求解方程的一类迭代公式

Using the forms of Newton iterative function, the iterative function of Newton's method to handle the problem of multiple roots and the Halley iterative function, we give a class of iterative formulae for solving equations in one variable in this paper and show that their convergence order is at least quadratic. At last we employ our methods to solve some non-linear equations and compare them with Newton's method and Halley's method. Numerical results show that our iteration schemes are convergent if we choose two suitable parametric functions λ(x) and μ(x). Therefore, our iteration schemes are feasible and effective.     

On Extraneous Fixed-Points of the Basic Family of Iteration Functions

Let K denote either the reals or the complex numbers. Consider the root-finding problem for an analytic function f from K into itself via an iteration function F. An extraneous fixed-point of F is a fixed-point different than a root of f. We prove that all extraneous fixed-points of any member of an infinite family of iteration functions, called the Basic Family in Kalantari et al. (1997). are repulsive. This generalizes a result of Vrscay and Gilbert (1988) who prove the property only for the second member of the family which coincides with the well-known Halley's method. Our result implies that a convergent orbit corresponding to any specific member of the Basic Family will necessarily converge to a zero of f. The Basic Family is a fundamental family with several different representations. It has been rediscovered by several authors using various techniques. The earliest derivation of this family is from an analysis of Schröder (1870). But in fact the Basic Family and its multipoint versions are all derivable from a determinantal generalization of Taylor's theorem (Kalantari (1997)).     

On the global convergence of Chebyshev's iterative method

In [A. Melman, Geometry and convergence of Euler's and Halley's methods, SIAM Rev. 39(4) (1997) 728–735] the geometry and global convergence of Euler's and Halley's methods was studied. Now we complete Melman's paper by considering other classical third-order method: Chebyshev's method. By using the geometric interpretation of this method a global convergence theorem is performed. A comparison of the different hypothesis of convergence is also presented.     

Root determination by use of Padé approximants

In this paper we describe a new technique for generating iteration formulas — of arbitrary order — for determining a zero (assumed simple) of a functionf, assumed analytic in a region containing the zero. The 1/p Padé Approximant (p0) to the functiong(t)f(z) is formed wherez=w+t, using the Taylor series forf at the pointw, an approxination to the zero off. The value oft for which the 1/p Padé Approximant vanishes provides the basis of iteration formulas of orderp+2.Some known iteration formulas, e.g., Newton-Raphson's, Halley's and Kiss's of order of convergence two, three and four, are directly obtained by settingp=0,1 and 2, respectively.     

A note on Halley's method

Summary We introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point. Making use of this concept we obtain a new theorem of global convergence for Halley's method.     

On the Order of Convergence of a Determinantal Family of Root-Finding Methods

Recently, we have shown that for each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. For each k the corresponding matrices are upper Hessenberg matrices. Additionally, for k = 1 these matrices are Toeplitz matrices. The goal of this paper is to analyze the order of convergence of this fundamental family. Newton's method, Halley's method, and their multi-point versions are members of this family. In this paper we also derive these special cases. We prove that for fixed m, as k increases, the order of convergence decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For fixed k, the order of convergence increases in m. The asymptotic error constant is also derived in terms of special determinants.     

The polyadic structure of factorable function tensors with applications to high-order minimization techniques

Factorable functions are shown to have arrays ofNth-order derivatives (tensors) which are naturally computed as sums of generalized outer product matrices (polyads). The computational implications of this for high-order minimization techniques (such as Halley's method of tangent hyperbolas) are investigated. A direct derivation of these high-order techniques is also given.This research was sponsored in part by the Office of Naval Research under Contract No. N00014-82-K.     

High order iterative methods for approximating square roots

Given any natural numberm 2, we describe an iteration functiong _m (x) having the property that for any initial iterate \sqrt \alpha $$ " align="middle" border="0"> , the sequence of fixed-point iterationx _k +1 =g _m (x _k ) converges monotonically to having anm-th order rate of convergence. Form = 2 and 3,g _m (x) coincides with Newton's and Halley's iteration functions, respectively, as applied top(x) =x ² – .This research is supported in part by the National Science Foundation under Grant No. CCR-9208371.     

The use of approximate factorization in stiff ODE solvers

We consider implicit integration methods for the numerical solution of stiff initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonstiff or mildly stiff with respect to the stepsize. In these nonstiff subintervals, we do not need the (expensive) Newton iteration process. This motivated us to look for an iteration process that converges in mildly stiff situations and is less costly than Newton iteration. The process we have in mind uses modified Newton iteration as the outer iteration process and a linear solver for solving the linear Newton systems as an inner iteration process. This linear solver is based on an approximate factorization of the Newton system matrix by splitting this matrix into its lower and upper triangular part. The purpose of this paper is to combine fixed point iteration, approximate factorization iteration and Newton iteration into one iteration process for use in initial-value problems where the degree of stiffness is changing during the integration.     

Frozen Landweber Iteration for Nonlinear Ill-Posed Problems

In this paper we propose a modification of the Landweber iteration termed frozen Landweberiteration for nonlinear ill-posed problems.A convergence analysis for this iteration is presented.The numericalperformance of this frozen Landweber iteration for a nonlinear Hammerstein integral equation is compared withthat of the Landweber iteration.We obtain a shorter running time of the frozen Landweber iteration based onthe same convergence accuracy.     

关于广义Φ-压缩映射几种迭代序列收敛的等价性

在广义Φ-压缩映射条件下,分别得到了Picard迭代序列与Krasnoselskii迭代序列以及Mann迭代序列与Ishikawa迭代序列收敛的等价性.     

MODIFIED NEWTON＇S ALGORITHM FOR COMPUTING THE GROUP INVERSES OF SINGULAR TOEPLITZ MATRICES

Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton＇s iteration can be done efficiently. We show that the convergence of the modified Newton iteration is still very fast. Numerical results are presented to demonstrate the fast convergence of the proposed method.     

一类弱非线性方程组的Picard-MHSS迭代方法

修正的Hermite/反Hermite分裂(MHSS)迭代方法是一类求解大型稀疏复对称线性代数方程组的无条件收敛的迭代算法.基于非线性代数方程组的特殊结构和性质,我们选取Picard迭代为外迭代方法,MHSS迭代作为内迭代方法,构造了求解大型稀疏弱非线性代数方程组的Picard-MHSS和非线性MHSS-like方法.这两类方法的优点是不需要在每次迭代时均精确计算和存储Jacobi矩阵,仅需要在迭代过程中求解两个常系数实对称正定子线性方程组.除此之外,在一定条件下,给出了两类方法的局部收敛性定理.数值结果证明了这两类方法是可行、有效和稳健的.     

On inexact Newton methods based on doubling iteration scheme for non‐symmetric algebraic Riccati equations

Newton iteration method can be used to find the minimal non‐negative solution of a certain class of non‐symmetric algebraic Riccati equations. However, a serious bottleneck exists in efficiency and storage for the implementation of the Newton iteration method, which comes from the use of some direct methods in exactly solving the involved Sylvester equations. In this paper, instead of direct methods, we apply a fast doubling iteration scheme to inexactly solve the Sylvester equations. Hence, a class of inexact Newton iteration methods that uses the Newton iteration method as the outer iteration and the doubling iteration scheme as the inner iteration is obtained. The corresponding procedure is precisely described and two practical methods of monotone convergence are algorithmically presented. In addition, the convergence property of these new methods is studied and numerical results are given to show their feasibility and effectiveness for solving the non‐symmetric algebraic Riccati equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Fourier-Laplace analysis of the multigrid waveform relaxation method for hyperbolic equations

The multigrid waveform relaxation (WR) algorithm has been fairly studied and implemented for parabolic equations. It has been found that the performance of the multigrid WR method for a parabolic equation is practically the same as that of multigrid iteration for the associated steady state elliptic equation. However, the properties of the multigrid WR method for hyperbolic problems are relatively unknown. This paper studies the multigrid acceleration to the WR iteration for hyperbolic problems, with a focus on the convergence comparison between the multigrid WR iteration and the multigrid iteration for the corresponding steady state equations. Using a Fourier-Laplace analysis in two case studies, it is found that the multigrid performance on hyperbolic problems no longer shares the close resemblance in convergence factors between the WR iteration for parabolic equations and the iteration for the associated steady state equations.     

Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems

By further generalizing the modified skew-Hermitian triangular splitting iteration methods studied in [L. Wang, Z.-Z. Bai, Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math. 44 (2004) 363-386], in this paper, we present a new iteration scheme, called the product-type skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this method. Moreover, when it is applied to precondition the Krylov subspace methods, the preconditioning property of the product-type skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that the product-type skew-Hermitian triangular splitting iteration method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.     

On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.     

迭代的计算与估计

对高次多项式函数这类非线性映射给出了一般 n次迭代的一个计算结果 ,还讨论了一维欧氏空间中一些非多项式型映射的迭代 .在二维欧氏空间中 ,我们给出了几类特殊的非线性迭代的结果 .对于一些难以精确计算迭代表达式的映射 ,我们给出了其迭代的估计 .     

The Quasi-Laguerre iteration

The quasi-Laguerre iteration has been successfully established, by the same authors, in the spirit of Laguerre's iteration for solving the eigenvalues of symmetric tridiagonal matrices. The improvement in efficiency over Laguerre's iteration is drastic. This paper supplements the theoretical background of this new iteration, including the proofs of the convergence properties.

     

Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

A generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian parts. We study the convergence of the generalized skew‐Hermitian triangular splitting iteration methods for non‐Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew‐Hermitian triangular splitting. Then the generalized skew‐Hermitian triangular splitting iteration method is applied to non‐Hermitian positive semidefinite saddle‐point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew‐Hermitian triangular splitting iteration methods are effective for solving non‐Hermitian saddle‐point linear systems with strong skew‐Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.