弦割法与Muller法收敛阶的新证明方法

弦割法、Muller法与牛顿法一样,都是求解非线性方程的著名算法之一.然而在目前众多优秀的数值分析教材或论著中.关于弦割法和Muller法收敛阶的证明过程都是比较复杂的,无一例外的都是借助于差分方程的求解.本文对这两个算法的收敛阶给出了一种新的简单、直接的证明方法,达到了与牛顿法收敛阶证明方法的统一,同时还能够方便地求...     

Finding Zeros of Analytic Functions: α Theory for Secant Type Methods

We present new results concerning the convergence of secant type methods with only conditions at a point. The radius of robustness of these methods is given, and we apply it to the study of the complexity of homotopy methods for approximating roots. In particular, we say how to use the secant type method to get an approximate zero relative to the Newton method.     

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.     

Weaker convergence conditions for the secant method

We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.     

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.     

一类三阶牛顿变形方法

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.     

The behavior of a multiplicity independent root-finding scheme in the presence of error

King and Kioustelidis independently proposed a derivative free scheme that permits root finding methods, such as the secant method to preserve high order convergence when the root in question is multiple. In this note it is shown that the scheme can fail to achieve the maximum accuracy that is attainable at a fixed precision of computation.This work was supported in part by the Office of Naval Research under contract No. N00014-76-C-0391.     

A modified secant method for unconstrained minimization

This paper presents a new globally convergent secant method for unconstrained optimization. The root rate of convergence of this algorithm is superlinear, between 1 and 2, decreasing with dimension of the problem. It is shown that on a class of problems, it is substantially more efficient than a number of other algorithms.     

MODIFIED NEWTON AND SECANT METHODS FOR SOLVING AN ORDER O(h~4) FINITE-DIFFERENCE PROBLEM

1IntroductionSolution0fn0nlineartwo-pointb0undaryvaIuepr0blems(NBVP)canoftenbefoundbythefinite-differenceappr0ach,wheref(t,y)isaconti-nuousfunction.Collatz[1]firstpresentedanapproximation0ffourthorderfwherey=(y1,''tyN)',g=(g1,'-,gN)'andtherelativepaperscanals0beseenin[2].Toestablishthesolutionof(1.l),thef0llowingmethodscanbeusedfnonlinearsuccessiverelaxati0n(NSOR)method[3],thedifferenceNewt0nmethod(0rNewtonmethod)[4],therelativesparsenonlinearequationpr0blemscanals0beseenin[5-8]-lnthisp…     

Directional secant method for nonlinear equations

We present a directional secant method, a secant variant of the directional Newton method, for solving a single nonlinear equation in several variables. Under suitable assumptions, we prove the convergence and the quadratic convergence speed of this new method. Numerical examples show that the directional secant method is feasible and efficient, and has better numerical behaviour than the directional Newton method.     

求複數根的舗D法

<正> 一. 引言 代數方程f(x)=0的實數根的逐步接近法已有多種,其中計算簡單收斂最快的是用牛頓公式     

牛顿方法的两个新格式

给出牛顿迭代方法的两个新格式,S im pson牛顿方法和几何平均牛顿方法,证明了它们至少三次收敛到单根,线性收敛到重根.文末给出数值试验,且与其它已知牛顿法做了比较.结果表明收敛性方法具有较好的优越性,它们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.     

An hybrid method that improves the accessibility of Steffensen’s method

Steffensen’s method is known for its fast speed of convergence and its difficulty in applying it in Banach spaces. From the analysis of the accessibility of this method, we see that we can improve it by using the simplified secant method for predicting the initial approximation of Steffensen’s method. So, from both methods, we construct an hybrid iterative method which guarantees the convergence of Steffensen’s method from approximations given by the simplified secant method. We also emphasize that the study presented in this work is valid for equations with differentiable operators and non-differentiable operators.     

Convergence theory for the structured BFGS secant method with an application to nonlinear least squares

In 1981, Dennis and Walker developed a convergence theory for structured secant methods which included the PSB and the DFP secant methods but not the straightforward structured version of the BFGS secant method. Here, we fill this gap in the theory by establishing a convergence theory for the structured BFGS secant method. A direct application of our new theory gives the first proof of local andq-superlinear convergence of the important structured BFGS secant method for the nonlinear least-squares problem, which is used by Dennis, Gay, and Welsh in the current version of the popular and successful NL2SOL code.This research was sponsored by SDIO/IST/ARO, AFOSR-85-0243, and DOE-DEFG05-86 ER-25017.A portion of this work is contained in the second author's doctoral thesis under the supervision of the other two authors in the Department of Mathematical Sciences, Rice University. The second author would like to thank Universidad del Valle, Cali, Columbia, for support during his graduate studies.An early draft of this work was presented at the SIAM 35th Anniversary Meeting, October 12–15, 1987, Denver, Colorado.     

Secant methods for semismooth equations

Some generalizations of the secant method to semismooth equations are presented. In the one-dimensional case the superlinear convergence of the classical secant method for general semismooth equations is proved. Moreover a new quadratically convergent method is proposed that requires two function values per iteration. For the n-dimensional cases, we discuss secant methods for two classes of composite semismooth equations. Most often studied semismooth equations are of such form. Received October 16, 1996 / Revised version received July 25, 1997     

Methods of the secant type for systems of equations with symmetric jacobian matrix

Symmetric methods (SS methods) of the secant type are proposed for systems of equations with symmetric Jacobian matrix. The SSI and SS2 methods generate sequences of symmetric matrices J and H which approximate the Jacobian matrix and inverse one, respectively. Rank-two quasi-Newton formulas for updating J and H are derived. The structure of the approximations J and H is better than the structure of the corresponding approximations in the traditional secant method because the SS methods take into account symmetry of the Jacobian matrix. Furthermore, the new methods retain the main properties of the traditional secant method, namely, J and H are consistent approximations to the Jacobian matrix; the SS methods converge superlinearly; the sequential (n + 1)-point SS methods have the R-order at least equal to the positive root of t_n+1-1=0.     

A note on the secant method

It is conjectured that the secant method converges linearly to a root of multiplicitym. In support of this, it is proved that the linearized secant method produces a bounded sequence of approximations except for a restricted set of values of the two initial approximations.     

The convergence ball and error analysis of the two-step Secant method

Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order, we obtain an estimate of the radius of the convergence ball for the two-step secant method. Moreover, we also provide an error estimate that matches the convergence order of the two-step secant method. At last, we give an application of the proposed theorem.     

Local and superlinear convergence of quasi-Newton methods based on modified secant conditions

For solving unconstrained minimization problems, quasi-Newton methods are popular iterative methods. The secant condition which employs only the gradient information is imposed on these methods. Several researchers paid attention to other secant conditions to get a better approximation of the Hessian matrix of the objective function. Recently, Zhang et al. [New quasi-Newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl. 102 (1999) 147–167] and Zhang and Xu [Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations, J. Comput. Appl. Math. 137 (2001) 269–278] proposed the modified secant condition which uses both gradient and function value information in order to get a higher order accuracy in approximating the second curvature of the objective function. They showed the local and q-superlinear convergence property of the BFGS-like and DFP-like updates based on their proposed secant condition. In this paper, we incorporate one parameter into this secant condition to smoothly switch the standard secant condition and the secant condition of Zhang et al. We consider a modified Broyden family which includes the BFGS-like and the DFP-like updates proposed by Zhang et al. We prove the local and q-superlinear convergence of our method.     

Runge-Kutta方法用于非线性方程求根

将Runge-Kutta方法用于非线性方程求根问题,给出二阶,三阶和四阶对应的三个新的方程求根公式,证明了它们至少三次收敛到单根,线性收敛到重根.文末给出数值试验,且与其它已知求根公式做了比较.结果表明此方法具有较好的优越性,它们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.