A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations

In this paper, we propose a family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations. They come from two modified conjugate gradient methods [W.Y. Cheng, A two term PRP based descent Method, Numer. Funct. Anal. Optim. 28 (2007) 1217–1230; L. Zhang, W.J. Zhou, D.H. Li, A descent modified Polak–Ribiére–Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal. 26 (2006) 629–640] recently proposed for unconstrained optimization problems. Under appropriate conditions, the global convergence of the proposed method is established. Preliminary numerical results show that the proposed method is promising.     

BFGS trust-region method for symmetric nonlinear equations

In this paper, we propose a BFGS trust-region method for solving symmetric nonlinear equations. The global convergence and the superlinear convergence of the presented method will be established under favorable conditions. Numerical results show that the new algorithm is effective.     

Scaled three-term derivative-free methods for solving large-scale nonlinear monotone equations

Numerical Algorithms - In this paper, two effective derivative-free methods are proposed for solving large-scale nonlinear monotone equations, in which the search directions are sufficiently...     

A monotone semismooth Newton type method for a class of complementarity problems

In this paper, we propose a modified semismooth Newton method for a class of complementarity problems arising from the discretization of free boundary problems and establish its monotone convergence. We show that under appropriate conditions, the method reduces to semismooth Newton method. We also do some preliminary numerical experiments to show the efficiency of the proposed method.     

A new smoothing quasi-Newton method for nonlinear complementarity problems

A new smoothing quasi-Newton method for nonlinear complementarity problems is presented. The method is a generalization of Thomas’ method for smooth nonlinear systems and has similar properties as Broyden's method. Local convergence is analyzed for a strictly complementary solution as well as for a degenerate solution. Presented numerical results demonstrate quite similar behavior of Thomas’ and Broyden's methods.     

A fourth-order derivative-free algorithm for nonlinear equations

A two-step derivative-free iterative algorithm is presented for solving nonlinear equations. Error analysis shows that the algorithm is fourth-order with efficiency index equal to 1.5874. A lot of numerical results show that the algorithm is effective and is preferable to some existing derivative-free methods in terms of computation cost.     

An iterative method for solving nonlinear equations

An iterative method for finding a solution of the equation f(x)=0$f(x)=0$ is presented. The method is based on some specially derived quadrature rules. It is shown that the method can give better results than the Newton method.     

A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations

This paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time-dependent problem and its corresponding steady-state problem. This method leads to a simple and yet efficient linear iterative algorithm. It yields two sequences of iterations that converge monotonically from above and below, respectively, to a unique solution of the system. The monotone property of the iterations gives concurrently improving upper and lower bounds for the solution. It is shown that the rate of convergence for the sum of the two sequences is quadratic. Under an additional requirement, quadratic convergence is attained for one of these two sequences. In contrast with the existing accelerated monotone iterative methods, our new method avoids computing local maxima in the construction of these sequences. An application using a model problem gives numerical results that illustrate the effectiveness of the proposed method.     

Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations

In this paper, two Chebyshev-like third order methods free from second derivatives are considered and analyzed for systems of nonlinear equations. The methods can be obtained by having different approximations to the second derivatives present in the Chebyshev method. We study the local and third order convergence of the methods using the point of attraction theory. The computational aspects of the methods are also studied using some numerical experiments including an application to the Chandrasekhar integral equations in Radiative Transfer.     

Frozen divided difference scheme for solving systems of nonlinear equations

The development of an inverse first-order divided difference operator for functions of several variables, as well as a direct computation of the local order of convergence of an iterative method is presented. A generalized algorithm of the secant method for solving a system of nonlinear equations is studied and the maximum computational efficiency is computed. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.     

Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations

In [YoonMee Ham etal., Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math., 222 (2008) 477–486], some higher-order modifications of Newton’s method for solving nonlinear equations are constructed. But if p=2$p=2$, then their main theorem did not hold. In this paper, we first give an example to show that YoonMee Ham etal.’s methods are not always correct in the case p=2$p=2$. Then, we present the condition that H(x,y)$H(x,y)$ should satisfy such that the order of convergence increases three or four or five units. Per iteration they only need two additional function evaluations to increase the order. Based on this and multi-step Newton’s scheme, we give further modifications of the method to obtain higher-order convergent iterative methods. Finally, several examples are given to demonstrate the efficiency and performance of our modified methods and compare them with some other methods.     

A globally convergent method for solving nonlinear equations without the differentiability condition

We give a general iterative method which computes the maximal real rootx _max of a one variable Lipschitzian function in a given interval. The method generates a monotonically decreasing sequence which converges towardsx _max or demonstrates the non-existence of a real root in the considered interval. We show that the method is globally convergent and locally linearly convergent. We also compute the number of iterations needed to reach the given accuracy.     

A modified inexact implicit method for mixed variational inequalities

In this paper, we suggest and analyze an inexact implicit method with a variable parameter for mixed variational inequalities by using a new inexactness restriction. Under certain conditions, the global convergence of the proposed method is proved. Some preliminary computational results are given to illustrate the efficiency of the new inexactness restriction. The results proved in this paper may be viewed as improvement and refinement of the previously known results.     

Solving nonconvex nonlinear programming problems via a new aggregate constraint homotopy method

In this paper, by introducing C² mappings ξ_i(x,z_i),i=1,…,m and using the idea of the aggregate function method, a new aggregate constraint homotopy method is proposed to solve the Karush-Kuhn-Tucker (KKT) point of nonconvex nonlinear programming problems. Compared with the previous results, the choice scope of initial points is greatly enlarged, so use of the new aggregate constraint homotopy method may improve the computational efficiency of reduced predictor-corrector algorithms.     

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.     

New higher order methods for solving nonlinear equations with multiple roots

For a nonlinear equation f(x)=0 having a multiple root we consider Steffensen’s transformation, T. Using the transformation, say, F_q(x)=T^qf(x) for integer q≥2, repeatedly, we develop higher order iterative methods which require neither derivatives of f(x) nor the multiplicity of the root. It is proved that the convergence order of the proposed iterative method is 1+2^q−2 for any equation having a multiple root of multiplicity m≥2. The efficiency of the new method is shown by the results for some numerical examples.     

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.     

Monotonically convergent iterative methods for nonlinear systems of equations

Summary This paper deals with discrete analogues of nonlinear elliptic boundary value problems and with monotonically convergent iterative methods for their numerical solution. The discrete analogues can be written asM(u)u+H(u)=0, whereM(u) is ann%n M-matrix for eachu ⁿ andH: ^{n n} . The numerical methods considered are the natural undeerrelaxation method, the successive underrelaxation method, and the Jacobi underrelaxation method. In the linear case and without underrelaxation these methods correspond to the direct, the Gauss-Seidel, and the Jacobi method for solving the underlying system of equations, resp. For suitable starting vectors and sufficiently strong underrelaxation, the sequence of iterates generated by any of these methods is shown to converge monotonically to a solution of the underlying system.     

On the global convergence of path-following methods to determine all solutions to a system of nonlinear equations

In this paper we prove a general theorem stating a sufficient condition for the inverse image of a point under a continuously differentiable map from ℝ ⁿ to ℝ ^k to be connected. This result is applied to the trajectories generated by the Newton flow. Several examples demonstrate the applicability of the results to nontrivial problems. This work was supported by the Deutsche Forschungsgemeinschaft.