Newton Generalized Hessenberg method for solving nonlinear systems of equations

In this paper, we give and analyze a Finite Difference version of the Generalized Hessenberg (FDGH) method. The obtained results show that applying this method in solving a linear system is equivalent to applying the Generalized Hessenberg method to a perturbed system. The finite difference version of the Generalized Hessenberg method is used in the context of solving nonlinear systems of equations using an inexact Newton method. The local convergence of the finite difference versions of the Newton Generalized Hessenberg method is studied. We obtain theoretical results that generalize those obtained for Newton-Arnoldi and Newton-GMRES methods. Numerical examples are given in order to compare the performances of the finite difference versions of the Newton-GMRES and Newton-CMRH methods. This revised version was published online in June 2006 with corrections to the Cover Date.     

A new smoothing Newton method for solving constrained nonlinear equations

In this paper, a new smoothing Newton method is proposed for solving constrained nonlinear equations. We first transform the constrained nonlinear equations to a system of semismooth equations by using the so-called absolute value function of the slack variables, and then present a new smoothing Newton method for solving the semismooth equations by constructing a new smoothing approximation function. This new method is globally and quadratically convergent. It needs to solve only one system of unconstrained equations and to perform one line search at each iteration. Numerical results show that the new algorithm works quite well.     

Globally convergent algorithm for solving nonlinear equations

A general iterative method is proposed for finding the maximal rootx _max of a one-variable equation in a given interval. The method generates a monotone-decreasing sequence of points converging tox _max or demonstrates the nonexistence of a real root. It is globally convergent. A concrete realization of the general algorithm is also given and is shown to be locally quadratically convergent. Computational experience obtained for eight test problems indicates that the new method is comparable to known methods claiming global convergence.     

Quasi-Newton ABS methods for solving nonlinear algebraic systems of equations

This paper is concerned with the solution of nonlinear algebraic systems of equations. For this problem, we suggest new methods, which are combinations of the nonlinear ABS methods and quasi-Newton methods. Extensive numerical experiments compare particular algorithms and show the efficiency of the proposed methods.The authors are grateful to Professors C. G. Broyden and E. Spedicato for many helpful discussions.     

A BFGS algorithm for solving symmetric nonlinear equations

In this article, we propose a BFGS method for solving symmetric nonlinear equations. The presented method possesses some favourable properties: (a) the generated sequence of iterates is norm descent; (b) the generated sequence of the quasi-Newton matrix is positive definite and (c) this method possesses the global convergence and superlinear convergence. Numerical results show that the presented method is interesting.     

A modified Newton method for solving non-symmetric algebraic Riccati equations arising in transport theory

Liang Bao ^{The non-symmetric algebraic Riccati equation arising in transporttheory can be rewritten as a vector equation and the minimalpositive solution of the non-symmetric algebraic Riccati equationcan be obtained by solving the vector equation. In this paper,we apply the modified Newton method to solve the vector equation.Some convergence results are presented. Numerical tests showthat the modified Newton method is feasible and effective, andoutperforms the Newton method.}     

An algebraic method exactly solving two high-dimensional nonlinear evolution equations

An algebraic method is applied to construct soliton solutions, doubly periodic solutions and a range of other solutions of physical interest for two high-dimensional nonlinear evolution equations. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solutions according to some parameters.     

谱HS投影算法求解非线性单调方程组

借助谱梯度法和HS共轭梯度法的结构, 建立一种求解非线性单调方程组问题的谱HS投影算法. 该算法继承了谱梯度法和共轭梯度法储存量小和计算简单的特征, 且不需要任何导数信息, 因此它适应于求解大规模非光滑的非线性单调方程组问题. 在适当的条件下, 证明了该算法的收敛性, 并通过数值实验表明了该算法的有效性.     

Maslov dequantization and the homotopy method for solving systems of nonlinear algebraic equations

The Maslov dequantization allows one to interpret the classical Gräffe-Lobachevski method for calculating the roots of polynomials in dimension one as a homotopy procedure for solving a certain system of tropical equations. As an extension of this analogy to systems of n algebraic equations in dimension n, we introduce a tropical system of equations whose solution defines the structure and initial iterations of the homotopy method for calculating all complex roots of a given algebraic system. This method combines the completeness and the rigor of the algebraicgeometrical analysis of roots with the simplicity and the convenience of its implementation, which is typical of local numerical algorithms.     

A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations

In this paper, a modified nonmonotone BFGS algorithm is developed for solving a smooth system of nonlinear equations. Different from the existent techniques of nonmonotone line search, the value of an algorithmic parameter controlling the magnitude of nonmonotonicity is updated at each iteration by the known information of the system of nonlinear equations such that the numerical performance of the developed algorithm is improved. Under some suitable assumptions, the global convergence of the algorithm is established for solving a generic nonlinear system of equations. Implementing the algorithm to solve some benchmark test problems, the obtained numerical results demonstrate that it is more effective than some similar algorithms available in the literature.     

Convergence of a generalized Newton and an inexact generalized Newton algorithms for solving nonlinear equations with nondifferentiable terms

In this paper, we consider two versions of the Newton-type method for solving a nonlinear equations with nondifferentiable terms, which uses as iteration matrices, any matrix from B-differential of semismooth terms. Local and global convergence theorems for the generalized Newton and inexact generalized Newton method are proved. Linear convergence of the algorithms is obtained under very mild assumptions. The superlinear convergence holds under some conditions imposed on both terms of equation. Some numerical results indicate that both algorithms works quite well in practice.      

Optimization of the quasi-Monte Carlo algorithm for solving systems of linear algebraic equations

An interesting conclusion about error reduction of the modified quasi-Monte Carlo method for solving systems of linear algebraic equations is suggested. The Monte Carlo method is compared with the quasi-Monte Carlo method and its modification. The optimal choice of the parameters of the Markov chain for the modified Monte Carlo method applied to solving systems of linear equations is substantiated.     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

Affine scaling inexact generalized Newton algorithm with interior backtracking technique for solving bound-constrained semismooth equations

We develop and analyze an affine scaling inexact generalized Newton algorithm in association with nonmonotone interior backtracking line technique for solving systems of semismooth equations subject to bounds on variables. By combining inexact affine scaling generalized Newton with interior backtracking line search technique, each iterate switches to inexact generalized Newton backtracking step to strict interior point feasibility. The global convergence results are developed in a very general setting of computing trial steps by the affine scaling generalized Newton-like method that is augmented by an interior backtracking line search technique projection onto the feasible set. Under some reasonable conditions we establish that close to a regular solution the inexact generalized Newton method is shown to converge locally p-order q-superlinearly. We characterize the order of local convergence based on convergence behavior of the quality of the approximate subdifferentials and indicate how to choose an inexact forcing sequence which preserves the rapid convergence of the proposed algorithm. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases.     

Numerical stability for solving nonlinear equations

Summary The concepts of the condition number, numerical stability and well-behavior for solving systems of nonlinear equationsF(x)=0 are introduced. Necessary and sufficient conditions for numerical stability and well-behavior of a stationary are given. We prove numerical stability and well-behavior of the Newton iteration for solving systems of equations and of some variants of secant iteration for solving a single equation under a natural assumption on the computed evaluation ofF. Furthermore we show that the Steffensen iteration is unstable and show how to modify it to have well-behavior and hence stability.This work was supported in part by the Office of Naval Research under Contract N 00014-67-0314-0010 NR 044-422 and by the National Science Foundation under Grant GJ 32111     

On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations

We study an iterative method with order for solving nonlinear operator equations in Banach spaces. Algorithms for specific operator equations are built up. We present the received new results of the local and semilocal convergence, in case when the first-order divided differences of a nonlinear operator are Hölder continuous. Moreover a quadratic nonlinear majorant for a nonlinear operator, according to the conditions laid upon it, is built. A priori and a posteriori estimations of the method’s error are received. The method needs almost the same number of computations as the classical Secant method, but has a higher order of convergence. We apply our results to the numerical solving of a nonlinear boundary value problem of second-order and to the systems of nonlinear equations of large dimension.     

Parallel regularized Newton method for nonlinear ill-posed equations

We introduce a regularized Newton method coupled with the parallel splitting-up technique for solving nonlinear ill-posed equations with smooth monotone operators. We analyze the convergence of the proposed method and carry out numerical experiments for nonlinear integral equations.     

A numerical method for solving partial differential algebraic equations

Linear systems of partial differential equations with constant coefficient matrices are considered. The matrices multiplying the derivatives of the sought vector function are assumed to be singular. The structure of solutions to such systems is examined. The numerical solution of initialboundary value problems for such equations by applying implicit difference schemes is discussed.