A Hybrid Learning Method for Multilayer Perceptrons

A Newton learning method for a neural network of multilayer perceptrons is proposed in this paper. Furthermore, a hybrid learning method id legitimately developed in combination of the backpropagation method proposed by Rumelhart et al with the Newton learning method. Finally, the hybrid learning algorithm is compared with the backpropagation algorithm by some illustrations, and the results show that this hybrid leaming algorithm bas the characteristics of rapid convergence.

A Fourth-order Covergence Newton-type Method

A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton＇s method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton and Newton-type methods. The results show that the proposed method has some more advantages than others. It enriches the methods to find the roots of non-linear equations and it is important in both theory and application.

Newton chaos iteration method and its application to mechanism kinematics synthesis

Many questions in natural science and engineering can be transformed into nonlinear equations. Newton iteration method is an important technique to one dimensional and multidimensional variables and iteration itself is very sensitive to initial guess point. This sensitive area is the Julia set of nonlinear discrete dynamic system which Newton iteration method forms. The Julia set, which is the boundaries of basins of attractions, displays the intricate fractal structures and chaos phenomena. By constructing repulsion two-cycle point function and making use of inverse image iteration method, a method to find Julia set point was introduced. For the first time, a new method to find all solutions was proposed based on utilizing sensitive fractal areas to locate the Julia set points to find all solutions of the nonlinear questions. The developed technique used an important feature of fractals to preserve shape of basins of attraction on infinitely small scales. The numerical examples in linkage synthesis showed that the method was effective and correct.

On Newton-Like Methods for Solving Nonlinear Equations

In this paper, we present a family of general New to n-like methods with a parametric function for finding a zero of a univariate fu nction, permitting f′(x)=0 in some points. The case of multiple roots is n ot treated. The methods are proved to be quadratically convergent provided the w eak condition. Thus the methods remove the severe condition f′(x)≠0. Based on the general form of the Newton-like methods, a family of new iterative meth ods with a variable parameter are developed.

On the approximate zero of Newton method

A judgment criterion to guarantee a point to be a Chen' s approximate zero of Newton method for solving nonlinear equation is sought by dominating sequence techniques. The criterion is based on the fact that the dominating function may have only one simple positive zero, assuming that the operator is weak Lipschitz continuous, which is much more relaxed and can be checked much more easily than Lipschitz continuous in practice. It is demonstrated that a Chen' s approximate zero may not be a Smale' s approximate zero. The error estimate obtained indicated the convergent order when we use |f(x) | < ε to stop computation in software.The result can also be applied for solving partial derivative and integration equations.

Solving constrained minimax problem via nonsmooth equations method

A new nonsmooth equations model of constrained minimax problem was de-rived. The generalized Newton method was applied for solving this system of nonsmooth equations system. A new algorithm for solving constrained minimax problem was established. The local superlinear and quadratic convergences of the algorithm were discussed.

A Class of Third-order Convergence Variants of Newton＇s Method

A class of third-order convergence methods of solving roots for non-linear equation,which are variant Newton＇s method, are given. Their convergence properties are proved. They are at least third order convergence near simple root and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton＇s methods. The results show that the proposed methods have some more advantages than others. They enrich the methods to find the roots of non-linear equations and they are important in both theory and application.

Smoothing Inexact Newton Method for Solving P_0-NCP Problems

Based on a smoothing symmetric disturbance FB-function,a smoothing inexact Newton method for solving the nonlinear complementarity problem with P0-function was proposed.It was proved that under mild conditions,the given algorithm performed global and superlinear convergence without strict complementarity.For the same linear complementarity problem(LCP),the algorithm needs similar iteration times to the literature.However,its accuracy is improved by at least 4 orders with calculation time reduced by almost 50%,and the iterative number is insensitive to the size of the LCP.Moreover,fewer iterations and shorter time are required for solving the problem by using inexact Newton methods for different initial points.

Newton-EGMSOR Methods for Solution of Second Order Two-Point Nonlinear Boundary Value Problems

The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. Apart from block iterative methods, the formulation of the MSOR （Modified Successive Over-Relaxation） method known as SOR method with red-black ordering strategy by using two accelerated parameters, ω and ω′, has also improved the convergence rate of the standard SOR method. Due to the effectiveness of these iterative methods, the primary goal of this paper is to examine the performance of the EG family without or with accelerated parameters in solving second order two-point nonlinear boundary value problems. In this work, the second order two-point nonlinear boundary value problems need to be discretized by using the second order central difference scheme in constructing a nonlinear finite difference approximation equation. Then this approximation equation leads to a nonlinear system. As well known that to linearize nonlinear systems, the Newton method has been proposed to transform the original system into the form of linear system. In addition to that, the basic formulation and implementation of 2 and 4-point EG iterative methods based on GS （Gauss-Seidel）, SOR and MSOR approaches, namely EGGS, EGSOR and EGMSOR respectively are also presented. Then, combinations between the EG family and Newton scheme are indicated as EGGS-Newton, EGSOR-Newton and EGMSOR-Newton methods respectively. For comparison purpose, several numerical experiments of three problems are conducted in examining the effectiveness of tested methods. Finally, it can be concluded that the 4-point EGMSOR-Newton method is more superior in accelerating the convergence rate compared with the tested methods.

NONLINEAR GALERKIN METHODS FOR SOLVING TWO DIMENSIONAL NEWTON－BOUSSINESQ EQUATIONS

The nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations are proposed.The existence and uniqueness of global generalized solution of these equations, and the convergence of approximate solutions are also obtained.

An improved nonlinear conjugate gradient method with an optimal property

Conjugate gradient methods have played a special role in solving large scale nonlinear problems. Recently, the author and Dai proposed an efficient nonlinear conjugate gradient method called CGOPT, through seeking the conjugate gradient direction closest to the direction of the scaled memoryless BFGS method. In this paper, we make use of two types of modified secant equations to improve CGOPT method. Under some assumptions, the improved methods are showed to be globally convergent. Numerical results are also reported.