply to smooth kernels. It is shown that, if the original kernel decomposes exactly, then the method produces a numerical solution which is as accurate as the method used to solve the corresponding differential system. If the kernel requires an approximation, the error is greater, but in examples seems to be around 0.5% for a reasonably small number of approximating terms. In any case, the problem of excessive kernel evaluations is circumvented by the conversion to the system of ordinary differential equations.     

Strong-norm error estimates for the projective-difference method for approximately solving abstract parabolic equations

Solutions continuously differentiable with respect to time of parabolic equations in Hilbert space are obtained by the projective-difference method approximately. The discretization of the problem is carried out in the spatial variables using Galerkin's method, and in the time variable using Euler's implicit method. Strong-norm error estimates for approximate solutions are obtained. These estimates not only allow one to establish the convergence of the approximate solutions to the exact ones but also yield numerical characteristics of the rates of convergence. In particular, order-sharp error estimates for finite element subspaces are obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 898–909, March, 1998. Translated by N. K. Kulman     

Superlinear convergence theorems for Newton-type methods for nonlinear systems of equations

In this paper, someQ-order convergence theorems are given for the problem of solving nonlinear systems of equations when using very general finitely terminating methods for the solution of the associated linear systems. The theorems differ from those of Dembo, Eisenstat, and Steihaug in the different stopping condition and in their applicability to the nonlinear ABS algorithm.Lecture presented at the University of Bergamo, Bergamo, Italy, October 1989.     

Projection method for solving degenerate first-order identification problem

In this paper we study an inverse problem for a degenerate differential equation on a Banach space. A projection method will be used to reduce the problem to a regular abstract inverse problem. To this end some conditions on the operators are assumed to avoid the calculations of some inverses of the involved operators.