a two step sor newton method for nonsmooth equations abstract in this paper we present a two step sor newton method for solving a system of nonlinear equations f x 0 where f is strongly monotone locally lipschitz continuous but not necessarily differentiable the convergence of the two step sor newton method is discussed for any starting point we give safe intervals such that for any parameters in these intervals the two step sor newton method converges numerical examples show that the two step sor newton method converges faster than the sor newton method given by chen in x chen on convergence of sor methods for nonsmooth equations numer linear algebra appl 9 2002 81 92

orientational ordering in two dimensional systems with long range interaction a variational method for calculating gaussian fluctuations of two dimensional 2d moment orientations in lattice systems with arbitrary anisotropic interaction is proposed it is asymptotically exact for very low temperatures and gives a good extrapolation for a certain low temperature range general relationships are obtained that enable one to represent from a unified standpoint and in some cases to refine the known results on 1 the existence of short range order in a degenerate 2d system of moments with isotropic short range interaction 2 long range order stabilization by weak local potentials and dipole interactions 3 the existence of long range order in a 2d system with interaction of the type v r r if 2 4 for systems with anisotropic long range dipole interaction of moments on arbitrary 2d lattices except square ones for which a short range phase takes place the existence of long range order is proved lower limits for temperatures of transitions into phases with long range order are evaluated the low values of the transition temperatures in the dipole systems can be explained due to the small values of interchain interaction in comparison with interchain ones the applicability area for the method used is discussed

one point newton type iterative methods in this paper a unified point of view that includes the most of one point newton type iterative methods for solving nonlinear equations is introduced a simple idea to design iterative methods with quadratic or cubic convergence is also described this idea is extended to construct one point iterative methods of order four in addition several numerical examples are given to illustrate and compare different known methods and some introduced by using this unifying idea

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

error analysis of a new transformation for multiple zeros finding free from derivative evaluations abstract in this paper we present the error analysis of a new transformation for multiple zeros finding without calculating derivatives which is a kind of version proposed in a recent paper of this journal see 1 we prove that the new derivative free transformation could succeed in the maximum accuracy that is attainable at fixed precision of computation and preserves the order of convergence of the iteration method in the case of multiple zeros and show the advantage of this new transformation compared with the other ones

construction of third order modifications of newton s method abstract in this paper an approach for constructing third order modifications of newton s method using second order iteration formulas is presented several numerical examples are given to compare some of the newly obtained methods with the other third order methods

variants of newton s method using fifth order quadrature formulas abstract some variants of newton s method are developed in this work in order to solve nonlinear equations depending on one or several variables based in rules of quadrature of fifth order we prove the third or fifth order of convergence of these methods for dimension one and the second or third order in several variables depending on the behaviour of the second derivative moreover different numeric tests confirm or improve theoretic results and allow us to compare these variants with newton s classical method

sixth order variants of chebyshev halley methods for solving non linear equations abstract in this paper we present a family of new variants of chebyshev halley methods with sixth order convergence compared with chebyshev halley methods the new methods require one additional evaluation of the function the numerical results presented show that the new methods compete with chebyshev halley methods

new iterative methods for non linear equations abstract in this paper some new iterative methods for solving non linear equations f x 0 are presented and analyzed by means of a new two step predictor corrector type iterative method the convergence criteria for these iterative methods are also discussed several numerical examples are given to illustrate the efficiency and performance of the new methods

a family of new newton like methods abstract in this paper we present a family of new newton like methods which removes the severe condition f x 0 in a neighborhood of the required root imposed on newton s method the new methods are quadratically convergent and moreover have the same error equation as newton s method the detailed analysis of convergence is supplied the numerical results show that the new methods have definite practical utility

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

note on the improvement of newton s method for system of nonlinear equations abstract in this note we are concerned with the further study for a significant improvement on newton s iterative method proposed by wu xinyuan wu a new continuation newton like method and its deformation appl math comput 112 2000 75 78 we present a natural extension and development of the improvement on newton s method for system of nonlinear equation the convergence is presented and the numerical results are given to show the efficiency of the extended method for system of nonlinear equations

on a family of halley like methods to find simple roots of nonlinear equations there are many methods for solving a nonlinear algebraic equation here we introduce a family of halley like methods and show that euler chebyshev and bsc are just members of the family we discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction

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 a class of quadratic convergence iteration formulae without derivatives a class of iterative formulae without derivatives are proposed in this paper these methods are shown to be quadratic convergence some numerical tests 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

a new iterative method to compute nonlinear equations the aim of this paper is to construct a new efficient iterative method to solve nonlinear equations the new method is based on the proposals of abbasbandy on improving the order of accuracy of newton raphson method s abbasbandy improving newton raphson method for nonlinear equations by modified adomian decomposition method applied mathematics and computation 145 2003 887 893 and on the proposals of babolian and biazar on improving the order of accuracy of adomian s decomposition method e babolian j biazar on the order of convergence of adomian method applied mathematics and computation 130 2002 383 387 the convergence of the new scheme is proved and at least the cubic order of convergence is established several examples are presented and compared to other methods showing the accuracy and fast convergence of this new method also it is shown in this paper that the modified adomian s method developed by babolian and biazar to solve nonlinear equations e babolian j biazar solution of nonlinear equations by modified adomian decomposition method applied mathematics and computation 132 2002 167 172 should be slightly modified due to the fact that convergence of adomian s method does not ensure convergence of the modified method an example illustrates this fact which unlike what is claimed by the authors does not converge with their method but with a simple different choice of the zero component becomes convergent

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

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

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

some second derivative free variants of super halley method with fourth order convergence in this paper we present some new variants of super halley method for solving nonlinear equations these methods are free from second derivatives and require one function and two first derivative evaluations per iteration analysis of convergence shows that the methods are fourth order several numerical examples are given to illustrate the performance of the presented methods

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

explicit auto transformations of integrable chains a construction scheme for explicit auto transformations of integrable discrete differential equations chains is presented these transformations are rather convenient to obtain the exact solutions for chains as well as associated partial differential systems on the other hand they exemplify new integrable discrete mappings their group properties are also of great interest the scheme is illustrated by several examples of integrable systems which contains the nonlinear schrodinger system and the landau lifshits model

a cubically convergent newton type method under weak conditions under weak conditions we present an iteration formula to improve newton s method for solving nonlinear equations the method is free from second derivatives permitting f x 0 in some points and per iteration it requires two evaluations of the given function and one evaluation of its derivative analysis of convergence demonstrates that the new method is cubically convergent some numerical examples illustrate that the algorithm is more efficient and performs better than classical newton s method

finite size scaling for the mean spherical model with inverse power law interaction a new analytical technique based on integral transformations with mittag leffler type kernels is used to derive the finite size scaling function for the free energy per particle of the mean spherical model with inverse power law asymptotics of the interaction potential the asymptotic formation of the singularities in the specific heat and magnetic susceptibility at the bulk critical point is studied

a one parameter family of second order iteration methods a new one parameter family of iteration methods for finding simple roots of nonlinear equations is developed error analysis providing the second order convergence is given numerical experience shows that the method is comparable to the well known newton s method

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

integrable cluster dynamics of directed networks and pentagram maps abstract the pentagram map was introduced by r schwartz more than 20 years ago in 2009 v ovsienko r schwartz and s tabachnikov established liouville complete integrability of this discrete dynamical system in 2011 m glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver using compatibility of poisson and cluster structures and poisson geometry of directed networks on surfaces we generalize glick s construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations the appendix relates the simplest of these discrete maps to the toda lattice and its tri hamiltonian structure

third order derivative free sph iterative method for solving nonlinear systems the majority of iterative methods to find roots of a function requires the evaluation of derivatives of this function in this paper based on the basic principle of the sph method s kernel approximation a kernel approximation was constructed to compute first and second order derivatives through taylor series expansion derivatives in our proposed method were replaced in a newton like iterative method to obtain a derivative free sph iterative method for solving nonlinear systems to illustrate that the new method has the same order of convergence as the considered iterative method some numerical examples are presented

new high order convergence iteration methods without employing derivatives for solving nonlinear equations abstract a family of new iteration methods without employing derivatives is proposed in this paper we have proved that these new methods are quadratic convergence their efficiency is demonstrated by numerical experiments the numerical experiments show that our algorithms are comparable to well known methods of newton and steffensen furthermore combining the new method with bisection method we construct another new high order iteration method with nice asymptotic convergence properties of the diameters b n a n

on a new class parametrized newton like method for semismooth equations an extension of some smooth parametrized newton like method to the semismooth case is presented in this paper thus the problems due to the fact the b differential is singular are solved these methods are shown to be superlinear convergence some numerical experiments are given

yang baxter maps and symmetries of integrable equations on quad graphs a connection between the yang baxter relation for maps and the multidimensional consistency property of integrable equations on quad graphs is investigated the approach is based on the symmetry analysis of the corresponding equations it is shown that the yang baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations we use the classification results by adler bobenko and suris to demonstrate this method some new examples of yang baxter maps are derived in this way from multifield integrable equations

dynamics of iterative families with memory based on weight functions procedure abstract in this paper we analyze the stability of a parametric family of iterative methods with memory for solving nonlinear equations this family is obtained from an optimal class of fourth order schemes without memory designed by means of weight functions procedure by studying the real fixed and critical points of the rational function resulting from the application of the family with memory on quadratic polynomials the best elements of the family in terms of absence of chaotic behavior are selected finally a numerical study is performed verifying the dynamical theoretical results

finite size scaling for systems with long range interactions abstract the present review is devoted to the fundamental problems of finite size scaling due to the presence of long range interactions the attention is focused on the precise formulation of critical finite size scaling in the case that the bulk correlation length is identically infinite the hypotheses based on the notion of a finite size scaling length are formulated in a way which treats the cases of short and long range interactions on equal grounds as well as the standard and modified above the upper critical dimensionality finite size scaling the general conjectures are tested on the exactly solvable mean spherical model with power law interaction special attention is paid to the adequate mathematical techniques the formulation of finite size scaling at first order phase transitions is also discussed since the finite size scaling length enters into the definition of the relevant scaled external field variable provided the system has d 0 infinite dimensions the review closes with a summary and short discussion of some open problems

newton like method with some remarks abstract this paper presents the new convergence analysis for the newton like method proposed by xinyuan wu a new continuation newton like method and its deformation appl math comput 112 2000 75 78 compared with the original version of the convergence analysis the restriction imposed on f x is removed thoroughly in order to guarantee the quadratic convergence of the newton like method it is suffices to suppose that f x 0 and f x is local lipschitz near x where x is a solution of nonlinear equation f x 0 moreover some comments are given with examples for the newton like method in comparison with newton s method it can be concluded that the new algorithm is more feasible effective than newton s method in particularly when it happens that the derivative of the function f x at an iterate is singular or almost singular the newton like method is vast superior to the classical newton method the numerical results of the paper strongly support the conclusion

a third order modification of newton method for systems of non linear equations in this paper we present a new modified newton method for finding a zero of a vector function it is proved that the order of convergence of the new method is three this new method permits that the jacobian is singular in some points thus the expected problem due to the fact the jacobian is numerically singular is solved by numerical examples we show that this method is vast superior in some cases

a new continuation newton like method and its deformation this paper presents a family of new continuation newton like methods and its deformations for computing approximate solutions of the non linear algebraic equation f x 0 from a practical point of view the new methods are vast superior

a sixth order transformation method for finding multiple roots of nonlinear equations and basin attractors for various methods in this contribution a sixth order transformation method is proposed and analyzed for finding multiple roots of nonlinear equations when the multiplicity of the root is not known explicitly the proposed method does not require the evaluation of second derivative the basins of attraction of existing transformation methods and the proposed method are presented to demonstrate their performance

new reductions of integrable matrix partial differential equations sp m invariant systems we propose a new type of reduction for integrable systems of coupled matrix partial differential equations this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix via this reduction we obtain a new integrable system of coupled derivative modified kdv equations and a new integrable variant of the massive thirring model in addition to the already known systems we also discuss integrable semidiscretizations of the obtained systems and present new soliton solutions to both continuous and semidiscrete systems as a by product a new integrable semidiscretization of the manakov model self focusing vector nonlinear schrodinger equation is obtained

efficient continuation newton like method for solving systems of non linear equations in this paper we present a new newton like method for finding a zero of a vector function permitting that the jacobian is singular in some points thus the problems due to the fact the jacobian is numerically singular are solved the method is showed to be quadratic convergence then we detail the continuation newton like method corresponding to this method by numerical examples we show that this method is vast superior and possible in global convergence

simply constructed family of a ostrowski s method with optimal order of convergence in this paper we propose a simple modification over chun s method for constructing iterative methods with at least cubic convergence 5 using iteration formulas of order two we now obtain several new interesting families of cubically or quartically convergent iterative methods the fourth order family of ostrowski s method is the main finding of the present work per iteration this family of ostrowski s method requires two evaluations of the function and one evaluation of its first order derivative therefore the efficiency index of this ostrowski s family is e 43 1 587 which is better than those of most third order iterative methods e 33 1 442 and newton s method e 2 1 414 the performance of ostrowski s family is compared with its closest competitors namely ostrowski s method jarratt s method and king s family in a series of numerical experiments

quadratically convergent multiple roots finding method without derivatives abstract in this paper an iteration method without derivatives for multiple roots is proposed this method proved to be quadratically convergent its efficiency and accuracy are illustrated by numerical experiments

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

ground state of an infinite two dimensional system of dipoles on a lattice with arbitrary rhombicity angle abstract a class of possible periodic orientational configurations of a system of dipole moments located at the sites of an infinite flat rhombic lattice with an arbitrary rhombicity angle is studied by using the luttinger and tisza method in the framework of the method it is obtained that for 80 the ground state is ferromagnetic at 60 being continuously degenerate in direction for 80 90 the ground state is antiferromagnetic at 90 being also continuously degenerate with respect to one parameter it is shown that the restriction of the interaction range leads to a change in the type of the ground state at 60 this takes place between the third and the fourth coordination spheres and at 30 at a distance of about 500 lattice constants from comparison with known results of numerical experiments on finite systems it is established that the type of the ground state of a finite and an infinite system may be essentially different

modified newton s method for systems of nonlinear equations with singular jacobian it is well known that newton s method for a nonlinear system has quadratic convergence when the jacobian is a nonsingular matrix in a neighborhood of the solution here we present a modification of this method for nonlinear systems whose jacobian matrix is singular we prove under certain conditions that this modified newton s method has quadratic convergence moreover different numerical tests confirm the theoretical results and allow us to compare this variant with the classical newton s method

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

a semi local convergence theorem for a robust revised newton s method it is well known that newton s iteration will abort due to the overflow if the derivative of the function at an iterate is singular or almost singular in this paper we study a robust revised newton s method for solving nonlinear equations which can be carried out with a starting point with a degenerate derivative at an iterative step it is proved that the method is convergent under the conditions of the newton kantorovich theorem which implies a larger convergence domain of the method we also show that our method inherits the fast convergence of newton s method numerical experiments are performed to show the robustness of the proposed method in comparison with the standard newton s method

classical dipoles on a finite triangular lattice the spherical model approximation abstract in the framework of the spherical model it is shown that a phase transition at finite temperature to a ferromagnetic phase in a two dimensional system with dipole dipole and exchange interactions on a lattice with arbitrary rhombicity angle exists if the ferromagnetic ground state is not continuously degenerate in spin direction when the degeneracy exists the system exhibits essential singularities in the thermodynamic functions at t 0 finite size effects in such systems are investigated in the example of the triangular latice under periodic boundary conditions the existence of only one relevant variable determining the crossover from infinite to finite size behavior is found as t 0 is approached despite that there are two characteristic lengths determining the scale of variation of the correlation function the validity of the general finite size scalling relations in terms of this variable is established

integrable deformations of a polygon abstract a new integrable mapping with a simple geometric interpretation is presented this mapping arises from the nonlinear superposition principle for the backlund transformations of some vector evolution equation