two new families of sixth order methods for solving non linear equations in this paper we developed two new families of sixth order methods for solving simple roots of non linear equations per iteration these methods require two evaluations of the function and two evaluations of the first order derivatives which implies that the efficiency indexes of our methods are 1 565 these methods have more advantages than newton s method and other methods with the same convergence order as shown in the illustration examples finally using the developing methodology described in this paper two new families of improvements of jarratt method with sixth order convergence are derived in a straightforward manner notice that kou s method in jisheng kou yitian li an improvement of the jarratt method appl math comput 189 2007 1816 1821 and wang s method in xiuhua wang jisheng kou yitian li a variant of jarratt method with sixth order convergence appl math comput 204 2008 14 19 are the special cases of the new improvements

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

a note on some recent methods for solving nonlinear equations abstract in this note we present some comments on the recent results concerning iterative methods for solving nonlinear equations in the first part we compare a new method n ujevic a method for solving nonlinear equations appl math comput 174 2006 1416 1426 with the newton and ostrowski method including an extensive analysis of numerical results together with the choice of initial approximations and the computational efficiency of the considered methods in the second part we give a list of recently derived root finding methods and we study the question of priority namely we find out that all reviewed methods were already derived at the beginning of the 60s of the last century one of them dates from the 18th century some remarks on the comparison analysis and the application of iteration methods are included

discontinuous galerkin isogeometric analysis of elliptic diffusion problems on segmentations with gaps we propose a new discontinuous galerkin dg isogeometric analysis iga technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with nonmatching interfaces due to an incorrect segmentation procedure it may happen that the interfaces of adjacent subdomains do not coincide in this way gap regions which are not present in the original physical domain are created in this paper the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains we apply a multipatch approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem the last formulation is further modified by deriving interface conditions without the presence of the solution in the gap finally the solution of this modified problem is approximated b

a family of iterative methods with higher order convergence in this paper we investigate the problem of solving nonlinear equations in banach spaces we present a new family of iterative methods with higher order convergence by using newton cotes quadrature formula

the improvements of modified newton s method abstract in this paper we improve some third order modifications of newton s method and obtain many new methods for solving non linear equations the new methods have the order of convergence five or six per iteration these methods require two evaluations of the function and two evaluations of its first derivative and therefore the efficiency of the new methods may also be improved these methods can compete with newton s method as we show in some examples

discontinuous galerkin isogeometric analysis of elliptic problems on segmentations with non matching interfaces abstract in this paper we develop a discontinuous galerkin isogeometric analysis method for solving elliptic problems on decompositions of the computational domain into volumetric patches with non matching parametrized interfaces we specially focus on high order numerical solutions for complex gap regions and extend ideas from our previous work on simple gap regions for the communication of the numerical solution between the subdomains which are separated by the gap region discontinuous galerkin numerical fluxes are constructed taking into account the diametrically opposite points on the boundary of the gap due to lack of information on the behavior of the solution in the gap region the fluxes coming from the interior of the gap are approximated by taylor expansions with respect to the adjacent subdomain solutions we follow the same ideas of our previous work and show a priori error estimates in the dg norm with respect to the mesh size and the gap distance numerical examples performed for two three and even four dimensional computational domains demonstrate the robustness of the proposed numerical method and validate the estimates predicted by the theory

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

inexact feti dp methods inexact feti dp domain decomposition methods are considered preconditioners based on formulations of feti dp as a saddle point problem are used which allow for an inexact solution of the coarse problem a positive definite reformulation of the preconditioned saddle point problem which also allows for approximate solvers is considered as well in the formulation that iterates on the original feti dp saddle point system it is also possible to solve the local neumann subdomain problems inexactly given good approximate solvers for the local and coarse problems convergence bounds of the same quality as for the standard feti dp methods are obtained numerical experiments which compare the convergence of the inexact methods with that of standard feti dp are shown for 2d and 3d elasticity using gmres and cg as krylov space methods based on parallel computations a comparison of one variant of the inexact feti dp algorithms and the standard feti dp method is carried out and similar parallel performance is achieved parallel scalability of the inexact variant is also demonstrated it is shown that for a very large number of subdomains and a very large coarse problem the inexact method can be superior copyright 2006 john wiley sons ltd

some variants of chebyshev halley methods free from second derivative abstract in this paper we present some new modifications of chebyshev halley methods free from second derivative the convergence analysis shows that the new methods are third order convergent and require one function and two first derivative evaluations per iteration several numerical examples are given to illustrate the performance of the presented methods

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

introduction to a newton type method for solving nonlinear equations in this paper we shall introduce a new newton type method namely rational newton for solving a nonlinear equation this new method is shown to converge cubically we shall examine the effectiveness of the rational newton method by comparing the performance with the well established methods namely the classical newton method the halley rational method the kou et al and the weerakoon and fernando method for approximating the root of a given nonlinear equation the approximate solution of the rational newton method is found to be substantially more accurate than the well established methods

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

a modified secant method for unconstrained optimization abstract in this study a modification of the classical secant method for solving nonlinear univariate and unconstrained optimization problems based on the development of the cubic approximation is presented the iteration formula including an approximation of the third derivative of f x by using the taylor series expansion is derived the performance of the new method is analyzed in terms of the number of iterations in comparison with the secant methods using six test functions

a new iteration method with cubic convergence to solve nonlinear algebraic equations abstract in this paper a new iteration scheme is proposed to solve the roots of an algebraic equation f x 0 given an initial guess x0 the roots of the equation can be obtained using the following iteration scheme x n 1 x n f x n f 2 x n 2 f x n f x n f x n this iteration scheme has unique convergence characteristics different from the well known newton s method it is shown that this iteration method has cubic local convergence in the neighborhood of the root using this scheme real or complex roots for specific algebraic equations can be found because there are two iteration directions for a given initial guess two solutions can be found for certain algebraic equations with multiple roots examples are presented and compared with other methods

a parallel implementation of dual primal feti methods for three dimensional linear elasticity using a transformation of basis dual primal feti methods for linear elasticity problems in three dimensions are considered these are nonoverlapping domain decomposition methods where some primal continuity constraints across subdomain boundaries are required to hold throughout the iterations whereas most of the constraints are enforced by lagrange multipliers an algorithmic framework for dual primal feti methods is described together with a transformation of basis to implement the primal constraints numerical results obtained from a parallel implementation of these algorithms applied to a model benchmark problem with structured meshes and to problems with more complicated geometries from industrial and biological applications using unstructured meshes are provided these results show that the presented dual primal feti algorithms are numerical and parallel scalable

higher order multi step jarratt like method for solving systems of nonlinear equations this paper proposes a multi step iterative method for solving systems of nonlinear equations with a local convergence order of 3m4 where m 2 is the number of steps the multi step iterative method includes two parts the base method and the multi step part the base method involves two function evaluations two jacobian evaluations one lu decomposition of a jacobian and two matrixvector multiplications every stage of the multi step part involves the solution of two triangular linear systems and one matrixvector multiplication the computational efficiency of the new method is better than those of previously proposed methods the method is applied to several nonlinear problems resulting from discretizing nonlinear ordinary differential equations and nonlinear partial differential equations

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

on modified newton methods with cubic convergence abstract in this paper we present two new modified newton methods for solving a non linear equation permitting f x 0 in some points these new methods are showed to be cubically convergent from the practical application we show that these new methods are vast superior and possible in global convergence

dynamics of a family of third order iterative methods that do not require using second derivatives the purpose of this article is to present results that amount to a description of the conjugacy classes of three third order root finding iterative methods that do not require the use of second derivatives for their formulation for complex polynomials of degrees two three and four for degrees two and three a full description of the conjugacy classes is accomplished in each case by a one parameter family of polynomials this is done in such a way that when one applies one of these three root finding iterative methods to the elements of these parametrized families a family of iterative methods is obtained in such a way that its dynamics represents up to conjugacy the dynamics of the corresponding iterative root finding method applied to any complex polynomial having the same degree for degree four analogous partial results are obtained

on the construction of iterative methods with at least cubic convergence using the iteration formulas of order two for solving nonlinear equations we present a basic tool for deriving new higher order iterative methods that do not require the computation of the second order or higher order derivatives the presented convergence analysis shows that the order of convergence of the obtained iterative methods are three or higher the comparison with other methods is given

a simple and efficient method with high order convergence for solving systems of nonlinear equations we propose an m 1 step modified newton method of convergence order m 2 to solve systems of nonlinear equations which are third frechet differentiable in a convex set containing the zero computational efficiency in the general form for a positive integer m is discussed which shows that the efficiency increases with m when applied to large systems moreover a comparison between the efficiency of this technique and some existing efficient methods is made which implies that the present method is more efficient particularly for solving large systems of equations theoretical results about order of convergence and computational efficiency are largely verified in numerical examples

several improvements of the 2 point third order midpoint iterative method using weight functions abstract in this paper we improve the order of the midpoint iterative method from three to four with same number of function evaluations using weight functions to obtain a class of 2 point fourth order jarratt type methods then we prove a general result to further improve the order of the old methods through an additional function evaluation using weight functions in this way we develop two 3 point sixth order midpoint methods using different weights a family of n point i f s of order 2 n is proposed using weight functions we further improve the methods to obtain a 5 point sixteenth order midpoint method with same efficiency index as the optimal 2 point fourth order jarratt type methods two numerical examples using different starting are provided to test the robustness of the methods

numerical simulation of singularly perturbed non linear elliptic boundary value problems using finite element method the present paper presents the finite element solution of two dimensional non linear singularly perturbed elliptic partial differential equation subject to appropriate dirichlet boundary conditions a new fifth order convergent newton type iterative method has been described and used to linearize the non linear problem the inclusion of this newton s method of fifth order convergence in finite element method for solving non linear system of equations reduces the number of iterations and hence the cost of computation to demonstrate the usefulness of the proposed scheme a non convex variational ginzburg landau equation is considered

certain improvements of newton s method with fourth order convergence in this paper we present two new schemes one is third order and the other is fourth order these are improvements of second order methods for solving nonlinear equations and are based on the method of undetermined coefficients we show that the fourth order method is more efficient than the fifth order method due to kou et al j kou y li x wang some modifications of newton s method with fifth order covergence j comput appl math 209 2007 146 152 numerical examples are given to support that the methods thus obtained can compete with other iterative methods

parallelization of continuous and discontinuous galerkin dual primal isogeometric tearing and interconnecting methods abstract in this paper we investigate the parallelization of dual primal isogeometric tearing and interconnecting ieti dp type methods for solving large scale continuous and discontinuous galerkin systems of equations arising from isogeometric analysis of elliptic boundary value problems these methods are extensions of the finite element tearing and interconnecting methods to isogeometric analysis the algorithms are implemented by means of energy minimizing primal subspaces we discuss how these methods can efficiently be parallelized in a distributed memory setting weak and strong scaling studies presented for two and three dimensional problems show an excellent parallel efficiency

on newton type methods with cubic convergence recently there has been some progress on newton type methods with cubic convergence that do not require the computation of second derivatives weerakoon and fernando appl math lett 13 2000 87 derived the newton method and a cubically convergent variant by rectangular and trapezoidal approximations to newton s theorem while frontini and sormani j comput appl math 156 2003 345 140 2003 419 derived further cubically convergent variants by using different approximations to newton s theorem homeier j comput appl math 157 2003 227 169 2004 161 independently derived one of the latter variants and extended it to the multivariate case here we show that one can modify the werrakoon fernando approach by using newton s theorem for the inverse function and derive a new class of cubically convergent newton type methods

isogeometric bddc preconditioners with deluxe scaling a balancing domain decomposition by constraints bddc preconditioner with a novel scaling introduced by dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling is extended to isogeometric analysis of scalar elliptic problems this new scaling turns out to be more powerful than the standard rho and stiffness scalings considered in a previous isogeometric bddc study our h analysis shows that the condition number of the resulting deluxe bddc preconditioner is scalable with a quasi optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings in particular for large isogeometric polynomial degree and high regularity

a one parameter square root family of two step root finders a one parameter family of iterative methods for solving nonlinear equations is constructed all the methods of the proposed family are cubically convergent for a simple root except one particular method which attains the fourth order without the increase of computational cost these methods belong to the class of two step methods and require three function evaluations per iteration the square root structure of the family provides finding a complex zero of real functions in some cases a comparison analysis shows that the presented family generates the methods which are comparable or even superior than the existing two step iterative methods of the third order

means based modifications of newton s method for solving nonlinear equations in this paper we consider a family of six sets of means based modifications of newton s method for solving nonlinear equations each set is a parametric class of methods some well known methods belong to our family for example the arithmetic mean newton s method s weerakoon t g i fernando a variant of newton s method with accelerated third order convergence appl math lett 13 2000 87 93 the harmonic mean newton s method a y ozban some new variants of newton s method appl math lett 17 2004 677 682 the geometric mean newton s method t lukic n m ralevic geometric mean newton s method for simple and multiple roots appl math lett 21 2008 30 36 the power mean newton s method x zhou a class of newton s methods with third order convergence appl math lett 20 2007 1026 1030 the convergence analysis shows third order of our family comparison of the family members shows that there are no big differences between them twelve numerical examples were tested and two characteristic ones are presented

a class of efficient quadrature based predictor corrector methods for solving nonlinear systems we extend a class of quadrature based predictor corrector techniques for root finding to multivariate systems they are found to have a rate of convergence of 1 2 regardless of the degree of precision for the quadrature technique from which they are derived provided it is at least one by reusing the linear system from the previous iterate this class incorporates a significant improvement in computational time relative to the standard class through the inclusion of an lu decomposition during the iteration complexity is equivalent to newton s method as they only require knowledge of f x and f x

nonlinear feti dp and bddc methods new nonlinear feti dp dual primal finite element tearing and interconnecting and bddc balancing domain decomposition by constraints domain decomposition methods are introduced in all these methods in each iteration local nonlinear problems are solved on the subdomains the new approaches can significantly reduce communication and show a significantly improved performance especially for problems with localized nonlinearities compared to a standard newton krylov feti dp or bddc approach moreover the coarse space of the nonlinear feti dp methods can be used to accelerate the newton convergence it is also found that the new nonlinear feti dp and nonlinear bddc methods are not as closely related as in the linear context numerical results for the p laplace operator are presented

a third order newton type method to solve systems of nonlinear equations in this paper we present a third order newton type method to solve systems of nonlinear equations in the first we present theoretical preliminaries of the method secondly we solve some systems of nonlinear equations all test problems show the third order convergence of our method

a numerical scheme for multi point special boundary value problems and application to fluid flow through porous layers abstract in this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi point boundary value problems over adjacent domains in each subdomain the solution is governed by a different equation the solutions are required to be smooth across the interface nodes the approach is based on using finite difference approximation of the derivatives at the interface nodes smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations a modified multi dimensional newton s method is proposed for solving the nonlinear system the accuracy of the proposed scheme is validated by examples whose exact solutions are known the proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media

recurrence relations for a newton like method in banach spaces the convergence of iterative methods for solving nonlinear operator equations in banach spaces is established from the convergence of majorizing sequences an alternative approach is developed to establish this convergence by using recurrence relations for example the recurrence relations are used in establishing the convergence of newton s method l b rall computational solution of nonlinear operator equations robert e krieger new york 1979 and the third order methods such as halley s chebyshev s and super halley s v candela a marquina recurrence relations for rational cubic methods i the halley method computing 44 1990 169 184 v candela a marquina recurrence relations for rational cubic methods ii the halley method computing 45 1990 355 367 j a ezquerro m a hernandez recurrence relations for chebyshev type methods appl math optim 41 2000 227 236 j m gutierrez m a hernandez third order iterative methods for operators with bounded second derivative j comput appl math 82 1997 171 183 j m gutierrez m a hernandez recurrence relations for the super halley method comput math appl 7 36 1998 1 8 m a hernandez chebyshev s approximation algorithms and applications comput math appl 41 2001 433 445 10 in this paper an attempt is made to use recurrence relations to establish the convergence of a third order newton like method used for solving a nonlinear operator equation f x 0 where f email protected x y be a nonlinear operator on an open convex subset w of a banach space x with values in a banach space y here first we derive the recurrence relations based on two constants which depend on the operator f then based on this recurrence relations a priori error bounds are obtained for the said iterative method finally some numerical examples are worked out for demonstrating our approach

toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations the solution of nonlinear problems e g in material science requires fast and highly scalable parallel solvers finite element tearing and interconnecting dual primal feti dp domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements recently nonlinear versions of the well known feti dp methods for linear problems have been introduced in these methods the nonlinear problem is decomposed before linearization this approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of feti dp methods toward extreme scale supercomputers here a recent nonlinear feti dp method is combined with an approach that allows an inexact solution of the feti dp coarse problem we combine the nonlinear feti dp domain decomposition method with an algebraic multigrid amg method and thus obtain a hybrid nonlinear domain decomposition multigrid method we consider scalar nonlinear problems as well as nonlinear hyp

a two parameter third order family of methods for solving nonlinear equations in this paper we present a new two parameter family of iterative methods for solving nonlinear equations which includes as a particular case the classical potra and ptak third order method per iteration the new methods require two evaluations of the function and one evaluation of its first derivative it is shown that each family member is cubically convergent several examples are given to illustrate the performance of the family members

some third order families of iterative methods for solving nonlinear equations abstract in this paper we present a convenient third order iteration formula improving newton s method for solving nonlinear equations with any two iteration functions of order two the formula provides one parameter third order family of methods each family member is free from second derivatives and per iteration it requires three evaluations of the given function and its derivative several numerical examples are given to illustrate the efficiency and performance of some of the presented methods

some variant of newton s method with third order convergence we present a new modification of the newton s method which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given and the best efficiency in term of function evaluations of two of this new methods is provided

construction of newton like iteration methods for solving nonlinear equations in this paper we present a simple and yet powerful and easily applicable scheme in constructing the newton like iteration formulae for the computation of the solutions of nonlinear equations the new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations it provides a tool to develop new newton like iteration methods or to improve the existing iteration methods which contains the well known newton iteration formula in logic those all improve the newton method the orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of maple some numerical tests are given to support the theory developed in this paper

third order methods from quadrature formulae for solving systems of nonlinear equations we extend to p dimensional problems a modification of the newton method based on quadrature formulas of order at least one which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given these new methods may be more efficient then other third order methods as they do not require the use of the second order frechet derivative

zero finder methods derived using runge kutta techniques abstract in this paper some families of zero finding iterative methods for nonlinear equations are presented the key idea to derive them is to solve an initial value problem applying runge kutta techniques more explicitly these methods are used to solve the problem that consists in a differential equation in what appears the inverse function of the one which zero will be computed and the condition given by the value attained by it at the initial approximation carrying out this procedure several families of different orders of local convergence are obtained furthermore the efficiency of these families are computed and two new families using like newton s methods that improve the most efficient one are also given

a feti dp preconditioner for a composite finite element and discontinuous galerkin method in this paper a nitsche type discretization based on a discontinuous galerkin dg method for an elliptic two dimensional problem with discontinuous coefficients is considered the problem is posed on a polygonal region omega which is a union of n disjoint polygonal subdomains omega i of diameter o h i the discontinuities of the coefficients possibly very large are assumed to occur only across the subdomain interfaces partial omega i inside each subdomain a conforming finite element space on a quasi uniform triangulation with mesh size o h i is introduced to handle the nonmatching meshes across the subdomain interfaces a dg discretization is applied only on the interfaces for solving the resulting discrete system a feti dp type method is proposed and analyzed it is established that the condition number of the preconditioned linear system is estimated by c 1 max i log h i h i 2 with a constant c independent of h i h i h j h i and the jumps of coefficients the

dual primal isogeometric tearing and interconnecting solvers for multipatch dg iga equations abstract in this paper we consider a new version of the dual primal isogeometric tearing and interconnecting ieti dp method for solving large scale linear systems of algebraic equations arising from discontinuous galerkin dg isogeometric analysis of diffusion problems on multipatch domains with non matching meshes the dg formulation is used to couple the local problems across patch interfaces the purpose of this paper is to present this new method and provide numerical examples indicating a polylogarithmic condition number bound for the preconditioned system and showing an incredible robustness with respect to large jumps in the diffusion coefficient across the interfaces

a family of composite fourth order iterative methods for solving nonlinear equations in this paper we present a family of new fourth order iterative methods for solving nonlinear equations per iteration the methods consisting of the family require only two evaluations of the function and one evaluation of its derivative several numerical examples are given to illustrate the efficiency and performance of some of the presented methods

letter to the editor third order modification of newton s method in this paper we present a new modification of newton s method for solving non linear equations analysis of convergence shows that the new method is cubically convergent numerical examples show that the new method can compete with the classical newton s method

adaptive selection of primal constraints for isogeometric bddc deluxe preconditioners isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of computer aided design cad software and the discretization of variational problems of continuum mechanics in contrast with the finite element case the basis functions of isogeometric analysis are often not nodal as a consequence there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains building on earlier work on the deluxe version of the bddc balancing domain decomposition by constraints family of domain decomposition algorithms several adaptive algorithms are developed in this paper for scalar elliptic problems in an effort to decrease the dimension of the global coarse component of these preconditioners numerical experiments provide evidence that this work can be successful yielding scalable and quasi optimal adaptive bddc algorithms for isogeometric discretizations

a uniparametric chebyshev type method free from second derivatives in this paper we present a family of new chebyshev type methods free from second derivatives for solving non linear equations analysis of convergence shows that these new methods are cubically convergent as particular cases we introduce two efficient methods in this family of the new methods the practical utility of which is demonstrated by numerical examples

achieving higher order of convergence for solving systems of nonlinear equations abstract in this paper we develop a class of third order methods which is a generalization of the existing ones and a method of fourth order method then introduce a technique that improves the order of convergence of any given iterative method for solving systems of nonlinear equations based on a given iterative method of order p 2 which uses the extended newton iteration as a predictor a new method of order p 2 is proposed with only one additional evaluation of the function moreover if the given iterative method of order p 3 uses the newton iteration as a predictor then a new method of order p 3 can be developed applying this procedure we obtain some new methods with higher order of convergence moreover computational efficiency is analyzed and comparisons are made between these new methods and the ones from which have been derived finally several numerical tests are performed to show the asymptotic behaviors which confirm the theoretical results

an analysis of the properties of the variants of newton s method with third order convergence for the last five years the variants of the newton s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non linear equations these methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives in this paper we investigate about the relationship between these methods which are in fact based on the approximation of the second order derivative present in the third order limited taylor expansion we also prove that they are different forms of the halley method and are all contractive iterative methods in a common neighbourhood we extend some of these variants to multivariate cases and prove their respective local cubic convergence from their corresponding linear models

some modifications of newton s method with fifth order convergence in this paper we present some new modifications of newton s method for solving non linear equations analysis of convergence shows that these methods have order of convergence five numerical tests verifying the theory are given and based on these methods a class of new multistep iterations is developed

overlapping schwarz methods for isogeometric analysis based on generalized b splines abstract generalized b splines gb splines are a special class of tchebycheff b splines that are smooth piecewise function with sections in more general spaces gb splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation in this paper we present overlapping schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on gb splines an h analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes numerical results in two and three dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters the domain deformation and the jumping coefficients

third order convergence theorem by using majorizing function for a modified newton method in banach space abstract following the ideas of frontini and sormani we present a modified newton method in banach space which is used to solve the nonlinear operator equation we establish the newton kantorovich convergence theorem for the modified newton method with third order convergence in banach space by using majorizing function we also get the error estimate finally two examples are provided to show the application of our theorem

characterizing strong pseudoconvexity obstructions to biholomorphisms and lyapunov exponents in this paper we consider the following question for bounded domains with smooth boundary can strong pseudoconvexity be characterized in terms of the intrinsic complex geometry of the domain our approach to answering this question is based on understanding the dynamical behavior of real geodesics in the kobayashi metric and allows us to prove a number of results for domains with low regularity for instance we show that for convex domains with c 2 epsilon boundary strong pseudoconvexity can be characterized in terms of the behavior of the squeezing function near the boundary the behavior of the holomorphic sectional curvature of the bergman metric near the boundary or any other reasonable measure of the complex geometry near the boundary the first characterization gives a partial answer to a question of fornaess and wold as an application of these characterizations we show that a convex domain with c 2 epsilon boundary which is biholomorphic to a strongly pseudoconvex domain is also strongly pseudoconvex

an efficient fifth order method for solving systems of nonlinear equations in this paper we present a three step iterative method of convergence order five for solving systems of nonlinear equations the methodology is based on the two step homeier s method with cubic convergence homeier 2004 computational efficiency in its general form is discussed and a comparison between the efficiency of proposed technique and existing ones is made the performance is tested through numerical examples moreover theoretical results concerning order of convergence and computational efficiency are verified in the examples it is shown that the present method has an edge over existing methods particularly when applied to large systems of equations

some modification of newton s method by the method of undetermined coefficients in this paper we construct some modifications of newton s method for solving nonlinear equations which is based on the method of undetermined coefficients it is shown by way of illustration that the method of undetermined coefficients is a promising tool for developing new methods and reveals its wide applicability by obtaining some existing methods as special cases two new sixth order methods are developed numerical examples are given to support that the methods thus obtained can compete with other iterative methods

the value of continuity refined isogeometric analysis and fast direct solvers abstract we propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers starting from a highly continuous isogeometric analysis iga discretization we introduce c 0 separators to reduce the interconnection between degrees of freedom in the mesh by doing so both the solution time and best approximation errors are simultaneously improved we call the resulting method refined isogeometric analysis riga to illustrate the impact of the continuity reduction we analyze the number of floating point operations flops computational times and memory required to solve the linear system obtained by discretizing the laplace problem with structured meshes and uniform polynomial orders theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3 with p being the polynomial order of the discretization numerical results indicate that our proposed refined isogeometric analysis delivers a speed up factor proportional to p 2 in a 2 d mesh with four million elements and p 5 the linear system resulting from riga is solved 22 times faster than the one from highly continuous iga in a 3 d mesh with one million elements and p 3 the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis

bounded imbeddings of open kahler manifolds in c n on caracterise intrinsequement des varietes complexes ouvertes qui sont biholomorphiquement equivalentes a des sous varietes analytiques bornees lisses dans c n avec des frontieres fortement pseudoconvexes lisses

a simply constructed third order modifications of newton s method in this paper we present a simple and easily applicable approach to construct some third order modifications of newton s method for solving nonlinear equations it is shown by way of illustration that existing third order methods can be employed to construct new third order iterative methods the proposed approach is applied to the classical chebyshev halley methods to derive their second derivative free variants numerical examples are given to support that the methods thus obtained can compete with known third order methods

on some third order iterative methods for solving nonlinear equations using the iteration formulas of second order mamta v kanwar v k kukreja sukhjit singh on a class of quadratically convergent iteration formulae appl math comput 2004 in press for solving single variable nonlinear equations two classes of third order multipoint methods without using second derivative are derived the main advantage of these classes is that they do not fail if the derivative of the function is either zero or very small in the vicinity of the required root further a new family of secant like method with guaranteed super linear convergence is obtained by discrete modifications and their comparison with respect to the existing classical methods is given

an optimal and efficient general eighth order derivative free scheme for simple roots abstract the main motivation of this study is to present an optimal scheme in a general way that can be applied to any existing optimal multipoint fourth order iterative scheme whose first substep employs steffensen s method or steffensen like method to further produce optimal eighth order iterative schemes a rational function approximation approach is used in the construction of proposed scheme in addition we also discussed the theoretical and computational properties of our scheme each member of the presented scheme satisfies the optimality conjecture for multipoint iterative methods without memory which was given by kung and traub in 1970 finally we also concluded on the basis of obtained numerical results that our methods have faster convergence in contrast to the existing methods of same order because they have minimum residual errors minimum error difference between two consecutive iterations and minimum asymptotic error constants corresponding to the considered test function

new third and fourth order nonlinear solvers for computing multiple roots abstract we present a new third order method for finding multiple roots of nonlinear equations based on the scheme for simple roots developed by kou et al j kou y li x wang a family of fourth order methods for solving non linear equations appl math comput 188 2007 1031 1036 further investigation gives rise to new third and fourth order families of methods which do not require second derivative the fourth order family has optimal order since it requires three evaluations per step namely one evaluation of function and two evaluations of first derivative the efficacy is tested on a number of relevant numerical problems computational results ascertain that the present methods are competitive with other similar robust methods

low rank tensor methods in galerkin based isogeometric analysis abstract the global patch wise geometry map which describes the computational domain is a new feature in isogeometric analysis this map has a global tensor structure inherited from the parametric spline geometry representation the use of this global structure in the discretization of partial differential equations may be regarded as a drawback at first glance as opposed to the purely local nature of high order classical finite elements in this work we demonstrate that it is possible to exploit the regularity of this structure and to identify the great potential for the efficient implementation of isogeometric discretizations first we formulate tensor product b spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure second we derive an algorithm for the separation of variables in the integrands arising in the discretization this is possible by means of low rank approximation of the integral kernels we arrive at a compact separated representation of the integrals the separated form implies an expression of galerkin matrices as kronecker products of matrix factors with small dimensions this representation is very appealing due to the reduction in both memory consumption and computation times our benchmarks performed using the c library g smo demonstrate that the use of tensor methods in isogeometric analysis possesses significant advantages

a matrix free quasi newton method for solving large scale nonlinear systems one of the widely used methods for solving a nonlinear system of equations is the quasi newton method the basic idea underlining this type of method is to approximate the solution of newton s equation by means of approximating the jacobian matrix via quasi newton update application of quasi newton methods for large scale problems requires in principle vast computational resource to form and store an approximation to the jacobian matrix of the underlying problem hence this paper proposes an approximation for newton step based on the update of approximation requiring a computational effort similar to that of matrix free settings it is made possible by approximating the jacobian into a diagonal matrix using the least change secant updating strategy commonly employed in the development of quasi newton methods under suitable assumptions local convergence of the proposed method is proved for nonsingular systems numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with newton s chord newton s and broyden s methods

a class of multi point iterative methods for nonlinear equations a family of multi point iterative methods for solving nonlinear equations is described and the order of convergence of its elements is studied the computational efficiency of some elements of this family in terms of the number of function evaluations is also provided in addition we present different numerical tests which confirm or improve the theoretical results and allow us to compare some methods of this family

a modification of newton method with third order convergence in this paper we present a new modification of newton method for solving non linear equations analysis of convergence shows that the new method is cubically convergent per iteration the new method requires two evaluations of the function and one evaluation of its first derivative thus the new method is preferable if the computational costs of the first derivative are equal or more than those of the function itself its practical utility is demonstrated by numerical examples

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

a geometric construction of iterative functions of order three to solve nonlinear equations in this paper we consider a geometric construction of iteration functions of order three to develop cubically convergent iterative methods for solving nonlinear equations this construction can be applied to any iteration function of order two to develop an iteration function of order three some examples are given of deriving several third order iteration methods and several numerical results follow to illustrate the performance of the derived methods

a class of exponential quadratically convergent iterative formulae for unconstrained optimization abstract for solving nonlinear univariate and unconstrained optimization problems newton method is an important and basic method which converges quadratically this paper presents a class of exponential iterative formulae convergence analyses show that the proposed methods converge quadratically the efficiencies of the methods are analyzed in terms of the most popular and widely used criterion in comparison with the classical newton method using five test functions numerical results indicate that one of the new exponential iterative formulae is effective and comparable to well known newton s method

third order modifications of newton s method based on stolarsky and gini means in this paper we consider third order modifications of newton s method for solving nonlinear equations considered methods are based on stolarsky and gini means stolarsky 1975 13 stolarsky 1980 14 chen 2008 12 and depend on two parameters some known methods are special cases of our methods for example the power mean newton s method zhou 2007 10 the arithmetic mean newton s method weerakoon and fernando 2000 5 the harmonic mean newton s method ozban 2004 8 and the geometric mean newton s method lukic and ralevic 2008 9 third order convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of two parameters numerical examples obtained using mathematica with high precision arithmetic support the theoretical results some numerical tests were performed and it was shown that our methods yield better numerical results i e a smaller error x email protected when compared to halley euler hansen patrick ostrowski and inverse interpolation methods

a new modified secant like method for solving nonlinear equations in this paper we present a new secant like method for solving nonlinear equations analysis of the convergence shows that the asymptotic convergence order of this method is 1 3 some numerical results are given to demonstrate its efficiency

on a family of second derivative free variants of chebyshev s method in this paper we present a family of new second derivative free variants of chebyshev s method in which f x 0 in some points is permitted analysis of convergence shows that the new methods are cubically convergent analysis of efficiency in term of function evaluations shows that the new methods have the great practical utility which is demonstrated by numerical examples

a sixth order method for nonlinear equations abstract in this paper a sixth order method is developed by extending a third order method of weerakoon and fernando s weerakoon t g i fernando a variant of newton s method with accelerated third order convergence appl math lett 13 2000 87 93 for finding the real roots of nonlinear equations in r starting with a suitably chosen x 0 the method generates a sequence of iterates converging to the root the convergence analysis is provided to establish its sixth order of convergence in terms of computational cost it requires evaluations of only two functions and two first derivatives per iteration this implies that efficiency index of our method is 1 565 our method is comparable with the methods of neta b neta a sixth order family of methods for nonlinear equations intern j computer math 7 1979 157 161 and that of kou and li jisheng kou yitian li an improvement of jarratt method appl math comput 189 2007 1816 1821 it does not require the evaluation of the second order derivative of the given function as required in the family of chebyshev halley type methods jisheng kou xiuhua wang sixth order variants of chebyshev halley methods for solving non linear equations appl math comput 190 2007 1839 1843 jisheng kou on chebyshev halley methods with sixth order convergence for solving non linear equations appl math comput 190 2007 126 131 the efficacy of the method is tested on a number of numerical examples it is observed that our method takes less number of iterations than newton s method and the method of weerakoon and fernando on comparison with the other sixth order methods it behaves either similarly or better for the examples considered

higher order multi step iterative method for computing the numerical solution of systems of nonlinear equations in the present study we consider multi step iterative method to solve systems of nonlinear equations since the jacobian evaluation and its inversion are expensive in order to achieve a better computational efficiency we compute jacobian and its inverse only once in a single cycle of the proposed multi step iterative method actually the involved systems of linear equations are solved by employing the lu decomposition rather than inversion the primitive iterative method termed base method has convergence order co five and then we describe a matrix polynomial of degree two to design a multi step method each inclusion of single step in the base method will increase the convergence order by three the general expression for co is 3 s 1 where s is the number of steps of the multi step iterative method computational efficiency is also addressed in comparison with other existing methods the claimed convergence rates proofs are established the new contribution in this article relies essentially in the increment of co by three for each added step with a comparable computational cost in comparison with existing multi steps iterative methods numerical assessments are made which justify the theoretical results in particular some systems of nonlinear equations associated with the numerical approximation of partial differential equations pdes and ordinary differential equations odes are built up and solved