semilocal and local convergence of a fifth order iteration with frechet derivative satisfying holder condition the semilocal and local convergence in banach spaces is described for a fifth order iteration for the solutions of nonlinear equations when the frechet derivative satisfies the holder condition the holder condition generalizes the lipschtiz condition the importance of our work lies in the fact that many examples are available which fail to satisfy the lipschtiz condition but satisfy the holder condition the existence and uniqueness theorem is established with error bounds for the solution the convergence analysis is finally worked out on different examples and convergence balls for each of them are obtained these examples include nonlinear hammerstein and fredholm integral equations and a boundary value problem it is found that the larger radius of convergence balls is obtained for all the examples in comparison to existing methods using stronger conditions

convergence of newton like iterative methods newton like methods in which the intermediate systems of linear equations are solved by iterative techniques are examined by applying the theory of inexact newton methods radius of convergence and rate of convergence results are easily obtained the analysis is carried out in affine invariant terms the results are applicable to cases where the underlying newton like method is for example a difference newton like or update newton method

on interpolation variants of newton s method for functions of several variables a generalization of the variants of newton s method based on interpolation rules of quadrature is obtained in order to solve systems of nonlinear equations under certain conditions convergence order is proved to be 2d 1 where d is the order of the partial derivatives needed to be zero in the solution moreover different numerical tests confirm the theoretical results and allow us to compare these variants with newton s classical method whose convergence order is d 1 under the same conditions

dynamics of a family of chebyshev halley type methods in this paper the dynamics of the chebyshev halley family is studied on quadratic polynomials a singular set that we call cat set appears in the parameter space associated to the family this set has interesting similarities with the mandelbrot set the parameter space has allowed us to find different elements of the family which have bad convergence properties since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method

a fast chebyshev s method for quadratic equations a multiresolution transform corresponding to interpolatory technique is used for fast application of chebyshev s method in designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the method a convergence theorem is performed finally some numerical results are presented

solving wiener hopf problems via an efficient iterative scheme abstract in the analysis of the fluid queues it is necessary to obtain the nonnegative solution of a nonsymmetric algebraic riccati matrix equation under suitable conditions this solution can be obtained transforming algebraic riccati equations into unilateral quadratic matrix equations in this paper we use an efficient iterative scheme to approximate a solution of this quadratic matrix equation we improve the efficiency and the accuracy of newton s method widely used in the literature moreover a local convergence result is proved finally we apply this efficient method to approximate the solution of a particular noisy wiener hopf problem and we compare it with newton s method moreover a predictor corrector iterative scheme is constructed that improve the accessibility of the aforesaid method

efficient high order methods based on golden ratio for nonlinear systems abstract we derive new iterative methods with order of convergence four or higher for solving nonlinear systems by composing iteratively golden ratio methods with a modified newton s method we use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

a modified chebyshev s iterative method with at least sixth order of convergence this paper is devoted to the construction and analysis of a high order variant of the classical chebyshev method the method has order of convergence at least six for simple roots the extension to system of equations and its semilocal convergence for nonlinear equations are presented finally an application to well known algebraic riccati equation is considered

extending the applicability of newton s method for a class of boundary value problems using the shooting method abstract we use newton s method to approximate locally unique solutions for a class of boundary value problems by applying the shooting method the utilized operator is frechet differentiable between banach spaces these conditions are more general than those that appear in previous works in particular we show that the old semilocal and local convergence criteria for newton s method involving banach space value operators can be replaced by weaker ones hence extending the applicability of the method several numerical examples are developed to test the new convergence criteria and also compare them to the old ones

a note on halley s method we introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point making use of this concept we obtain a new theorem of global convergence for halley s method

an optimal reconstruction of chebyshev halley type methods for nonlinear equations having multiple zeros abstract establishment of a new optimal higher order iterative scheme for multiple zeros with known multiplicity m 1 of univariate function is one of the hard and demanding task in the area of computational mathematics and numerical analysis we write this paper with the aim to propose a new higher order reconstruction of chebyshev halley type iteration functions for multiple zeros in a simple way minimum order of convergence of our scheme is seven and maximum reaches to eight without using any extra functional evaluation with regard to computational cost each member of our scheme needs four functional evaluations at each step so the maximum efficiency index of our scheme is 1 6818 for 1 in addition we thoroughly study the convergence analysis which confirms the theoretical order of convergence moreover we check the efficiency and compare them to the existing ones on four real life problems namely fractional conversion chemical engineering continuous stirred tank reactor and eigen value problems and two well known academic problems finally we confirm on the basis of obtained results that our methods have better residual error errors between two consecutive iterations and stable computational order of convergence as compared to the other existing ones

newton like methods for the computation of fixed points abstract the celebrated banach fixed point theorem provides conditions which assure that the method of successive substitution is convergent the convergence however may take place very slowly so that it may be desirable to use a newton like method for the computation of the fixed point if newton s method itself is applied one ignores the additional information that the problem arises from a frixed point problem with a contraction mapping in the present note some variants of newton s method are discussed which make use of this contraction information it turns out that the convergence of newton s method can be accelerated without any relevant additional computational labour

on a characterization of some newton like methods of r order at least three in this paper we extend to banach spaces the result given by gander in amer math monthly 92 1985 131 to obtain a characterization of newton like iterative process with r order of convergence at least three to do this we consider the construction and the study of the semilocal convergence of a multiparametric family of iterative processes in banach spaces for solving the nonlinear equation f x 0

image restoration via simultaneous sparse coding where structured sparsity meets gaussian scale mixture in image processing sparse coding has been known to be relevant to both variational and bayesian approaches the regularization parameter in variational image restoration is intrinsically connected with the shape parameter of sparse coefficients distribution in bayesian methods how to set those parameters in a principled yet spatially adaptive fashion turns out to be a challenging problem especially for the class of nonlocal image models in this work we propose a structured sparse coding framework to address this issue more specifically a nonlocal extension of gaussian scale mixture gsm model is developed using simultaneous sparse coding ssc and its applications into image restoration are explored it is shown that the variances of sparse coefficients the field of scalar multipliers of gaussians if treated as a latent variable can be jointly estimated along with the unknown sparse coefficients via the method of alternating optimization when applied to image restoration our experimental results have shown that the proposed ssc gsm technique can both preserve the sharpness of edges and suppress undesirable artifacts thanks to its capability of achieving a better spatial adaptation ssc gsm based image restoration often delivers reconstructed images with higher subjective objective qualities than other competing approaches

on the convergence of a higher order family of methods and its dynamics in this paper we present the study of the local convergence of a higher order family of methods moreover the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family by means of studying the associated rational function parameter spaces are shown and the study of the stability of all the fixed points is presented

an iterative method with cubic convergence for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations we show that this new iterative method has third order convergence several numerical examples are given to illustrate the efficiency and performance of this new method new method can be viewed as an improvement of the previously known iterative methods

enlarging the convergence domain in local convergence studies for iterative methods in banach spaces in this work we introduce a new form of setting the general assumptions for the local convergence studies of iterative methods in banach spaces that allows us to improve the convergence domains specifically a local convergence result for a family of higher order iterative methods for solving nonlinear equations in banach spaces is established under the assumption that the frechet derivative satisfies the lipschitz continuity condition for some values of the parameter these iterative methods are of fifth order the importance of our work is that it avoids the usual practice of boundedness conditions of higher order derivatives which is a drawback for solving some practical problems the existence and uniqueness theorem that establishes the convergence balls of these methods is obtained we have considered a number of numerical examples including a nonlinear hammerstein equation and computed the radii of the convergence balls it is found that the radius of convergence ball obtained by our approach is much larger when compared with some other existing methods the global convergence properties of the family are explored by analyzing the dynamics of the corresponding operator on complex quadratic polynomials

accelerated methods of order 2p for systems of nonlinear equations we present a new iterative method of order of convergence 5 for solving nonlinear systems by composing the midpoint method with newton s method and using an approximation for the jacobian matrix in order to reduce the required number of functional evaluations per iteration in addition we compare the efficiency index of these methods with that of newton s method and present several numerical tests which confirm the theoretical results and allow us to compare these variants with newton s method

higher order iterative methods for approximating zeros of analytic functions abstract iterative methods with extremely rapid convergence in floating point arithmetic and circular arithmetic for simultaneously approximating simple zeros of analytic functions inside a simple smooth closed contour in the complex plane are presented the r order of convergence of the basic total step and single step methods as well as their improvements which use newton s and halley s corrections is given some hybrid algorithms that combine the efficiency of ordinary floating point iterative methods with the accuracy control provided by interval arithmetic are also considered

chebyshev type methods and preconditioning techniques recently a newton s iterative method is attracting more and more attention from various fields of science and engineering this method is generally quadratically convergent in this paper some chebyshev type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system ax b theoretic analysis and numerical experiments show that chebyshev s method is more effective than newton s one in the case of constructing approximate inverse preconditioners

global dynamics of one class of nonlinear nonautonomous systems with time varying delays abstract a class of nonautonomous systems of nonlinear delay differential equations was studied via construction of matrix inequalities and comparison techniques the results for the nonautonomous systems with time varying delays are novel e g the global stability of differential equations with nonlinear casual volterra operators is considered for the first time in the literature criteria obtained for permanence and global attractivity are explicit and hence are convenient for applying verifying in practice we illustrate applications of the results obtained to the nonautonomous and asymptotically autonomous nicholson type models