块二级迭代法的近似最优内迭代次数

本文讨论线性方程组定常块二级迭代法内迭代次数的选择.对于单调矩阵,证明了块Jacobi矩阵的谱半径ρp(T)为非定常块二级迭代法R_1-因子的下界.对于M-矩阵,用某个单调范数给出了ρ(T_p)的关于p单调下降且收敛于ρ(T)的上界.于是,当系数矩阵为M-矩阵时,我们定义了定常块二级迭代法的近似最优内迭代次数.所定义的近似最优值与模型问题数值计算的实际最优值非常吻合.本文分析表明,实际计算中应该把内迭代次数控制在较小的数目.     

一类T单调算子障碍问题的块单调迭代法

本文对一类T单调算子的障碍问题提出了几个块迭代法．所得的迭代序列为上解序列和下解序列,它们均单调收敛于问题的准确解,本文建立了这些算法的比较定理．     

关于PageRank的广义二级分裂迭代方法

本文研究计算PageRank的迭代法,在Gleich等人提出的内/外迭代方法的基础上,提出了具有三个参数的广义二级分裂迭代法,该方法包含了内/外迭代法和幂迭代法,并研究了该方法的收敛性.基于该方法的收缩因子的计算公式,讨论了迭代参数可能的选择,通过参数的选择能有效提高内/外迭代法的收敛效率.     

牛顿迭代法优于预测式迭代法

从两个方面说明牛顿迭代法优于预测式迭代法：1．牛顿迭代法的收敛阶数高于预测式迭代法的收敛阶数。2·从算法复杂性出发，采用Ostrowski给出的“迭代过程有效性指标的概念，得到牛顿迭代法的有效性指标是2^1/3，预测式迭代法的有效性指标是3^1/3．     

块H－矩阵的进一步推广

对Ｎａｂｂｅｎ［２］提出的块Ｈ－矩阵做进一步推广，使得非对角块矩阵不必是Ｈｅｒｍｉｔｅ矩阵．但仍保留其基本特征不变．对块Ｈ－矩阵提出块Ｈ－分裂及块相容Ｈ－分裂．证明了矩阵的任意块相容Ｈ－分裂都是收敛分裂．对ＪＯＲ迭代法给出松驰参数的上界．     

求解混合型Lyapunov矩阵方程的参数迭代法

采用参数迭代法求一类混合型Lyapunov矩阵方程A~TX XA B~TXB=C的对称解.在方程相容的条件下,给出了迭代法收敛的充要条件和一些充分条件,以及参数的选取方法.最后,利用数值算例对有关结果进行了验证.     

修正的三阶收敛的牛顿迭代法

给出了牛顿迭代法的两种修正形式,证明了它们是三阶收敛的,数值实验表明,与其它已知的三阶收敛的牛顿迭代法相比,修正的牛顿迭代法具有一定的优势.     

非Hermitian正定线性方程组的外推的HSS迭代方法

为了高效地求解大型稀疏非Hermitian正定线性方程组，在白中治、Golub和Ng提出的Hermitian和反Hermitian分裂（HSS）迭代法的基础上，通过引入新的参数并结合迭代法的松弛技术，对HSS迭代方法进行加速，提出了一种新的外推的HSS迭代方法（EHSS），并研究了该方法的收敛性.数值例子表明：通过参数值的选择，新方法比HSS方法具有更快的收敛速度和更少的迭代次数，选择了合适的参数值后，可以提高HSS方法的收敛效率.     

关于Katz指标的二级分裂迭代方法

本文研究复杂网络中计算Katz指标的迭代法,基于网络拓扑结构,在快速Katz指标算法的基础上,运用二级分裂迭代思想,提出了具有两个参数的二级分裂迭代法,并研究了该方法的收敛性.基于该方法的收缩因子的计算公式,讨论了迭代参数可能的选择,通过参数的选择能有效提高二级迭代法的收敛效率.最后通过数值实例验证了此方法的有效性.     

Jacobi和Gauss-Seidel迭代法收敛性的判定

§1 引言 解线代数方程组 AX=b 的Jacobi迭代法和Gauss-Seidel迭代法收敛的充要条件是Jacobi迭代矩阵B=D(-1)(E F)的谱半径ρ(B)小于1,但验证这一充要条件需要求阵B的特征值,使用很不方便。因此促使人们去寻找使用方便、计算简单判定两迭代法收敛的充分条件。如大家所熟知,两迭代法收敛的一充分条件是:     

TWO ITERATION METHODS FOR SOLVING LINEAR ALGEBRAIC SYSTEMS WITH LOW ORDER MATRIX A AND HIGH ORDER MATRIX B:Y=(AB)Y Ф

1. IntroductionIt is wen known that for IvP of stiff ODEsy, = f(y), to < t 5 T y(to) ~ yo E R", f: fi E R" E R", m >> 0 (1.1)implicit method with good stability have to be used, e.g., IRK methods[7], implicitblock methods[4'12--lv,18], etc. At each illtegral step, each of all these methods bringsabout solving block nonlinear equation systemsY = h(A @ Im)F(Y) afl, A E R'*,, Y,F(y), 4 E Rm,, ms >> 0, (1.2)where h is the stepsize, @ kronecker product, Im e Rm identity matrix, Y = (y…     

Two Iteration Methods for Solving Linear Algebraic Systems with Low Order Matrix $A$ and High Order Matrix $B:Y=(A \otimes B)Y+\Phi$

This paper presents optimum a one-parameter iteration (OOPI) method and a multi-parameter iteration direct (MPID) method for efficiently solving linear algebraic systems with low order matrix $A$ and high order matrix $B: Y = (A \otimes B)Y+\Phi$. On parallel computers (also on serial computer) the former will be efficient, even very efficient under certain conditions, the latter will be universally very efficient.     

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.     

A basic family of iteration functions for polynomial root finding and its characterizations

Let p(x) be a polynomial of degree n?2 with coefficients in a subfield K of the complex numbers. For each natural number m?2, let L_m(x) be the m×m lower triangular matrix whose diagonal entries are p(x) and for each j=1,…,m−1, its jth subdiagonal entries are . For i=1,2, let L_m^i)(x) be the matrix obtained from L_m(x) by deleting its first i rows and its last i columns. L₁⁽¹⁾(x)≡1. Then, the function B_m(x)=x−p(x) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, B_m(x) coincides with Newton's and Halley's, respectively. The function B_m(x) is a member of S(m,m+n−2), where for any M?m, S(m,M) is the set of all rational iteration functions g(x) ∈ K(x) such that for all roots θ of p(x), then g(x)=θ+∑_i=m^Mγ_i(x)(θ−x)ⁱ, with γ_i(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gⁱ(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is . For B_m(x) we obtain . If all roots of p(x) are simple, B_m(x) is the unique member of S(m,m + n − 2). By making use of the identity , we arrive at two recursive formulas for constructing iteration functions within the S(m,M) family. In particular, the family of B_m(x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schröder, whose mth order member belongs to S(m,mn), m>2. The iteration functions within S(m,M) can be extended to any arbitrary smooth function f, with the uniform replacement of p^(j) with f^(j) in g as well as in γ_m(θ).     

A Liouville-type theorem for the stationary MHD equations

We establish a new Liouville-type theorem for solutions of the stationary MHD equations imposing asymmetric oscillation growth conditions on the tensor-valued functions for the velocity and the magnetic field.     

An efficient numerical method for preconditioned saddle point problems

In this paper, we consider the solution of linear systems of saddle point type by a preconditioned numerical method. We first transform the original linear system into two sub-systems with small size by a preconditioning strategy, then employ the conjugate gradient (CG) method to solve the linear system with a SPD coefficient matrix, and a splitting iteration method to solve the other sub-system, respectively. Numerical experiments show that the new method can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime and iteration steps.     

A numerical method of structure-preserving model updating problem and its perturbation theory

A method is presented to update a special finite element (FE) analytical model, based on matrix approximation theory with spectral constraint. At first, the model updating problem is treated as a matrix approximation problem dependent on the spectrum data from vibration test and modal parameter identification. The optimal approximation is the first modified solution of FE model. An algorithm is given to preserve the sparsity of the model by multiple correction. The convergence of the algorithm is investigated and perturbation of the modified solution is analyzed. Finally, a numerical example is provided to confirm the convergence of the algorithm and perturbation theory.     

A new primal-dual path-following interior-point algorithm for semidefinite optimization

In this paper we present a new primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only full Nesterov-Todd step. Moreover, we obtain the currently best known iteration bound for the algorithm with small-update method, namely, , which is as good as the linear analogue.     

Positive definite solutions of the nonlinear matrix equation X + AXA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A^∗X^qA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.