on the existence of solutions for a class of fourth order differential equations abstract we study the existence of the homoclinic solution of a class of nonlinear fourth order ordinary differential equations which involve particular cases of the extended fisher kolmogorov and swift hohenberg equations we address the case when the source term possesses a nonlinear dependence on the first order derivative of the solution the main result follows from an iterative method using the sequence of solutions obtained through the mountain pass technique in order to illustrate this method some numerical examples are shown

a modified quasi newton method for nonlinear equations abstract in this paper a modified quasi newton method is proposed for solving the nonlinear equation f x 0 which is based on a new quasi newton approach the usual quasi newton equation is b k 1 s k y k where s k x k 1 x k y k f x k 1 f x k the new quasi newton equation is b k 1 s k y k in which s k is based on the iterates x k 1 x k x k 1 and y k is based on the function values f x k 1 f x k f x k 1 the new quasi newton equation exploits additional information by assuming a quadratic relationship between the information from the last three iterates the modified quasi newton method is based on the new quasi newton equation and possess local superlinear convergence properties numerical experiments show that the modified quasi newton method is promising

a kantorovich type convergence analysis for the gauss newton method we present a semilocal kantorovich type convergence analysis for the gauss newton method which reduces to the wellknown newton kantorovich theorem for the newton method in a natural way additionnally a classification of the nonlinear regression problem into adequate and not adequate models is obtained

computing singular solutions to nonlinear analytic systems a method to generate an accurate approximation to a singular solution of a system of complex analytic equations is presented since manyreal systems extend naturally tocomplex analytic systems this porvides a method for generating approximations to singular solutions to real systems examples include systems of polynomials and systems made up of trigonometric exponential and polynomial terms the theorem on which the method is based is proven using results from several complex variables no special conditions on the derivatives of the system such as restrictions on the rank of the jacobian matrix at the solution are required the numerical method itself is developed from techniques of homotopy continuation and 1 dimensional quadrature a specific implementation is given and the results of numerical experiments in solving five test problems are presented

deflation algorithm for the multiple roots of a system of nonlinear equations abstract t ojika s watanabe and t mitsui in preparation have been developing a subroutine package naes nonlinear algebraic equations solver for the numerical solutions of the system of nonlinear equations an algorithm in the package termed the deflation algorithm for determining multiple roots for a system of nonlinear equations is presented and its effectiveness is shown by solving a numerical example

robustness of discrete systems under structured uncertainties we give a sufficient condition for the robust stability of the discrete system x k 1 f e x k where the perturbation e is bounded e u a necessary and sufficient condition is derived in the case where f is a positive or a negative matrix and u f the stability of interval matrices is also considered numerical examples are given for illustration

convergence rates for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root the convergence of the newton iterates to the root is linear rather than quadratic in this paper we give a detailed analysis of the linear convergence rates for several types of singular problems for some of these problems we describe modifications of newton s method which will restore quadratic convergence

a high order modified levenberg marquardt method for systems of nonlinear equations with fourth order convergence fan 2014 presented an accelerated modified levenberg marquardt method for nonlinear equations at every iteration the accelerated modified lm method computed not only a lm trial step but also an additional approximate lm step which employed a line search in this paper based on the accelerated modified lm method we compute the approximate lm step one more time at every iteration and obtain a high order accelerating modified levenberg marquardt method under the local error bound condition which is weaker than nonsingularity the convergence order of this new method is shown to be fourth a globally convergence is also given by the trust region technique numerical results show that the new method is efficient and could save many calculations of the jacobian

qualitative analysis of newton s flow the newton flow of a parameter dependent mapping is analysed in a neighbourhood of a singular point it is shown that the liapunov schmidt reduction process yields an invariant exponentially attracting manifold of the newton flow if the singular point is a fold then a normal form of the reduced 2 d newton flow is given moreover numerical experiments suggest that discrete versions of the normal form seem to be useful for an understanding of the performance of the ordinary newton method in a neighbourhood of the fold

approximate solution of the trust region problem by minimization over two dimensional subspaces the trust region problem minimization of a quadratic function subject to a spherical trust region constraint occurs in many optimization algorithms in a previous paper the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two dimensional trust region problem they showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem this paper reports computational results showing that the two dimensional minimization approach gives nearly optimal reductions in then dimension quadratic model over a wide range of test cases we also show that there is very little difference in efficiency and reliability between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization these results may encourage the application of similar approximate trust region techniques in other contexts

existence results for nonlinear elliptic equations with leray lions operator and dependence on the gradient the existence of positive solutions for nonlinear elliptic problems under dirichlet boundary condition is studied as well as the compactness and directness of the solution set the main novelties consist in the presence of a leray lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution our approach relies on the method of sub supersolution nonlinear regularity theory and strong maximum principle

tensor methods for large sparse systems of nonlinear equations this paper introduces tensor methods for solving large sparse systems of nonlinear equations tensor methods for nonlinear equations were developed in the context of solving small to medium sized dense problems they base each iteration on a quadratic model of the nonlinear equations where the second order term is selected so that the model requires no more derivative or function information per iteration than standard linear model based methods and hardly more storage or arithmetic operations per iteration computational experiments on small to medium sized problems have shown tensor methods to be considerably more efficient than standard newton based methods with a particularly large advantage on singular problems this paper considers the extension of this approach to solve large sparse problems the key issue considered is how to make efficient use of sparsity in forming and solving the tensor model problem at each iteration accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the jacobian whether the jacobian is nonsingular or singular we develop such an approach and based upon it an efficient tensor method for solving large sparse systems of nonlinear equations test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse newton based method in terms of iterations function evaluations and execution time 1998 the mathematical programming society inc published by elsevier science b v

a modified two steps levenberg marquardt method for nonlinear equations the modified levenberg marquardt mlm method to solve nonlinear equations was introduced by fan 2012 this method uses an addition of the levenberg marquardt step and an approximate lm step as the trial step at every iteration using a trust region technique the global and cubic convergence of the mlm method under the local error bound condition is proved fan 2012 recently fan proposed an accelerated mlm algorithm by using a line search strategy to generate a modified lm step and showed that the convergence rate of the algorithm is min 1 2 3 which results the cubic convergence for 1 fan 2014 in this paper by introducing an adaptive lm parameter for amlm algorithm we propose an efficient amlm algorithm the cubic convergence of the new algorithm is presented while numerical experiments show the new algorithm is promising

superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems we propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular compared to other techniques for computing singular solutions a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation based reformulations of the nonlinear complementarity problems reformulations of complementarity problems serve both as illustration of and motivation for our approach and one of them we consider in detail we show that the proposed method possesses local superlinear quadratic convergence under reasonable assumptions we further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent newton type algorithms for solving complementarity problems which are typically based on nonsmooth reformulations therefore our approach appears to be an interesting complement to the existing ones

interval finite element method for dynamic response of closed loop system with uncertain parameters in practical engineering it is difficult to obtain all possible solutions of dynamic responses with sharp bounds even if an optimum scheme is adopted where there are many uncertain parameters in this paper using the interval finite element ife method and precise time integration pti method we discuss the dynamic response of vibration control problem of structures with interval parameters with matrix perturbation theory and interval arithmetic the algorithm for estimating upper and lower bounds of dynamic response of the closed loop system is developed directly from the interval parameters two numerical examples are given to illustrate the application of the present method the example 1 is used to show the applicability of the present method the example 2 is used to show the validity of the present method by comparing the results with those obtained by the classical random perturbation method copyright 2006 john wiley sons ltd

positive solutions of quasilinear elliptic equations with exponential nonlinearity combined with convection term abstract we establish the existence of positive solutions for a nonlinear elliptic dirichlet problem in dimension n involving the n laplacian the nonlinearity considered depends on the gradient of the unknown function and an exponential term in such case variational methods cannot be applied our approach is based on approximation scheme where we consider a new class of normed spaces of finite dimension as a particular case we extended the result achieved by de araujo and montenegro 2016 for any n 2

local convergence analysis of tensor methods for nonlinear equations tensor methods for nonlinear equations base each iteration upon a standard linear model augmented by a low rank quadratic term that is selected in such a way that the mode is efficient to form store and solve these methods have been shown to be very efficient and robust computationally especially on problems where the jacobian matrix at the root has a small rank deficiency this paper analyzes the local convergence properties of two versions of tensor methods on problems where the jacobian matrix at the root has a null space of rank one both methods augment the standard linear model by a rank one quadratic term we show under mild conditions that the sequence of iterates generated by the tensor method based upon an ideal tensor model converges locally and two step q superlinearly to the solution with q order 3 2 and that the sequence of iterates generated by the tensor method based upon a practial tensor model converges locally and three step q superlinearly to the solution with q order 3 2 in the same situation it is known that standard methods converge linearly with constant converging to 1 2 hence tensor methods have theoretical advantages over standard methods our analysis also confirms that tensor methods converge at least quadratically on problems where the jacobian matrix at the root is nonsingular

an accelerated newton method for equations with semismooth jacobians and nonlinear complementarity problems we discuss local convergence of newton s method to a singular solution x of the nonlinear equations f x 0 for f mathbb r n rightarrow mathbb r n it is shown that an existing proof of griewank concerning linear convergence to a singular solution x from a starlike domain around x for f twice lipschitz continuously differentiable and x satisfying a particular regularity condition can be adapted to the case in which f is only strongly semismooth at the solution further newton s method can be accelerated to produce fast linear convergence to a singular solution by overrelaxing every second newton step these results are applied to a nonlinear equations reformulation of the nonlinear complementarity problem ncp whose derivative is strongly semismooth when the function f arising in the ncp is sufficiently smooth conditions on f are derived that ensure that the appropriate regularity conditions are satisfied for the nonlinear equations reformulation of the ncp at x

comparison of fully implicit and impes formulations for simulation of water injection in fractured and unfractured media in this work we compare the fully implicit fi and implicit pressure explicit saturation impes formulations for the simulation of water injection in fractured media the system of partial differential equations is discretized within the discrete fracture framework using a control volume method a unique feature of the methodology is that there is no need for the computation of matrix fracture transfer terms the non linear system of equations resulting from the fi formulation is solved with state of the art newton and tensor methods direct and krylov iterative methods are employed to solve the system resulting from the newton linearization the performance of the fi and impes formulations is compared with numerical testing results show that the contrast between matrix and fracture properties affects the performance of both impes and fi formulations and that the tensor method outperforms all the newton solvers for the near singular jacobian matrix copyright 2006 john wiley sons ltd

existence of positive solutions for nonlinear elliptic equations with convection terms abstract we prove the existence of positive solutions for the equation i 1 n x i a i x u u f x u u in under the dirichlet boundary condition where the essential point is the dependence of the terms of the elliptic equation on the solution u and its gradient u we develop an approach based on approximate solutions and on a new strong maximum principle

a levenberg marquardt algorithm with correction for singular system of nonlinear equations in this paper we propose a new lm algorithm for the singular system of nonlinear equations f x 0 where f x r n r n is continuously differentiable and f x is lipschitz continuous at every iteration not only a lm step but also a correction step are computed the lm parameter is taken as m email protected f email protected d where email protected 0 2 and m k is updated by the trust region technique under the local error bound condition which is weaker than nonsingularity we prove the convergence rate of the algorithm is min 2 email protected numerical results show that the new lm algorithm performs very well

interval uncertain method for multibody mechanical systems using chebyshev inclusion functions this study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on chebyshev inclusion functions the interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain but bounded parameters which only requires lower and upper bounds of uncertain parameters without having to know probability distributions a chebyshev inclusion function based on the truncated chebyshev series rather than the taylor inclusion function is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions to effectively handle the overestimation caused by the wrapping effect intrinsic to interval computations the mehler integral is used to evaluate the coefficients of chebyshev polynomials in the numerical implementation the multibody dynamics of mechanical systems are governed by index 3 differential algebraic equations daes including a combination of differential equations and algebraic equations responsible for the dynamics of the system subject to certain constraints the proposed interval method with chebyshev inclusion functions is applied to solve the daes in association with appropriate numerical solvers this study employs hht i3 as the numerical solver to transform the daes into a series of nonlinear algebraic equations at each integration time step which are solved further by using the newton raphson iterative method at the current time step two typical multibody dynamic systems with interval parameters the slider crank and double pendulum mechanisms are employed to demonstrate the effectiveness of the proposed methodology the results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect as cosine functions are incorporated to sharpen the range of non monotonic interval functions copyright 2013 john wiley sons ltd

robust partial quadratic eigenvalue assignment with time delay using the receptance and the system matrices abstract in this paper we consider the robust partial quadratic eigenvalue assignment problem in vibration by active feedback control based on the receptance measurements and the system matrices we propose an optimization method for the robust and minimum norm partial quadratic eigenvalue assignment problem we provide a new cost function and the closed loop eigenvalue sensitivity and the feedback norms can be minimized simultaneously our method is also extended to the case of time delay between measurements of state and actuation of control numerical tests demonstrate the effectiveness of our method

on the resolution of the generalized nonlinear complementarity problem minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem gncp defined on a polyhedral cone it is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems theoretical results that relate stationary points of the function that is minimized to the solutions of the gncp are presented perturbations of the gncp are also considered and results are obtained related to the resolution of gncps with very general assumptions on the data these theoretical results show that local methods for box constrained optimization applied to the associated problem are efficient tools for solving the gncp numerical experiments are presented that encourage the use of this approach

on precise integration method the numerical integration for the ordinary differential equations is extremely important for applications so far almost all the numerical integration methods apply the finite difference approximation even for a time invariant system the precise integration method pim solves the time step integration for the time invariant system first for such a problem precise integration gives a highly precise numerical result which approaches the full computer precision after the integration of time invariant system is solved various approximate methods can be applied to other problems such as time variant or nonlinear time integration the numerical integration of two point boundary value problems tpbvp is also very important in applications such as wave propagation optimal control structural mechanics electro magnetic wave guide problems etc the pim can also be applied to solve the tpbvp the tpbvp induced initial value problems such as the lyapunov differential equation the riccati differential equation and the kalman bucy filter equation etc can also be solved along the same way

the time response of structures with bounded parameters and interval initial conditions uncertainty plays an important role in the performance of structures in this paper we focus on the dynamic response of structures with bounded parameters and interval initial conditions and present a new method to determine the supremum and infimum of the time response the method is based on the vertex solution theorem for the first order deviation of the dynamic response from its central value and avoids interval extension problems present in current methods where the length of the interval increases significantly due to the intermediate calculations the method is more accurate than existing perturbation methods and provides tighter bounds on the response the approach neglects the second order terms in the equation of motion and care should be exercised when the parameter variations are large the other advantage of this method is its ability to solve problems with uncertainties in the initial conditions

tensor methods for unconstrained optimization using second derivatives a new type of method for unconstrained optimization called a tensor method is introduced it is related in its basic philosophy to the tensor methods for nonlinear equations for schnabel and frank siam j numer anal 21 1984 pp 815 843 but beyond that the methods have significant differences the tensor method for unconstrained optimization bases each iteration upon a fourth order model of the objective function this model consists of the quadratic portion of the taylor series plus low rank third and fourth order terms that cause the model to interpolate already calculated function and gradient values from one or more previous iterates this paper also shows that the costs of forming storing and solving the tensor model are not significantly more than these costs for a standard method based upon a quadratic taylor series model test results are presented for sets of problems where the hessian at the minimizer is nonsingular and where it is singular on all the test sets the tensor method

a higher order levenberg marquardt method for nonlinear equations in this paper we present a high order levenberg marquardt method for nonlinear equations at every iteration not only a lm step is computed but also two approximate lm steps are computed which use the previous calculated jacobian under the local error bound condition which is weaker than nonsingularity the new method has biquadratic convergence a globally convergent lm algorithm is also given by the trust region technique numerical results show that the new fourth order lm algorithm performs very well and could save many jacobian calculations especially for large scale problems

parameter perturbation method for dynamic responses of structures with uncertain but bounded parameters based on interval analysis the study was intended to evaluate the range of dynamic responses of structures with uncertain but bounded parameters by using the parameter perturbation method the uncertain parameters were modeled as an interval vector the first order perturbation quantities of responses of the perturbed system were obtained through the parameter perturbation method and then taking advantage of interval mathematics a new algorithm to estimate the response interval was presented comparisons between the parameter perturbation method and the probabilistic approach from mathematical proofs and numerical simulations were performed the numerical results are in agreement with the mathematical proofs the response range given by the parameter perturbation method encloses that obtained by the probabilistic approach the results also show good robustness of the proposed method

jacobian smoothing brown s method for ncp abstract jacobian smoothing brown s method for nonlinear complementarity problems ncp is studied in this paper this method is a generalization of classical brown s method it belongs to the class of jacobian smoothing methods for solving semismooth equations local convergence of the proposed method is proved in the case of a strictly complementary solution of ncp furthermore a locally convergent hybrid method for general ncp is introduced some numerical experiments are also presented

existence and uniqueness results for double phase problems with convection term abstract in this paper we consider quasilinear elliptic equations with double phase phenomena and a reaction term depending on the gradient under quite general assumptions on the convection term we prove the existence of a weak solution by applying the theory of pseudomonotone operators imposing some linear conditions on the gradient variable the uniqueness of the solution is obtained

positive solutions to nonlinear nonhomogeneous inclusion problems with dependence on the gradient abstract the goal of the paper is to study a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with the dirichlet boundary condition and a convection multivalued term an existence theorem for positive solutions of the problem is established by applying the method of subsolution supersolution together with truncation and comparison techniques

a modified brown algorithm for solving singular nonlinear systems with rank defects a modified brown algorithm for solving a class of singular nonlinear systems f x 0 where x f rn is presented this method is constructed by combining the discreted brown algorithm with the space transforming method the second order information of f x at a point is not required calculating which is different from the tensor method and the hoy s method the q quadratic convergence of this algorithm and some numerical examples are given as well

characterization and computation of generalized turning points a minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency applications to de

tensor methods for minimizing convex functions with holder continuous higher order derivatives in this paper we study p order methods for unconstrained minimization of convex functions that are p times differentiable p geq 2 with nu holder continuous p th derivatives we propose

an efficient two step trust region algorithm for exactly determined consistent systems of nonlinear equations abstract in this paper we propose a modified two step trust region algorithm for solving nonlinear systems two step trust region algorithms at every iteration use a trust region step and an approximate step by saving the jacobian matrix to have fewer computations we introduce a convex combination to modify the trust region subproblems along with an additional criterion to verify whether the first step is accepted or not we establish global and quadratic convergence of the algorithm under some mild assumptions numerical results show that the modified algorithm is efficient and promising

finding a multiple zero by transformations and newton like methods many methods for finding a multiple zero x of a function f are based on transforming f to a function t for which x is a simple zero we give and analyze a general transformation which i

convergence property of interval matrices and interval polynomials in association with robust control system design and analysis the hurwitz property of interval matrices and interval polynomials has recently been actively investigated however its discrete counterpart the convergence property has seemingly not been much discussed in this paper this property is studied in comparison with the hurwitz counterpart some conditions under which interval matrices or interval polynomials are convergent are derived

modal optimal control procedure for near defective systems abstract this study discusses a modal optimal control procedure for defective systems with repeated eigenvalues from the view point of mathematics although near defective close eigenvalues are distinct the characteristic of the system is also defective therefore we have to transform near defective systems into the defective one and then modal optimal control procedure for the defective systems can be extended to deal with the corresponding problems for near defective systems with close eigenvalues because of the defective characteristic of the system we have to use an invariant sub space recursive method with numerical stability to calculate the generalized modes of the defective and near defective systems the potter s approach is extended to solve the riccati equations in the generalized model subspace of the defective system because the order of the jordan block matrix of the defective eigenvalues m is much smaller than that of the state matrix n i e m n the present modal optimal control procedure is very simple and reduces the computing effort for the complex system with large number of degrees of freedom a numerical example is given to illustrate and verify the validity of the procedure

convergence of newton like methods for singular operator equations using outer inverses we present a semilocal kantorovich type analysis for newton like methods for singular operator equations using outer inverses we establish sharp generalizations of the kantorovich theory and the mysovskii theory for operator equations when the derivative is not necessarily invertible the results reduce in the case of an invertible derivative to well known theorems of kantorovich and mysovskii with no additional assumptions unlike earlier theorems which impose strong conditions the strategy of the analysis is based on banach type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses to a given bounded linear operator admits selections that behave like bounded linear inverses in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator we give two examples to illustrate our results and compare them with earlier results and another numerical example to relate our results to computational issues

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

existence and multiplicity of solutions for quasilinear nonhomogeneous problems an orlicz sobolev space setting abstract we study the boundary value problem div log 1 u q u p 2 u f u in u 0 on where is a bounded domain in r n with smooth boundary we distinguish the cases where either f u u p 2 u u r 2 u or f u u p 2 u u r 2 u with p q 1 p q min n r and r n p n p n p in the first case we show the existence of infinitely many weak solutions for any 0 in the second case we prove the existence of a nontrivial weak solution if is sufficiently large our approach relies on adequate variational methods in orlicz sobolev spaces

quantitative measures of modal controllability and observability in vibration control of defective and near defective systems in this paper we focus on the control problem of defective systems with repeated eigenvalues using the singular value decomposition of input matrix b and output matrix c in the modal subspace the quantitative measures of modal controllability and observability of defective systems are discussed for a near defective system with close eigenvalues we first transform it into a defective one and then apply the same method to deal with near defective systems numerical examples for a general damping system and an airfoil are given to illustrate the applications of the present method

a compact limited memory method for large scale unconstrained optimization abstract a compact limited memory method for solving large scale unconstrained optimization problems is proposed the compact representation of the quasi newton updating matrix is derived to the use in the form of limited memory update in which the vector y k is replaced by a modified vector y k so that more available information about the function can be employed to increase the accuracy of hessian approximations the global convergence of the proposed method is proved numerical tests on commonly used large scale test problems indicate that the proposed compact limited memory method is competitive and efficient

on high order model regularization for constrained optimization in two recent papers regularization methods based on taylor polynomial models for minimization were proposed that only rely on holder conditions on the higher order employed derivatives grapiglia and nesterov considered cubic regularization with a sufficient descent condition that uses the current gradient and resembles the classical armijo s criterion cartis gould and toint used taylor models with arbitrary order regularization and defined methods that tackle convex constraints employing the descent criterion that compares actual reduction with predicted reduction the methods presented in this paper consider general not necessarily taylor models of arbitrary order employ a very mild descent criterion and handle general nonnecessarily convex constraints complexity results are compatible with the ones presented in the papers mentioned above

design optimization for dynamic response of vibration mechanical system with uncertain parameters using convex model the concept of uncertainty plays an important role in the design of practical mechanical system the most common methods for solving uncertainty problems are to model the parameters as a random vector a natural way to handle the randomness is to admit that a given probability density function represents the uncertainty distribution however the drawback of this approach is that the probability distribution is difficult to obtain in this paper we use the non probabilistic convex model to deal with the uncertain parameters in which there is no need for probability density functions using the convex model theory a new method to optimize the dynamic response of mechanical system with uncertain parameters is derived because the uncertain parameters can be selected as the optimization parameters the present method can provide more information about the optimization results than those obtained by the deterministic optimization the present method is implemented for a torsional vibration system the numerical results show that the method is effective

a shamanskii like acceleration scheme for nonlinear equations at singular roots on applique une variante de la methode de shamanskii pour realiser une methode a convergence superlineaire de resolution d une classe d equations non lineaires ayant une derivee de frechet singuliere a la racine

parallel quasi newton methods for unconstrained optimization we discuss methods for solving the unconstrained optimization problem on parallel computers when the number of variables is sufficiently small that quasi newton methods can be used we concentrate mainly but not exclusively on problems where function evaluation is expensive first we discuss ways to parallelize both the function evaluation costs and the linear algebra calculations in the standard sequential secant method the bfgs method then we discuss new methods that are appropriate when there are enough processors to evaluate the function gradient and part but not all of the hessian at each iteration we develop new algorithms that utilize this information and analyze their convergence properties we present computational experiments showing that they are superior to parallelization either the bfgs methods or newton s method under our assumptions on the number of processors and cost of function evaluation finally we discuss ways to effectively utilize the gradient values at unsuccessful trial points that are available in our parallel methods and also in some sequential software packages

a new acceleration method for newton s method at singular points when newton s method is used to find a root of a map from a banach space into itself and the derivative is singular at that root convergence of the newton sequence is in general linear in this paper we give a modification of newton s method that under certain conditions converges superlinearly our method is applicable under more general conditions than other techniques in particular it may be used for certain quadratic problems and problems in which the dimension of the null space of the derivative is larger than one

convergence acceleration for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root linear convergence of the newton sequence to the root is the best

discretization of implicit odes for singular root finding problems this paper addresses the use of dynamical system theory to tackle singular root finding problems the use of continuous time methods leads to implicit differential systems when applied to singular nonlinear equations the analysis is based on a taxonomy of singularities and uses previous stability results proved in the context of quasilinear implicit odes the proposed approach provides a framework for the systematic formulation of quadratically convergent iterations to singular roots the scope of the work includes also the introduction of discrete time analysis techniques for singular problems which are based on continuous time stability and numerical stability some numerical experiments illustrate the applicability of the proposed techniques

geometrically isolated nonisolated solutions and their approximation a solution x x 0 of f x 0 is said to be isolated if the frechet derivative f x 0 is nonsingular it is said to be geometrically isolated if no other solution is in x x 0 leqq rho for some rho 0 isolated solutions are always geometrically isolated sufficient conditions are obtained to insure that a nonisolated solution is also geometrically isolated we then study the application of approximation methods in the general form f h x h 0 to approximate nonisolated solutions which are geometrically isolated under strong consistency conditions the results are somewhat negative the approximations may have an even number including zero or an odd number of roots near x 0 depending upon the multiplicity cf x 0 as a root if the accuracy is o h p and the multiplicity is n then the approximations have error o h p n the relation of these results to limit and bifurcation points is discussed briefly

on a quasilinear elliptic problem with convection term and nonlinear boundary condition abstract the first part of this paper deals with existence of solutions to the quasilinear elliptic problem p div a x u f x u u in a x u g x u u p 2 u on involving a general nonhomogeneous differential operator namely div a and caratheodory functions f r r n r and g r r under appropriate conditions on the perturbations we show that p possesses a bounded solution in the second part we consider the special case when div a is the p q laplacian with a parameter 0 and study the asymptotic behavior of solutions as goes to zero or to infinity a uniqueness result is also provided

analysis of newton s method at irregular singularities recent work on newton s method at singularities of the jacobian has established linear convergence under certain regularity assumptions here newton s method is analyzed in the neighborhood of irregular singularities which include all minimizers at which the hessian has a one dimensional null space depending on a parameter specific to the underlying optimization problem or system of simultaneous equations newton s method is found either to converge with a limiting ratio of about frac 2 3 or to diverge from arbitrarily close starting points or to behave in a certain sense chaotically

local dynamics of a family of quasilinear odes with folded singular equilibria abstract the local behavior of newton s method near singular equilibria can be addressed via normal form theory in a continuous time framework in this paper we analyze the dynamics of a recently obtained normal form for the continuous newton method near so called folded equilibrium points these can be classified in terms of a morse index which characterizes the nearby structure of pseudoequilibria and impasse points extremal values of this index yield the simplest cases in which all surrounding singularities are either forward or backward impasse points in more involved isotropic cases the behavior of trajectories will be distinguished by two types of separatrices one of them undergoing singularity crossing phenomena this approach leads to a complete local description of the different attraction domains which can be displayed at singular solutions

fixed point quasi newton methods this article studies iterative methods defined by x k 1 phi x k e k where x k in mathbb r n and e k lie on a space of parameters sufficient conditions are established for local convergence and for convergence at an ideal linear or superlinear rate a theory of least change secant update methods for this class of processes is developed several examples are given showing a wide range of applications of the new theory key words quasi

convergence analysis of a variant of the newton method for solving nonlinear equations the paper presents a convergence analysis of a modified newton method for solving nonlinear systems of equations the convergence results show that this method converges cubically in the nonsingular case and linearly with the rate 3 8 under some sufficient conditions when the jacobian is singular at the root the convergence theory is used to analyze the convergence behavior when the modified newton method is applied to a nonsymmetric algebraic riccati equation arising in transport theory numerical experiment confirms the theoretical results

root locus approach to the stability analysis of interval matrices a new approach called the root locus approach is developed for the stability analysis of perturbed matrices the basic idea of this approach is to ensure that the root loci of a continuously perturbed matrix remain in the open left half complex s plane based on this approach new techniques are presented to analyse the stability of interval matrices examples are given to demonstrate the merit of the proposed theorems

a shamanskii like self adaptive levenberg marquardt method for nonlinear equations abstract in this paper we are concerned with the shamanskii like self adaptive levenberg marquardt methods for nonlinear equations we consider two choices of levenberg marquardt parameter one of them is the standard self adaptive levenberg marquardt parameter the other is nonmonotone self adaptive levenberg marquardt parameter by using the nonmonotone technique of grippo lampariello and lucidi under the error bound condition which is weaker than nonsingularity we show that the shamanskii like self adaptive levenberg marquardt methods converge with q order m 1 numerical experiments show the new algorithms are efficient and promising

tensor methods for nonlinear equations a new class of methods for solving systems of nonlinear equations called tensor methods is introduced tensor methods are general purpose methods intended especially for problems where the jacobian matrix at the solution is singular or ill conditioned they base each iteration on a quadratic model of the nonlinear function the standard linear model augmented by a simple second order term the second order term is selected so that the model interpolates function values from several previous iterations as well as the current function value and jacobian the tensor method requires no more function and derivative information per iteration and hardly more storage or arithmetic per iteration than a standard method based on newton s method in extensive computational tests a tensor algorithm is significantly more efficient than a similar algorithm based on the standard linear model both on standard nonsingular test problems and on problems where the jacobian at the solution is singular

computing isolated singular solutions of polynomial systems case of breadth one we present a symbolic numeric method to refine an approximate isolated singular solution hat mathbf x hat x 1 ldots hat x n of a polynomial system f f 1 ldots f n when the jacobian matrix of f evaluated at hat mathbf x has corank one approximately our new approach is based on the regularized newton iteration and the computation of differential conditions satisfied at the approximate singular solution the size of matrices involved in our algorithm is bounded by n times n the algorithm converges quadratically if hat mathbf x is close to the isolated exact singular solution

interval static displacement analysis for structures with interval parameters in this paper a new method to solve the uncertain static displacement problem of structures with interval parameters is presented it is difficult to obtain all possible solutions with sharp bounds even if an optimum scheme is adopted when there are many uncertain parameters with the interval mathematics the interval finite element equation is developed based on the perturbation and the interval extension the upper and lower bounds of the static displacements are obtained in which the sharp bounds are guaranteed by the interval calculation operators two numerical examples a box cantilever beam and an automobile frame are given to illustrate the validity of the present method copyright 2001 john wiley sons ltd

on the local classification of smooth maps induced by newton s method abstract the classification of germs of smooth maps f r n r n induced by the continuous newton method f x x f x is addressed in this paper this classification problem is shown to rely on an equivalence notion located between right and contact equivalences driving the classification problem from the setting of quasilinear odes to the singularity theory framework one dimensional problems and regular n dimensional cases are easily characterized and normal forms for them are given folded zeros in r n display a much richer behavior an invariant and a preliminary normal form are derived for these cases

tensor methods for large sparse nonlinear least squares problems this paper introduces tensor methods for solving large sparse nonlinear least squares problems where the jacobian either is analytically available or is computed by finite difference approximations tensor methods have been shown to have very good computational performance for small to medium sized dense nonlinear least squares problems in this paper we consider the application of tensor methods to large sparse nonlinear least squares problems where a sparse factorization of the jacobian can be stored this involves an entirely new way of solving the tensor model that is efficient for sparse problems a number of interesting linear algebraic implementation issues are addressed the test results of the tensor method applied to a set of sparse nonlinear least squares problems compared with those of the standard gauss newton method reveal that the tensor method is significantly more robust and efficient than the standard gauss newton method

sublinear convergence of the chord method at singular points we give sufficient conditions for convergence of the chord method for a class of singular problems the rate of convergence is sublinear

an example of irregular convergence in some constrained optimization methods that use the projected hessian in this paper we give examples illustrating the behavior of the coleman conn horizontal vertical method and of successive quadratic programming with a hessian approximation exact on the tangent space of the constraints one example shows that these methods in general are not one step superlinearly convergent

properties and numerical performance of quasi newton methods with modified quasi newton equations quasi newton qn equation plays a core role in contemporary nonlinear optimization the usual qn equation employs only the gradients but ignores the available function value information in this paper we derive a class of modified qn equations with a vector parameter which use both available gradient and function value information the modified quasi newton methods maintain most properties of the usual quasi newton methods meanwhile they achieve a higher order accuracy in approximating the second order curvature of the problem functions than the usual ones do numerical experiments are reported which support the theoretical analyses and show the advantages of the modified qn methods over the usual ones

verified error bounds for isolated singular solutions of polynomial systems case of breadth one in this paper we describe how to improve the performance of the symbolic numeric method in 19 20 for computing the multiplicity structure and refining approximate isolated singular solutions in the breadth one case by introducing a parameterized deflated system with smoothing parameters we generalize the algorithm in 33 to compute verified error bounds such that a slightly perturbed polynomial system is guaranteed to have a breadth one multiple root within the computed bounds

stability robustness analysis of digital control systems in state space models the stability robustness of discrete linear time invariant systems in state space models is analysed based on a root locus approach and from the stable eigenvalue viewpoint fundamental criteria for testing the stability robustness of autonomous systems are derived and applied to the robustness analysis of multivariable feedback systems both the norm bound and the element bounds for the allowed perturbations are obtained a stability robustness index is denned which is useful both for the analysis and synthesis of control systems

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

a note on the levenberg marquardt parameter abstract in c ma l jiang some research on levenberg marquardt method for the nonlinear equations appl math comput 184 2007 1032 1040 the lm parameter at the kth iteration is chosen as k f k 2 1 j k t f k 2 where f is the residual function j is the jacobi of f and 0 1 is a constant in this note we point out that the lm parameter can be any combination of f k 2 and j k t f k 2 provided it is positive furthermore we give a more general choice of the lm parameter and show that the lm method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity

newton s method with deflation for isolated singularities of polynomial systems we present a modification of newton s method to restore quadratic convergence for isolated singular solutions of polynomial systems our method is symbolic numeric we produce a new polynomial system which has the original multiple solution as a regular root using standard bases a tool for the symbolic computation of multiplicities we show that the number of deflation stages is bounded by the multiplicity of the isolated root our implementation performs well on a large class of applications

dealing with singularities in nonlinear unconstrained optimization we propose a new trust region based optimization algorithm for solving unconstrained nonlinear problems whose second derivatives matrix is singular at a local solution we give a theoretical characterization of the singularity in this context and we propose an iterative procedure which allows to identify a singularity in the objective function during the course of the optimization algorithm and artificially adds curvature to the objective function numerical tests are performed on a set of unconstrained nonlinear problems both singular and non singular results illustrate the significant performance improvement compared to classical trust region and filter algorithms proposed in the literature the approach is also shown to be competitive with tensor methods in terms of efficiency while reaching a higher level of robustness

multiple existence results of solutions for the neumann problems via super and sub solutions by variational methods we provide existence results of multiple solutions for quasilinear elliptic equations under the neumann boundary condition our main result shows the existence of two constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the unknown function the studied equations contain the p laplacian problems as a special case moreover we give a result concerning a local minimizer in c1 versus w1 p auxiliary results of independent interest are also obtained a density property for the space w1 p a strong maximum principle of zhang s type and a moser s iteration scheme depending on a parameter

a corrected levenberg marquardt algorithm with a nonmonotone line search for the system of nonlinear equations in this paper we propose a corrected levenberg marquardt method for the system of nonlinear equations in which not only an l m step and a corrected step are computed at every iteration but also a nonmonotone line search to find a new iteration point will be performed if a trial step is not accepted to ensure the global convergence of the new method a new nonmonotone line search technique is introduced for the merit function the cubic convergence of the new method is proved under the local error bound condition which is weaker than nonsingularity some numerical results are reported which shows that the algorithm is quite effective

nonlinear programming via an exact penalty function asymptotic analysis in this paper we consider the final stage of a global method to solve the nonlinear programming problem we prove 2 step superlinear convergence in the process of analyzing this asymptotic behavior we compare our method theoretically to the popular successive quadratic programming approach

three steps modified levenberg marquardt method with a new line search for systems of nonlinear equations three steps modified levenberg marquardt method for nonlinear equations was introduced by yang 2013 this method uses the addition of the levenberg marquardt lm step and two approximate lm steps as the trial step at every iteration using trust region technique the global and biquadratic convergence of the method is proved by yang the main aim of this paper is to introduce a new line search strategy while investigating the convergence properties of the method with this line search technique since the search direction of yang method may be not a descent direction standard line searches cannot be used directly in this paper we propose a new nonmonotone third order armijo type line search technique which guarantees the global convergence of this method while we use an adaptive lm parameter it is proved that the convergence order of the new method is biquadratic numerical results show the new algorithm is efficient and promising