两类五阶解非线性方程组的迭代算法

本文首先根据Runge-Kutta方法的思想,结合Newton迭代法,提出了一类带参数的解非线性方程组F(x)=0的迭代算法,然后基于解非线性方程f(x)=0的King算法,给出第二类解非线性方程组的迭代算法,收敛性分析表明这两类算法都是五阶收敛的.其次给出了本文两类算法的效率指数,以及一些已知算法的效率指数,并且将本文算法的效率指数与其它方法进行详细的比较,通过效率比率R_(i,j)可知本文算法具有较高的计算效率.最后给出了四个数值实例,将本文两类算法与现有的几种算法进行比较,实验结果说明本文算法收敛速度快,迭代次数少,有明显的优势.     

三阶奇异边值问题的正解

本文在较弱的条件下,研究了三阶奇异边值问题{x'+a(t)F(t,x)=0 0＜t＜1,x(0)=x'(0)=x(1)=0正解的存在性.允许非线性项a(t),F(t,x)在t=0,t=1及x=0处奇异.     

适用于任意初始值条件的两侧逼近区间迭代法

本文提出了一个解非线性方程组f(x)=x的两侧逼近区间迭代法,该算法在任意的初值条件下都可使用,并在f(x)较弱的条件下,线性收敛到f(x)=x的解.     

解非线性方程的一个非线性迭代法

1 引 言 用常微分方程及其数值解的理论和方法(简称ODE方法)来构造解非线性方程组的方法见Branin.F.H.等,但未能讨论收敛性。其后对线性方程组A_x=b和非线性方程f(x)=0都有专门的论述且论证了方法的大范围收敛性,对于求非线性方程f(x)=0在[a,b]内的根x~·的不使用导数的大范围收敛的算法使我们容易想到两分法和试位法,是否有其它更为有效的不使用导数的大范围收敛的方法,下面我们来讨论基于ODE方法原理的非线性迭代方法。     

函数因子法——非线性方程组求解中处理奇异问题的一种新方法

§1.引言 非线性方程组 F(x)=0,F:D?R~n→R~n (1.1)嵌入参数t,构成同伦H:[0,T]×D?R~(n 1)→R~n,使得 H(0,x~0)=0,H(T,x)=F(x),(1.2)这里T可以是有限的或 ∞,当T为 ∞时以极限过程代替求值.若 H(t.x)=0(1.3)存在连续解x(t):[0,T]→D,则非线性方程组(1.1)的解x~*=x(T).若(1.3)的解     

再论Broyden方法的收敛性

本文用Smale提出的点估计理论,建立了在点估计条件下的求解非线性方程组F(x)=0的著名的Broyden方法的收敛性及解的存在唯一性定理,从而在R~n空间的解析映射类上,解除了由于F′的区域性Lipschitz条件带来的Broyden方法收敛判据之间的相互制约性。它为一大类修正算法点估计理论的建立,提供了新的途径。     

四阶微分方程的迭代解

利用一个构造性的方法,在假设边值问题存在上解α和下解β,满足β≤α的前提下,给出了两个单调序列它们一致收敛于如下两类边值问题的极值解u(4)(x)-Mu″(x)=f(x,u(x),u'(x),u″(x),u″'(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0;u(4)(x)-Mu″(x)=g(x,u(x),u'(x),u″(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0.     

Newton-like方法的收敛性及其应用

1 引言我们考虑非线性方程组 F(x)=0, (1．1) 其中F:Rn→Rn是给定的非线性向量函数,并具有如下性质: (1)存在x*使得F(x*)=0; (2)F(x)在x*的邻域内是连续可微的; (3)F′(x*)是非奇异的． Newton法是求(1．1)的数值解的经典算法:     

再论Broyden方法的收敛性

本文用 Smale 提出的点估计理论,建立了在点估计条件下的求解非线性方程组 F(x)=0的著名的 Broyden 方法的收敛性及解的存在唯一性定理.从而在 R~n 空间的解析映射类上,解除了由于 F′的区域性 Lipschitz 条件带来的 Broyden 方法收敛判据之间的相互制约性.它为一大类修正算法点估计理论的建立,提供了新的途径.     

用于解算子方程的Newton法的松弛收敛性

本文研究在适当条件下用于求算子方程F(x)=0的解的Newton法的收敛性。一方面,给出了解所在的区域和在该区域中解的唯一性,以及逼近解的新的收敛率估计。另一方面,也举例说明了理论结果应用于解Hammerstein型的非线性积分方程时,可给出解的存在唯一性与逼近解的收敛判据方面的结果。     

The Separation of Zeros of Solutions of Higher Order Linear Differential Equations With Entire Coefficients

Suppose that a homogeneous linear differential equation has entire coefficients of finite order and a fundamental set of solutions each having zeros with finite exponent of convergence. Upper bounds are given for the number of zeros of these solutions in small discs in a neighbourhood of infinity.     

A note on some improvements of the simultaneous methods for determination of polynomial zeros

Applying Gauss-Seidel approach to the improvements of two simultaneous methods for finding polynomial zeros, presented in [9], two iterative methods with faster convergence are obtained. The lower bounds of the R-order of convergence for the accelerated methods are given. The improved methods and their accelerated modifications are discussed in view of the convergence order and the number of numerical operations. The considered methods are illustrated numerically in the example of an algebraic equation.     

Efficient methods for the inclusion of polynomial zeros

Using a suitable zero-relation and the inclusion isotonicity property, new interval iterative methods for the simultaneous inclusion of simple complex zeros of a polynomial are derived. These methods produce disks in the complex plane that contain the polynomial zeros in each iteration, providing in this manner an information about upper error bounds of approximations. Starting from the basic method of the fourth order, two accelerated methods with Newton’s and Halley’s corrections, having the order of convergence five and six respectively, are constructed. This increase of the convergence rate is obtained without any additional operations, which means that the methods with corrections are very efficient. The convergence analysis of the basic method and the methods with corrections is performed under computationally verifiable initial conditions, which is of practical importance. Two numerical examples are presented to demonstrate the convergence behavior of the proposed interval methods.     

On the Rate of Overconvergence of the Generalized Eneström-Kakeya Functional for Polynomials

The classical Eneström-Kakeya Theorem, which provides an upper bound for the moduli of zeros of any polynomial with positive coefficients, has been recently extended by Anderson, Saff and Varga to the case of any complex polynomial having no zeros on the ray [0,$+∞$). Their extension is sharp in the sense that, given such a complex polynomials $p_n(z)$ of degree $n≥1$, a sequence of multiplier polynomial can be found for which the Eneström-Kakeya upper bound, applied to the products $Q_{mi}(z)$ · $p_n(z)$, converges, in the limit as $i$ tends to $∞$, to the maximum of the moduli of the zeros of $p_n(z)$. Here, the rate of convergence of these upper bounds is studied. It is shown that the obtained rate of convergence is best possible.     

Enclosing clusters of zeros of polynomials

Lagrange interpolation and partial fraction expansion can be used to derive a Gerschgorin-type theorem that gives simple and powerful a posteriori error bounds for the zeros of a polynomial if approximations to all zeros are available. Compared to bounds from a corresponding eigenvalue problem, a factor of at least two is gained.The accuracy of the bounds is analyzed, and special attention is given to ensure that the bounds work well not only for single zeros but also for multiple zeros and clusters of close zeros.A Rouché-type theorem is also given, that in many cases reduces the bound even further.     

Lower bounds for the zeros of analytic functions

Summary In the present work the problem of finding lower bounds for the zeros of an analytic function is reduced by a Hilbert space technique to the well-known problem of finding upper bounds for the zeros of a polynomial. Several lower bounds for all the zeros of analytic functions are thus found, which are always better than the well-known Carmichael-Mason inequality. Several numerical examples are also given and a comparison of our bounds with well-known bounds in literature and/or the exact solution is made.     

A new simultaneous method of fourth order for finding complex zeros in circular interval arithmetic

Starting from disjoint disks which contain polynomial complex zeros, the new iterative interval method for simultaneous finding of inclusive disks for complex zeros is formulated. The convergence theorem and the conditions for convergence are considered, and the convergence is shown to be fourth. Numerical examples are included.     

On the refinement of Cauchy's theorem and Pellet's theorem

Using the relationship of a polynomial and its associated polynomial, we derived a necessary and sufficient condition for determining all roots of a given polynomial on the circumference of a circle defined by its associated polynomial. By employing the technology of analytic inequality and the theory of distribution of zeros of meromorphic function, we refine two classical results of Cauchy and Pellet about bounds of modules of polynomial zeros. Sufficient conditions are obtained for the polynomial whose Cauchy's bound and Pellet's bounds are strict bounds. The characteristics is given for the polynomial whose Cauchy's bound or Pellet's bounds can be achieved by the modules of zeros of the polynomial.     

Newton's method for a class of nonsmooth functions

This paper presents and justifies a Newton iterative process for finding zeros of functions admitting a certain type of approximation. This class includes smooth functions as well as nonsmooth reformulations of variational inequalities. We prove for this method an analogue of the fundamental local convergence theorem of Kantorovich including optimal error bounds.The research reported here was sponsored by the National Science Foundation under Grants CCR-8801489 and CCR-9109345, by the Air Force Systems Command, USAF, under Grants AFOSR-88-0090 and F49620-93-1-0068, by the U. S. Army Research Office under Grant No. DAAL03-92-G-0408, and by the U. S. Army Space and Strategic Defense Command under Contract No. DASG60-91-C-0144. The U. S. Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.     

Interpolatory product quadratures for Cauchy principal value integrals with Freud weights

Summary. We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions. Received July 15, 1997 / Revised version received August 25, 1998