A variant of Steffensen’s method of fourth-order convergence and its applications

In this paper, a variant of Steffensen’s method of fourth-order convergence for solving nonlinear equations is suggested. Its error equation and asymptotic convergence constant are proven theoretically and demonstrated numerically. The derivative-free method only uses three evaluations of the function per iteration to achieve fourth-order convergence. Its applications on systems of nonlinear equations and boundary-value problems of nonlinear ODEs are showed as well in the numerical examples.     

A class of nonlinear Lagrangians for nonconvex second order cone programming

This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function.     

A study on the convergence of homotopy analysis method

In this study, a reliable approach for convergence of the homotopy analysis method when applied to nonlinear problems is discussed. First, we present an alternative framework of the method which can be used simply and effectively to handle nonlinear problems. Then, mainly, we address the sufficient condition for convergence of the method. The convergence analysis is reliable enough to estimate the maximum absolute truncated error of the homotopy series solution. The analysis is illustrated by investigating the convergence results for some nonlinear differential equations. The study highlights the power of the method.     

m步非线性多重分裂Newton法的收敛性

本文讨论了多重分裂算法在求解一类非线性方程组的全局收敛性和单侧收敛性．当用研步Newton法来代替求得每个非线性多重分裂子问题的近似解时，同样给出相应收敛性结论．数值算例证实了算法的有效性．     

Convergence analysis of the nonlinear block scaled ABS methods

In this paper, by a further investigation of the algorithm structure of the nonlinear block scaled ABS methods, we convert it into an inexact Newton method. Based on this equivalent version, we establish the semilocal convergence theorem of the nonlinear block scaled ABS methods and obtain convergence conditions that mainly depend on the behavior of the mapping at the initial point. This complements the convergence theory of the nonlinear block scaled ABS methods.     

Implicit Scheme for Stochastic Parabolic Partial Diferential Equations Driven by Space-Time White Noise

We extend Rothe's method of solving linear parabolic PDEs to the case of nonlinear SPDEs driven by space-time white noise. When the nonlinear terms are Lipschitz functions we prove almost sure convergence of the approximations uniformly in time and space. When the nonlinear drift term is only measurable we obtain the convergence in probability, by using Malliavin calculus.     

Modular filter convergence theorems for Urysohn integral operators and applications

We prove some versions of modular convergence theorems for nonlinear Urysohn-type integral operators with respect to filter convergence. We consider pointwise filter convergence of functions giving also some applications to linear and nonlinear Mellin operators. We show that our results are strict extensions of the classical ones.     

Stability and convergence of a system of nonlinear difference equations

We investigate stability and convergence of solutions of a system of nonlinear difference equations approximating a system of nonlinear parabolic equations. A linear system of similar structure is also considered. An energy norm is constructed for the linear system, and stability and convergence in this norm are proved under certain necessary conditions. Stability and convergence of solutions of the nonlinear system of difference equations are proved in a similar norm.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 120–127, 1986.     

Uniform quadratic convergence of monotone iterates for nonlinear singularly perturbed parabolic problems

This paper deals with a monotone iterative method for solving nonlinear singularly perturbed parabolic problems. Monotone sequences, based on the method of upper and lower solutions, are constructed for a nonlinear difference scheme which approximates the nonlinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. The monotone sequences possess quadratic convergence rate. An analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and to the continuous problem is given. Numerical experiments are presented.     

Variants of Steffensen-secant method and applications

In this paper, a parametric variant of Steffensen-secant method and three fast variants of Steffensen-secant method for solving nonlinear equations are suggested. They achieve cubic convergence or super cubic convergence for finding simple roots by only using three evaluations of the function per step. Their error equations and asymptotic convergence constants are deduced. Modified Steffensen’s method and modified parametric variant of Steffensen-secant method for finding multiple roots are also discussed. In the numerical examples, the suggested methods are supported by the solution of nonlinear equations and systems of nonlinear equations, and the application in the multiple shooting method.     

PARALLEL NONLINEAR MULTISPLITTING RELAXATION METHODS

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Nonlinear Lagrangian Functions and Applications to Semi-Infinite Programs

In this paper a nonlinear penalty method via a nonlinear Lagrangian function is introduced for semi-infinite programs. A convergence result is established which shows that the sequence of optimal values of nonlinear penalty problems converges to that of semi-infinite programs. Moreover a conceptual convergence result of a discretization method with an adaptive scheme for solving semi-infinite programs is established. Preliminary numerical experiments show that better optimal values for some nonlinear semi-infinite programs can be obtained using the nonlinear penalty method.     

On the convergence of Newton-like methods using restricted domains

We present a new semi-local convergence analysis for Newton-like methods in order to approximate a locally unique solution of a nonlinear equation containing a non-differentiable term in a Banach space setting. The new idea uses more precise convergence domains. This way the new sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, are also provided in this study.     

一类四阶牛顿变形方法

给出非线性方程求根的一类四阶方法,也是牛顿法的变形方法.证明了方法收敛性,它们至少四次收敛到单根,线性收敛到重根.文末给出数值试验,且与牛顿法及其它牛顿变形法做了比较.结果表明方法具有很好的优越性,它丰富了非线性方程求根的方法,在理论上和应用上都有一定的价值.     

从常步长梯度方法的视角看不可微凸优化增广Lagrange方法的收敛性

增广Lagrange方法是求解非线性规划的一种有效方法.从一新的角度证明不等式约束非线性非光滑凸优化问题的增广Lagrange方法的收敛性.用常步长梯度法的收敛性定理证明基于增广Lagrange函数的对偶问题的常步长梯度方法的收敛性,由此得到增广Lagrange方法乘子迭代的全局收敛性.     

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White Noise I

We approximate quasi-linear parabolic SPDEs substituting the derivatives in the space variable with finite differences. When the nonlinear terms in the equation are Lipschitz continuous we estimate the rate of L^p convergence of the approximations and we also prove their almost sure uniform convergence to the solution. When the nonlinear terms are not Lipschitz continuous we obtain this convergence in probability, if the pathwise uniqueness for the equation holds.     

Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces

In this paper, we first prove a general fixed point theorem for nonlinear mappings in a Banach space. Then we prove a nonlinear mean convergence theorem of Baillon??s type and a weak convergence theorem of Mann??s type for 2-generalized nonspreading mappings in a Banach space.     

A class of two‐stage iterative methods for systems of weakly nonlinear equations

The discretizations of many differential equations by the finite difference or the finite element methods can often result in a class of system of weakly nonlinear equations. In this paper, by applying the two-tage iteration technique and in accordance with the special properties of this weakly nonlinear system, we first propose a general two-tage iterative method through the two-tage splitting of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a two-tage AOR method, which particularly uses the AOR iteration as the inner iteration and is substantially a relaxed variant of the afore-presented method. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only B-differentiable. When the system matrix is either a monotone matrix or an H-matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new two-tage iteration methods, and investigate the influence of the matrix splittings as well as the relaxation parameters on the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for solving the system of weakly nonlinear equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

用非精确的平行割线法求解奇异问题

奇异方程经常出现在很多实际非线性问题中,如反应扩散系统等.因此,研究奇异非线性方程的求解具有十分重要的意义.平行割线法是一种经典的求解非线性方程的迭代方法,它收敛阶较高,计算量较少.但在解决实际问题时,一方面,抽象出的数学模型与实际问题总是存在着一定的偏差,另外,在数据的计算中难免存在着一定的计算误差,所以研究用非精确的平行割线法求解非线性奇异问题具有很重要的现实意义,使得求解奇异问题具有更高的实用性和可行性.采用在平行割线法的迭代公式中加入摄动项的方法,构造出新的加速迭代格式,证明了新的迭代格式的收敛性,给出了收敛速率,得到了误差估计.     

A weak condition for secant method to solve systems of nonlinear equations

In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.