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:     

凝固过程的温度场数值模拟

通过对铸件凝固过程中各换热边界条件的研究,建立了凝固过程的二维非稳态温度场计算数学模型;并运用了有限差分方法对模型进行离散,得到大型方程组,并利用超松驰迭代法(即SOR法)解该方程组,据此,利用Turbo C编制了计算机程序.上机运行结果表明,可较满意地模拟凝固过程温度场的分布.     

一类新的求解双边障碍问题的数值方法

针对双边障碍问题的离散互补形式,提出了一类新的格式将其等价转化为方程组的形式,并采用牛顿迭代法进行求解.实验结果显示所提算法能快速,有效地计算出数值解和接触集.     

分裂型单调迭代方法在非线性抛物型微分方程组初边值问题的应用

由于物理学,生物学,工程等学科中存在着大量没有解决的非线性问题,这些众多的非线性问题常常归结为各种可以近似求解的方程组。因此,研究非线性方程组解的近似与构造可解性理论已成为目前引起人们重视的课题,例如单调迭代法就是其中最为重要的课题之一,它具有大范围收敛性,迭代过程的每一步单调包含未知解,对方程组解的存在性给出了计算检验等优点。     

电厂冷却水在湖泊和水库中热扩散过程的数学模拟

本文对电厂冷却水排入湖泊和水库后的热扩散过程进行了数学模拟,建立了三维扩散方程,给出了一维、二维和三维计算湖泊和水库中任意点任意时刻水温上升量的计算公式,并通过实例对一维情形的计算公式进行了验证,结果很好.     

解强刚性块线代数方程组的两类L-收敛迭代法

对解强刚性块线代数方程组X=(A(?)J)X φ,本文提出了L-收敛的最佳单参数迭代法(L-OOPI)和L-收敛的多参数迭代直接法(L-MPID),并给出了数值例子．数例表明,对于强刚性块线代数方程组,该二迭代法是有效的．     

关于线性代数方程的一种迭代解法的收敛性问题

用迭代法求解线性代数方程组,已有大量的文献与专著,例如[4、6、7]。最常用的是逐次超松弛,及其种种变形。但是,许多情况表明这些方法并非完全令人满意的,特别对病态线性代数方程组,即方程组的系数矩阵有大的条件数,用这些方法求解时,收敛得相当慢。 [1]对求解病态常微分方程初值问题构造了一种恒稳格式。从线性代数方程组的解,等价于某一常微分方程组初值问题的稳态解,这一事实出发,从而构造了一种新的求解线性代数方程组的迭代解法。[1、2]某些计算实例表明,此迭代法特别适合于求解病态线性     

关于Jain的隐式迭代法的注

本文利用嵌入法思想构造了一类求解非线性方程组的隐式迭代法，分析了方法的收敛阶，给出了具体的计算格式，最后的计算结果表明了方法的有效性．     

Neumann边值问题多格子方法的收缩数

迄今为止对线性多格子方法收敛性的分析大多限于所得的离散方程组是非奇的情况。本文应用奇异线性方程组迭代法的收敛理论对Neumann边值问题的Braess 2格子算法进行模型问题分析,计算了2格子迭代的谱半径(渐近收缩数)以及谱范数和能量范数下的收缩数。所得结果与对Dirichlet边值问题所得的结果基本上是相同的,     

一类高阶牛顿迭代法及其在线性互补问题中的应用

通过等价转换,把线性互补问题转化为一个不可微的非线性方程组,进而采用光滑函数处理,得到一个光滑非线性方程组,利用高阶牛顿迭代法进行求解.该方法不再区分线性互补问题是否单调,因此扩大了线性互补问题的求解范围.计算结果表明,方法计算速度快,对线性互补问题求解较为有效.     

解线性方程组的一种新方法

本文提出一种求解线性方程组的快速Ｊａｃｏｂｉ迭代方法，该方法在通常的串行计算机上比Ｇａｕｓｓ－Ｓｅｉｄｅｌ方法快，而且精度高，它对收敛慢的大型线性计算特别有效。     

求解带非线性边界条件的反应扩散方程数值解的组合迭代方法

1　引　　言我们首先考虑如下抛物型方程ｕｔ-ＤΔｕ =ｆ(ｘ ,ｔ ,ｕ) (ｔ∈ ( 0 ,Ｔ],ｘ∈Ω ) ｕ/ ν+ βｕ =ｇ(ｘ ,ｔ ,ｕ) (ｔ∈ ( 0 ,Ｔ],ｘ∈ Ω )ｕ(ｘ ,0 ) =ψ(ｘ) (ｘ∈Ω )( 1 .1 )其中Ｔ为正常数 ,Ω 是ＲＰ 空间的有界区域 记ＱＴ=Ω × ( 0 ,Ｔ],ＳＴ= Ω × ( 0 ,Ｔ],假设在ＱＴ上Ｄ≡ｄ(ｘ ,ｔ) >0 ,在ＳＴ 上β≡β(ｘ ,ｔ)≥ 0 又设 ｆ(ｘ ,ｔ,ｕ) ,ｇ(ｘ ,ｔ,ｕ)为关于ｕ的非线性函数 ,且对ｘ ,ｔ各参数满足Ｈ¨ｏｌｄｅｒ连续条件 将 ( 1 .1 )离散化之后我们得到相应的有限差分系统 ,当 ｇ(ｘ ,ｔ,ｕ)为ｕ的线性…     

On a new iterative method for solving linear systems and comparison results

In Ujević [A new iterative method for solving linear systems, Appl. Math. Comput. 179 (2006) 725–730], the author obtained a new iterative method for solving linear systems, which can be considered as a modification of the Gauss–Seidel method. In this paper, we show that this is a special case from a point of view of projection techniques. And a different approach is established, which is both theoretically and numerically proven to be better than (at least the same as) Ujević's. As the presented numerical examples show, in most cases, the convergence rate is more than one and a half that of Ujević.     

Gauss–Seidel Estimation of Generalized Linear Mixed Models With Application to Poisson Modeling of Spatially Varying Disease Rates

Generalized linear mixed models (GLMMs) are often fit by computational procedures such as penalized quasi-likelihood (PQL). Special cases of GLMMs are generalized linear models (GLMs), which are often fit using algorithms like iterative weighted least squares (IWLS). High computational costs and memory space constraints make it difficult to apply these iterative procedures to datasets having a very large number of records.

We propose a computationally efficient strategy based on the Gauss–Seidel algorithm that iteratively fits submodels of the GLMM to collapsed versions of the data. The strategy is applied to investigate the relationship between ischemic heart disease, socioeconomic status, and age/gender category in New South Wales, Australia, based on outcome data consisting of approximately 33 million records. For Poisson and binomial regression models, the Gauss–Seidel approach is found to substantially outperform existing methods in terms of maximum analyzable sample size. Remarkably, for both models, the average time per iteration and the total time until convergence of the Gauss–Seidel procedure are less than 0.3% of the corresponding times for the IWLS algorithm. Platform-independent pseudo-code for fitting GLMS, as well as the source code used to generate and analyze the datasets in the simulation studies, are available online as supplemental materials.     

Splitting iterative methods for fuzzy system of linear equations

A class of splitting iterative methods is considered for solving fuzzy system of linear equations, which cover Jacobi, Gauss–Seidel, SOR, SSOR, and their block variants proposed by others before. We give a convergence theorem for a regular splitting, where the corresponding iterative methods converge to the strong fuzzy solution for any initial vector and fuzzy right-hand vector. Two schemes of splitting are given to illustrate the theorem. Numerical experiments further show the efficiency of the splitting iterative methods.     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

A generalized preconditioned HSS method for non-Hermitian positive definite linear systems

Based on the HSS (Hermitian and skew-Hermitian splitting) and preconditioned HSS methods, we will present a generalized preconditioned HSS method for the large sparse non-Hermitian positive definite linear system. Our method is essentially a two-parameter iteration which can extend the possibility to optimize the iterative process. The iterative sequence produced by our generalized preconditioned HSS method can be proven to be convergent to the unique solution of the linear system. An exact parameter region of convergence for the method is strictly proved. A minimum value for the upper bound of the iterative spectrum is derived, which is relevant to the eigensystem of the products formed by inverse preconditioner and splitting. An efficient preconditioner based on incremental unknowns is presented for the actual implementation of the new method. The optimality and efficiency are effectively testified by some comparisons with numerical results.     

Some extensions of the improved modified Gauss–Seidel iterative method for H‐matrices

In this paper, first we present a convergence theorem of the improved modified Gauss–Seidel iterative method, referred to as the IMGS method, for H‐matrices and compare the range of parameters α_i with that of the parameter ω of the SOR iterative method. Then with a more general splitting, the convergence analysis of this method for an H‐matrix and its comparison matrix is given. The spectral radii of them are also compared. Copyright © 2006 John Wiley & Sons, Ltd.     

Method of penalization for the state equation for an elliptical optimal control problem

We solve by finite difference method an optimal control problem of a system governed by a linear elliptic equation with pointwise control constraints and non-local state constraints. A discrete optimal control problem is approximated by a minimization problem with penalized state equation. We derive the error estimates for the distance between the exact and regularized solutions. We also prove the rate of convergence of block Gauss–Seidel iterative solution method for the penalized problem. We present and analyze the results of the numerical experiments.