improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

the study of existence of equilibria for generalized games without lower semicontinuity in locally topological vector spaces the aim of this paper is to establish general existence results of maximal elements for l majorized mappings which are in turn used to establish the general existence theorems of equilibria for generalized games resp abstract economics without lower semicontinuity for both constraint and preference mappings correspondences and in which strategic resp commodity spaces may not be compact the set of players resp agents may not be countable and underlying spaces are either finite or infinite dimensional locally topological vector spaces our results unify or improve corresponding results in the literature

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

minimizing neumann fundamental tones of triangles an optimal poincare inequality abstract the first nonzero eigenvalue of the neumann laplacian is shown to be minimal for the degenerate acute isosceles triangle among all triangles of given diameter hence an optimal poincare inequality for triangles is derived the proof relies on symmetry of the neumann fundamental mode for isosceles triangles with aperture less than 3 antisymmetry is proved for apertures greater than 3

study on gas kinetic unified algorithm for flows from rarefied transition to continuum the modified bgk equation adapted to various flow regimes can be presented by the aid of the basic characteristics on molecular movement and collision approaching to equilibrium the discrete velocity ordinate method is developed and applied to the velocity distribution function to remove its continuous dependency on the velocity space and then the velocity distribution function equation is cast into hyperbolic conservation law form with nonlinear source terms based on the unsteady time splitting method and the non oscillatory containing no free parameters and dissipative nnd scheme the gas kinetic finite difference second order scheme is constructed for the computation of the discrete velocity distribution functions the mathematical model on the interaction of molecules with solid surface is studied and used in the numerical method four types of numerical quadrature rules such as the modified gauss hermite formula the composite newton cotes integration method the gauss legendre numerical quadrature rule and the golden section number theoretic integral method are developed and applied to the discretized velocity space to evaluate the macroscopic flow parameters at each point in the physical space as a result a unified simplified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime based on analyzing the inner parallel degree of the unified algorithm the parallel strategy adapted to the gas kinetic numerical algorithm is studied and then the hpf parallel processing software for the unified algorithm is developed to test the present method the one dimensional shock tube problems the flows past two dimensional circular cylinder and the flows around three dimensional sphere and spacecraft shape with various knudsen numbers are simulated the computational results are found in high resolution of the flow fields and good agreement with the theoretical dsmc n s and experimental results

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

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

an extension of gander s result for quadratic equations in the study of iterative methods with high order of convergence gander provides a general expression for iterative methods with order of convergence at least three in the scalar case taking into account an extension of this result we define a family of iterations in banach spaces with r order of convergence at least four for quadratic equations

improved iterative methods for solving nonlinear equations in this paper we use the decomposition technique of noor m aslam noor numerical analysis and optimization lecture notes comsats institute of information technology islamabad pakistan 2006 to suggest and analyze some new iterative methods for solving nonlinear equations our method of developing these iterative methods is very simple as compared with other methods several numerical examples are given to illustrate the efficiency and performance of the new method our method can be considered as an improvement of the existing methods

efficient iterative methods applied to the solution of transonic flows we investigate the use of an inexact newton s method to solve the potential equations in the transonic regime as a test case we solve the two dimensional steady transonic small disturbance equation approximate factorization adi techniques have traditionally been employed for implicit solutions of this nonlinear equation instead we apply newton s method using an exact analytical determination of the jacobian with preconditioned conjugate gradient like iterative solvers for solution of the linear systems in each newton iteration two iterative solvers are tested a block s step version of the classical orthomin k algorithm called orthogonal s step orthomin osomin and the well known gmres method the preconditioner is a vectorizable and parallelizable version of incomplete lu ilu factorization efficiency of the newton iterative method on vector and parallel computer architectures is the main issue addressed in vectorized tests on a single processor of the cray c 90 the performance of newton osomin is superior to newton gmres and a more traditional monotone af adi method maf for a variety of transonic mach numbers and mesh sizes newton gmres is superior to maf for some cases the parallel performance of the newton method is also found to be very good on multiple processors of the cray c 90 and on the massively parallel thinking machine cm 5 where very fast execution rates up to 9 gflops are found for large problems

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

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

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

a selection theorem for quasi lower semicontinuous mappings in hyperconvex spaces we prove a continuous selection theorem for quasi lower semicontinuous mappings with values that are closed sub admissible subsets of a hyperconvex metric space and apply this result to obtain fixed point theorems in these spaces

nonconforming fems for an optimal design problem some optimal design problems in topology optimization eventually lead to a degenerate convex minimization problem e v int omega w nabla v dx int omega f v dx for v in h 0 1 omega with possibly multiple minimizers u but with a unique stress sigma dw nabla u this paper proposes the discrete raviart thomas mixed finite element method drt mfem and establishes its equivalence with the crouzeix raviart nonconforming finite element method the convergence analysis combines the a priori convergence rate of the conforming fem with the efficient a posterior error control of mfem numerical experiments provide empirical evidence that the proposed drt mfem overcomes the reliability efficiency gap for the first time

modified families of newton halley and chebyshev methods this paper presents new families of newton type iterative methods newton halley and chebyshev methods for finding simple zero of univariate non linear equation permitting f x 0 in the vicinity of the root newton type iterative methods have well known geometric interpretation and admit their geometric derivation from a parabola these algorithms are comparable to the well known powerful classical methods of newton halley and chebyshev respectively and these can be seen as special cases of these families the efficiency of the presented methods is demonstrated by numerical examples

hyperconvexity and nonexpansive multifunctions it is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection this implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps the result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex

remarks on approximate fixed points abstract a mapping f from a topological space x to a metric space m d is said to be r continuous r 0 if for any x x there exists a neighborhood u x of x such that diam f u x r in 1961 victor klee proved that an r continuous self mapping of a compact convex subset of a banach space always has a invariant point for any r recently h x phu extended klee s result to mappings that are almost r continuous in this note we extend phu s result to compact hyperconvex spaces and to geodesically bounded complete r trees we also extend klee s stability result to certain geodesic spaces

approximate selection theorems and their applications abstract we prove three approximate selection theorems and give an improved version of the michael selection theorem as their applications new fixed point theorems and equilibrium theorems for generalized games are established

approximate selections fixed points almost fixed points of multivalued mapping and generalized quasi variational inequalities in h spaces a fixed point theorem for continuous multivalued mapping with closed h convex values is analyzed using the almost fixed point theorem new approximate selection theorems a new almost fixed point theorem for a lower semicontinuous multivalued mapping and a new existence theorem for solutions of a generalized quasi variational inequality in h spaces are discussed

michael s selection theorem under an assumption weaker than lower semicontinuous in h spaces a new continuous selection theorem is proved which unifies and generalizes some known results

numerical solution of the scalar double well problem allowing microstructure the direct numerical solution of a non convex variational problem p typically faces the difficulty of the finite element approximation of rapid oscillations although the oscillatory discrete minimisers are properly related to corresponding young measures and describe real physical phenomena they are costly and difficult to compute in this work we treat the scalar double well problem by numerical solution of the relaxed problem rp leading to a degenerate convex minimisation problem the problem rp has a minimiser u and a related stress field dw u which is known to coincide with the stress field obtained by solving p in a generalised sense involving young measures if u h is a finite element solution h dw u h is the related discrete stress field we prove a priori and a posteriori estimates for h in l 4 3 and weaker weighted estimates for u u h the a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments

two level additive schwarz preconditioners for nonconforming finite element methods two level additive schwarz preconditioners are developed for the nonconforming p1 finite element approximation of scalar second order symmetric positive definite elliptic boundary value problems the morley finite element approximation of the biharmonic equation and the divergence free nonconforming p1 finite element approximation of the stationary stokes equations the condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap

new results on equilibria of abstract fuzzy economies abstract in this paper we consider the model of an abstract fuzzy economy and we obtain new equilibrium theorems for the situations when the involved correspondences satisfy properties which are weaker than upper semicontinuity in our case the constraint or preference correspondences are almost weakly upper semicontinuous or have the e upper semicontinuous selection property the obtained results either generalize or improve corresponding ones present in the literature

guaranteed lower eigenvalue bounds for the biharmonic equation the computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate s vibrations this paper shows that the eigenvalue provided by the nonconforming morley finite element analysis which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense is not always a lower bound a fully explicit error analysis of the morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound this paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower order terms

a convergent adaptive finite element method for an optimal design problem the optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation it eventually leads to a degenerate convex minimisation problem the numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives dw d u even fine a posteriori error estimates still suffer from the reliability efficiency gap however it motivates a simple edge based adaptive mesh refining algorithm afem that is not a priori guaranteed to refine everywhere its convergence proof is therefore based on energy estimates and some refined convexity control numerical experiments illustrate even nearly optimal convergence rates of the proposed afem

best approximation coincidence and fixed point theorems for quasi lower semicontinuous set valued maps in hyperconvex metric spaces abstract suppose x is a compact admissible subset of a hyperconvex metric spaces m and suppose f x m is a quasi lower semicontinuous set valued map whose values are nonempty admissible suppose also g x x is a continuous onto quasi convex set valued map with compact admissible values then there exists an x 0 x such that d g x 0 f x 0 inf x x d x f x 0 as applications we give some coincidence and fixed point results for weakly inward set valued maps our results generalize some well known results in literature

a further generalization of yannelis prabhakar s continuous selection theorem and its applications abstract in this article we first prove one improved version of the yannelis prabhakar continuous selection theorem and next as its applications a fixed point theorem in noncompact product spaces a nonempty intersection theorem some existence theorems of solutions for the generalized quasi variational inequalities and some equilibrium existence theorems for the abstract economies are given

on the convergence of adaptive nonconforming finite element methods for a class of convex variational problems we formulate and analyze an adaptive nonconforming finite element method for the solution of convex variational problems the class of minimization problems we admit includes highly singular problems for which no euler lagrange equation or inequality is available as a consequence our arguments only use the structure of the energy functional we are nevertheless able to prove convergence of an adaptive algorithm using even refinement indicators that are not reliable error indicators

on the convergence of finite element approximations of a relaxed variational problem the accuracy of finite element approximations to a convex but not strictly convex variational problem is considered convergence is proved for a finite element approximation of a particular vector field related to the solution in a special one dimensional case 0 h convergence is shown for a piecewise linear approximation of the derivative h denotes the size of each element domain numerical results are also presented for this one dimensional case

on a two step optimal steffensen type method relaxed local and semi local convergence analysis and dynamical stability abstract in this paper we study on the local and semi local convergence and dynamic of an iterative optimal two step steffensen s method we study local convergence by using lipschitz conditions and continuity of differentiable function f we also obtain semi local convergence by using majorizing sequences by this approach we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions moreover we analyze the stability of the method via julia sets and parameter planes that is a novelty for steffensen type methods we extend this method for obtaining the sign function of a matrix numerical examples are given to confirm our theoretical results

on the existence of equilibrium for abstract economies abstract in this paper several existence theorems for abstract economies have been established they are natural modifications strict generalizations when strategy spaces are metrizable of the main results of yannelis and prabhakar 23 tulcea 22 and ding kim and tan 10 for infinite dimensional casas for the finite dimensional case we show that the topological conditions on correspondences in shafer and sonnenschein s result 20 can be much weaker along the proofs several propositions are established which are of independent interest we also show that many existence results in fixed point theory maximization theory and generalized quasi variational inequalities can be encompassed

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner