陀螺系统特征值问题的收缩Jacobi-Davidson方法

本文研究陀螺系统特征值问题的Jacobi-Davidson方法. 利用陀螺系统的结构性质,给出了求解Jacobi-Davidson方法中校正方程的有效方法. 基于非等价低秩收缩技术,给出了计算陀螺系统一些特征值的收缩Jacobi-Davidson方法. 数值结果表明本文所给算法是有效的.     

非线性特征值问题的二次近似方法

本文研究求解非线性特征值问题的数值方法.基于矩阵值函数的二次近似,将非线性特征值问题转化为二次特征值问题,提出了求解非线性特征值问题的逐次二次近似方法,分析了该方法的收敛性.结合求解二次特征值问题的Arnoldi方法和Jacobi-Davidson方法,给出求解非线性特征值问题的一些二次近似方法.数值结果表明本文所给算法是有效的.     

解大规模矩阵特征问题的复合正交投影方法 *

对于求解大规模矩阵特征问题的经典正交投影类方法 ,当矩阵非Hermite时 ,Ritz向量收敛比Ritz值收敛要困难得多 .已有一类新的精化正交投影类方法 ,它们用精化的近似特征向量取代标准的Ritz向量来逼近所求的特征向量 .证明了在某种意义下 ,每个精化方法是两个经典方法的复合 ,精化近似特征向量满足某个Her mite半正定矩阵在同一个子空间上的经典正交投影 ,进而 ,用特征向量到子空间的距离建立了精化近似特征向量的先验误差界 .结果表明 ,精化的近似特征向量和对应的Ritz值收敛的充分条件相同 .     

求解右定两参数特征值问题的精化Jacobi-Davidson方法(英文)

在文献[1]中,作者M E Hochstenbach和B Plestenjak认为精化的方法不适合两参数特征值问题,原因是求解两参数特征值问题的精化方法存在着三个问题:即精化Ritz向量收敛性差,运算量大,不能计算多个特征值.本文指出,事实并非如此.针对右定两参数特征值问题,本文提出了一种有效的精化数值方法.并通过理论证明和数值实验说明了Ritz值的收敛性,以及精化Ritz向量具有比通常的Ritz向量更好的收敛性.     

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.     

调和Arnoldi方法的一种变形

讨论求解大规模非对称矩阵内部特征问题的一种方法,与标准的调和A rnold i方法相比,该方法仍用调和R itz值作为特征值的近似,而在近似特征向量选取方面,我们充分利用A rnold i过程所提供的最末一个基向量的信息,在多1维K ry lov子空间中选取一个向量-称之为改进的调和R itz向量-作为所求的特征向量的近似.理论分析和数值试验均表明这种变形的调和A rnold i方法的可行性和有效性.     

计算大规模矩阵最大最小奇异值和奇异向量的两个精化Lanczos算法

1.引言 在科学工程计算中经常需要计算大规模矩阵的少数最大或最小的奇异值及其所对应的奇异子空间。例如图像处理中要计算矩阵端部奇异值之比作为图像的分辨率,诸如此类的问题还存在于最小二乘问题、控制理论、量子化学中等等。然而大多实际问题中的矩阵是大型稀疏矩阵,且需要的是矩阵的部分奇异对。如果计算A的完全奇异值分解(SVD),则运算量和存储量极大,甚至不可能。因此必须寻求其它有效可靠的算法。 假设A的SVD为     

KdV方程的特征线混合间断有限元方法

考虑KdV方程的两种特征线性混合间断有限元方法,一种方法建立在标准特征线修正方法的基础上,另一种方法是带有对流项修正的特征线修正方法.利用具有实际物理意义的特征线追踪技巧处理时间导数项和对流项,采用混合间断有限元方法处理三阶导数项,分别证明了两种方法有限元解的唯一性、稳定性和误差估计.     

计算最小奇异组的一个精化调和 Lanczos双对角化方法

在很多实际应用中需要计算大规模矩阵的若干个最小奇异组.调和投影方法是计算内部特征对的常用方法,其原理可用于求解大规模奇异值分解问题.本文证明了,当投影空间足够好时,该方法得到的近似奇异值收敛,但近似奇异向量可能收敛很慢甚至不收敛.根据第二作者近年来提出的精化投影方法的原理,本文提出一种精化的调和Lanczos双对角化方法,证明了它的收敛性.然后将该方法与Sorensen提出的隐式重新启动技术相结合,开发出隐式重新启动的调和Lanczos双对角化算法(IRHLB)和隐式重新启动的精化调和Lanczos双对角化算法(IRRHLB).位移的合理选取是算法成功的关键之一,本文对精化算法提出了一种新的位移策略,称之为"精化调和位移".理论分析表明,精化调和位移比IRHLB中所用的调和位移要好,且可以廉价可靠地计算出来.数值实验表明,IRRHLB比IRHLB要显著优越,而且比目前常用的隐式重新启动的Lanczos双对角化方法(IRLB)和精化算法IRRLB更有效.     

半直线上BBM方程初边值问题的修正有理谱耦合方法

本文考虑使用修正的有理谱方法处理半直线上的BBM方程初边值问题.对非线性项使用Chebyshev有理插值显式处理,而线性项使用修正Legendre有理谱方法隐式处理.这种处理既可以节约运算又可以保持良好的稳定性.数值例子表明了算法的有效性     

Composite orthogonal projection methods for large matrix eigenproblems

For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. Apriori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones. Project supported by the China State Major Key Projects for Basic Researches, the National Natural Science Foundation of China (Grant No. 19571014), the Doctoral Program (97014113), the Foundation of Excellent Young Scholors of Ministry of Education, the Foundation of Returned Scholars of China and the Liaoning Province Natural Science Foundation.     

Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems

In this paper we will show how the Jacobi-Davidson iterative method can be used to solve generalized eigenproblems. Similar ideas as for the standard eigenproblem are used, but the projections, that are required to reduce the given problem to a small manageable size, need more attention. We show that by proper choices for the projection operators quadratic convergence can be achieved. The advantage of our approach is that none of the involved operators needs to be inverted. It turns out that similar projections can be used for the iterative approximation of selected eigenvalues and eigenvectors of polynomial eigenvalue equations. This approach has already been used with great success for the solution of quadratic eigenproblems associated with acoustic problems.Our friend Albert died on November 12, 1995     

Some theoretical comparisons of refined Ritz vectors and Ritz vectors

Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (A, x) of a large matrix A. Given a subspace ω that contains an approximation to x, these two methods compute approximations (μ,x) and (μ,x) to (λ, x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector x is unique as the     

计算部分奇异值分解的隐式重新启动的双对角化Lanczos方法和精化的双对角化Lanczos方法

The singular value decomposition problem is mathematically equivalent to the eigenproblem of an argumented matrix. Golub et al. give a bidiagonalization Lanczos method for computing a number of largest or smallest singular values and corresponding singular vertors, but the method may encounter some convergence problems. In this paper we analyse the convergence of the method and show why it may fail to converge. To correct this possible nonconvergence, we propose a refined bidiagonalization Lanczos method and apply the implicitly restarting technique to it, and we then present an implicitly restarted bidiagonalization Lanczos algorithm(IRBL) and an implicitly restarted refined bidiagonalization Lanczos algorithm (IRRBL). A new implicitly restarting scheme and a reliable and efficient algorithm for computing refined shifts are developed for this special structure eigenproblem.Theoretical analysis and numerical experiments show that IRRBL performs much better than IRBL.     

Riccati-based preconditioner for computing invariant subspaces of large matrices

Summary. This paper introduces and analyzes the convergence properties of a method that computes an approximation to the invariant subspace associated with a group of eigenvalues of a large not necessarily diagonalizable matrix. The method belongs to the family of projection type methods. At each step, it refines the approximate invariant subspace using a linearized Riccati's equation which turns out to be the block analogue of the correction used in the Jacobi-Davidson method. The analysis conducted in this paper shows that the method converges at a rate quasi-quadratic provided that the approximate invariant subspace is close to the exact one. The implementation of the method based on multigrid techniques is also discussed and numerical experiments are reported. Received June 15, 2000 / Revised version received January 22, 2001 / Published online October 17, 2001     

Alternative correction equations in the Jacobi‐Davidson method

The correction equation in the Jacobi‐Davidson method is effective in a subspace orthogonal to the current eigenvector approximation, whereas for the continuation of the process only vectors orthogonal to the search subspace are of importance. Such a vector is obtained by orthogonalizing the (approximate) solution of the correction equation against the search subspace. As an alternative, a variant of the correction equation can be formulated that is restricted to the subspace orthogonal to the current search subspace. In this paper, we discuss the effectiveness of this variant. Our investigation is also motivated by the fact that the restricted correction equation can be used for avoiding stagnation in the case of defective eigenvalues. Moreover, this equation plays a key role in the inexact TRQ method [18]. Copyright © 1999 John Wiley & Sons, Ltd.     

A Jacobi-Davidson method for two real parameter nonlinear eigenvalue problems arising from delay differential equations

The critical delays of a delay-differential equation can be computed by solving a nonlinear two-parameter eigenvalue problem. For large scale problems, we propose new correction equations for a Jacobi-Davidson type method, that also forces real valued critical delays. We present two different equations: one complex valued equation using a direct linear system solver, and one Jacobi-Davidson style correction equation which is suitable for an iterative linear system solver. A numerical example of a large scale problem arising from PDEs shows the effectiveness of the method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

精化的二次残量迭代法

According to the refined projection principle advocated by Jia[8], we improve the residual iteration method of quadratic eigenvalue problems and propose a refined residual iteration method. We study the restarting issue of the method and develop a practical algorithm. Preliminary numerical examples illustrate the efficiency of the method.