一类非线性方程组的Newton-PSS迭代法

正定反Hermite分裂(PSS)方法是求解大型稀疏非Hermite正定线性代数方程组的一类无条件收敛的迭代算法.将其作为不精确Newton方法的内迭代求解器,我们构造了一类用于求解大型稀疏且具有非Hermite正定Jacobi矩阵的非线性方程组的不精确Newton-PSS方法,并对方法的局部收敛性和半局部收敛性进行了详细的分析.数值结果验证了该方法的可行性与有效性.     

非线性方程组的非交替Newton-PHSS迭代法

大型稀疏非Hermite正定Jacobi矩阵对应的非线性方程组的迭代求解历来受到重视.结合不精确Newton法和非交替PHSS迭代法,提出了迭代求解非线性方程组的NewtonNPHSS方法,给出了迭代法的局部收敛定理,并演算了数值例子,阐明了Newton-NPHSS是有效的迭代法.     

离散空间分数阶非线性薛定谔方程的MHSS型迭代方法

利用隐式守恒型差分格式来离散空间分数阶非线性薛定谔方程,可得到一个离散线性方程组.该离散线性方程组的系数矩阵为一个纯虚数复标量矩阵、一个对角矩阵与一个对称Toeplitz矩阵之和.基于此,本文提出了用一种\textit{修正的埃尔米特和反埃尔米特分裂}(MHSS)型迭代方法来求解此离散线性方程组.理论分析表明,MHSS型迭代方法是无条件收敛的.数值实验也说明了该方法是可行且有效的.     

非Hermite线性方程组的若干预处理迭代算法

非Hermite线性方程组在科学和工程计算中有着重要的理论研究意义和使用价值,因此如何高效求解该类线性方程组,一直是研究者所探索的方向.通过提出一种预处理方法,对非Hermite线性方程组和具有多个右端项的复线性方程组求解的若干迭代算法进行预处理,旨在提高原算法的收敛速度.最后通过数值试验表明,所提出的若干预处理迭代算法与原算法相比较,预处理算法迭代次数大大降低,且收敛速度明显优于原算法.除此之外,广义共轭A-正交残量平方法(GCORS2)的预处理算法与其他算法相比,具有良好的收敛性行为和较好的稳定性.     

四阶微分方程Hermite有限元的预处理共轭梯度法

对一类四阶微分方程两点边值问题的Hermite有限元方法进行了研究.首先讨论了该方程通常意义下的Galerkin有限元离散,考虑到有限元离散得到的线性方程组的对称正定性,文中采用了预处理最速下降法和共轭梯度方法求解线性方程组,通过选择不同的预处理器,使得求解该方程组的迭代次数有了很大的改观.     

有限元线代数方程组的杂交算法及相应的节点编号方法

在有限元问题的计算中,大家知道耗时最多的是求解线代数方程组的部分。其计算方法一般可分为直接法和迭代法两大类。这两类方法从有限元发展历史看,基本上是以直接法占优势,特别是大型计算机的发展更加强了这个优势。但是从七十年代中后期起,有限元的发展由线性问题向非线性问题过渡,由二维问题向三维问题过渡,特别是微机     

Hermite四点插指公式

文章利用Hermite插值基函数,将求解Hermite四点插指问题转换为求解8个派生出来的多项式插值问题,证明了Hermite四点插指公式的存在唯一性,并用两种方法构造出Hermite四点插指公式,最后给出了一个算例.     

六次PH曲线C~1 Hermite插值

本文讨论六次PH(pythagorean hodograph)曲线的Hermite插值问题.六次PH曲线可以分为两种类型,本文使用参数曲线的复数表示形式,分别给出这两类曲线的构造方法.在给定C1连续的Hermite条件下,需要指定一个自由参数以确定插值曲线,本文进一步阐述这个自由参数的几何意义.由于六次PH曲线是非正则曲线,对于第一类曲线,不易控制奇异点在曲线中的位置;而对于第二类曲线,奇异点可以在构造过程中显式地被指定,因此可以有效地避免其在特定曲线段上的出现.     

非精确Rayleigh商迭代和非精确的简化Jacobi-Davidson方法的收敛性分析

非精确的Rayleigh商迭代被用于计算大型Hermite矩阵的最小特征值和对应的特征向量. 已有文献证明了方法二次收敛. 解决了两个问题: 第一, 证明文献中的原条件不能保证方法二次收敛和收敛到所要求的特征对,更糟的是, 方法可能会错误收敛到其他不要求的特征对. 给出了方法二次收敛的新条件, 称之为一致正条件. 证明在此条件下, 非精确的Rayleigh商迭代可以克服错误收敛的问题,且保证二次收敛到要求的特征值和特征向量. 第二, 不带子空间加速的Jacobi-Davidson~(JD)方法是求解该问题的另一种方法, 给出关于非精确的Jacobi-Davidson方法线性收敛的新证明, 得到一个更紧致的界. 所得的所有理论结果都用数值实验做了验证和分析.     

一类求解非线性代数方程组的并行多分裂AOR算法

本文提出了求解大型非线性代数方程组Aф(x)+Bψ(x)=b的并行多分裂AOR(Accelerated Overrela Xation)算法。在一定的条件下,证明了非线性代数方程组解的存在唯一性,并建立了新算法的全局收敛性理论。     

On semi-convergence of modified HSS iteration methods

We discuss semi-convergence of the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving a broad class of complex symmetric singular linear systems. The semi-convergence theory of the MHSS iteration method is established. In addition, numerical examples show the effectiveness of the MHSS iteration method when it is used as a solver or as a preconditioner (for the restarted GMRES method).     

Complex-extrapolated MHSS iteration method for singular complex symmetric linear systems

In this paper, by extrapolating the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method with a complex relaxation parameter, a complex-extrapolated MHSS (CMHSS) iteration method is present for solving a class of complex singular symmetric of linear equations. We study the semi-convergence properties of the CMHSS iteration method and the extent of the optimal iterative parameters. Furthermore, the convergence conditions also hold for solving nonsingular complex systems. Numerical experiments are given to verify the effectiveness of the CMHSS iteration method for solving both singular and nonsingular complex symmetric systems.     

On semi-convergence of modified HSS method for a class of complex singular linear systems

In this paper, we discuss the semi-convergence of the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving a broad class of complex singular linear systems. Some semi-convergence theories of the MHSS iteration method are established and are weaker than those appeared in previously published works.     

Accelerated PMHSS iteration methods for complex symmetric linear systems

In this paper, we introduce and analyze an accelerated preconditioning modification of the Hermitian and skew-Hermitian splitting (APMHSS) iteration method for solving a broad class of complex symmetric linear systems. This accelerated PMHSS algorithm involves two iteration parameters α,β and two preconditioned matrices whose special choices can recover the known PMHSS (preconditioned modification of the Hermitian and skew-Hermitian splitting) iteration method which includes the MHSS method, as well as yield new ones. The convergence theory of this class of APMHSS iteration methods is established under suitable conditions. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. Theoretical analyses show that the upper bound σ₁(α,β) of the asymptotic convergence rate of the APMHSS method is smaller than that of the PMHSS iteration method. This implies that the APMHSS method may converge faster than the PMHSS method. Numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method.     

MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices

Preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) method is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equation. Motivated by the PMHSS method, we develop a new method of solving a class of linear equations with block two-by-two complex coefficient matrix by introducing two coefficients, noted as DPMHSS. By making use of the DPMHH iteration as the inner solver to approximately solve the Newton equations, we establish modified Newton-DPMHSS (MN-DPMHSS) method for solving large systems of nonlinear equations. We analyze the local convergence properties under the Hölder continuous conditions, which is weaker than Lipschitz assumptions. Numerical results are given to confirm the effectiveness of our method.     

Shifted skew-symmetric iteration methods for nonsymmetric linear complementarity problems

We present a shifted skew-symmetric iteration method for solving the nonsymmetric positive definite or positive semidefinite linear complementarity problems. This method is based on the symmetric and skew-symmetric splitting of the system matrix, which has been adopted to establish efficient splitting iteration methods for solving the nonsymmetric systems of linear equations. Global convergence of the method is proved, and the corresponding inexact splitting iteration scheme is established and analyzed in detail. Numerical results show that the new methods are feasible and effective for solving large sparse and nonsymmetric linear complementarity problems.     

On HSS-based iteration methods for weakly nonlinear systems

Based on separable property of the linear and the nonlinear terms and on the Hermitian and skew-Hermitian splitting of the coefficient matrix, we present the Picard-HSS and the nonlinear HSS-like iteration methods for solving a class of large scale systems of weakly nonlinear equations. The advantage of these methods over the Newton and the Newton-HSS iteration methods is that they do not require explicit construction and accurate computation of the Jacobian matrix, and only need to solve linear sub-systems of constant coefficient matrices. Hence, computational workloads and computer memory may be saved in actual implementations. Under suitable conditions, we establish local convergence theorems for both Picard-HSS and nonlinear HSS-like iteration methods. Numerical implementations show that both Picard-HSS and nonlinear HSS-like iteration methods are feasible, effective, and robust nonlinear solvers for this class of large scale systems of weakly nonlinear equations.     

The semi-convergence properties of MHSS method for a class of complex nonsymmetric singular linear systems

We use the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method to solve a class of complex nonsymmetric singular linear systems. The semi-convergence properties of the MHSS method are studied by analyzing the spectrum of the iteration matrix. Moreover, after investigating the semi-convergence factor and estimating its upper bound for the MHSS iteration method, an optimal iteration parameter that minimizes the upper bound of the semi-convergence factor is obtained. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the MHSS method served both as a preconditioner for GMRES method and as a solver.     

ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS

This paper discusses the accelerating of nonlinear parabolic equations. Two iterative methods for solving the implicit scheme new nonlinear iterative methods named by the implicit-explicit quasi-Newton （IEQN） method and the derivative free implicit-explicit quasi-Newton （DFIEQN） method are introduced, in which the resulting linear equations from the linearization can preserve the parabolic characteristics of the original partial differential equations. It is proved that the iterative sequence of the iteration method can converge to the solution of the implicit scheme quadratically. Moreover, compared with the Jacobian Free Newton-Krylov （JFNK） method, the DFIEQN method has some advantages, e.g., its implementation is easy, and it gives a linear algebraic system with an explicit coefficient matrix, so that the linear （inner） iteration is not restricted to the Krylov method. Computational results by the IEQN, DFIEQN, JFNK and Picard iteration methods are presented in confirmation of the theory and comparison of the performance of these methods.     

Symmetric modified AOR method to solve systems of linear equations

We propose a class of symmetric modified accelerated overrelaxation (SMAOR) methods for solving large sparse linear systems. The convergence region of the method has been investigated. Numerical examples indicate that the SMAOR method is better than other methods such as accelerated overrelaxation(AOR) and modified accelerated overrelaxation(MAOR) methods, since the spectral radius of iteration matrix in SMAOR method is less than that of the other methods. Also, we apply the method to solve a real boundary value problem.