On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.     

A refined iterative algorithm based on the block arnoldi process for large unsymmetric eigenproblems

When the matrix in question is unsymmetric, the approximate eigenvectors or Ritz vectors obtained by orthogonal projection methods including Arnoldi's method and the block Arnoldi method cannot be guaranteed to converge in theory even if the corresponding approximate eigenvalues or Ritz values do. In order to circumvent this danger, a new strategy has been proposed by the author for Arnoldi's method. The strategy used is generalized to the block Arnoldi method in this paper. It discards Ritz vectors and instead computes refined approximate eigenvectors by small-sized singular-value decompositions. It is proved that the new strategy can guarantee the convergence of refined approximate eigenvectors if the corresponding Ritz values do. The resulting refined iterative algorithm is realized by the block Arnoldi process. Numerical experiments show that the refined algorithm is much more efficient than the iterative block Arnoldi algorithm.     

A geometric theory for preconditioned inverse iteration applied to a subspace

The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration, i.e., the well-known inverse iteration procedure, where the associated system of linear equations is solved approximately by using a preconditioner. This step is followed by a Rayleigh-Ritz projection so that preconditioned inverse iteration is always applied to the Ritz vectors of the actual subspace of approximate eigenvectors. The given theory provides sharp convergence estimates for the Ritz values and is mainly built on arguments exploiting the geometry underlying preconditioned inverse iteration.     

IMPROVING EIGENVECTORS IN ARNOLDI''''S METHOD

1. IntroductionArnoldi's method [1, 12] is used for computing.,a few selected eigenpairs of largeunsymmetric matrices. It hajs been investigated since the 1980s; see, e-g., [3--15].It is well known that the m--step Arnoldi processt as described in detail in Section 2,generates an orthonormal basis {yi}7=1 of the Krylov subspace Km(vi, A) spanned byvil Avi,... 5 Am--'v,. Here yi is an initial unit norm vector. The projected matrix ofA onto Km(vi, A) is represented by an m x m upper Hessenb…     

Generalized block Lanczos methods for large unsymmetric eigenproblems

Generalized block Lanczos methods for large unsymmetric eigenproblems are presented, which contain the block Arnoldi method, and the block Arnoldi algorithms are developed. The convergence of this class of methods is analyzed when the matrix A is diagonalizable. Upper bounds for the distances between normalized eigenvectors and a block Krylov subspace are derived, and a priori theoretical error bounds for Ritz elements are established. Compared with generalized Lanczos methods, which contain Arnoldi's method, the convergence analysis shows that the block versions have two advantages: First, they may be efficient for computing clustered eigenvalues; second, they are able to solve multiple eigenproblems. However, a deep analysis exposes that the approximate eigenvectors or Ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix A even if corresponding approximate eigenvalues or Ritz values do, since the convergence of Ritz vectors needs more sufficient conditions, which may be impossible to satisfy theoretically, than that of Ritz values does. The issues of how to restart and to solve multiple eigenproblems are addressed, and some numerical examples are reported to confirm the theoretical analysis. Received July 7, 1994 / Revised version received March 1, 1997     

A VARIATION ON THE BLOCK ARNOLDIMETHOD FOR LARGE UNSYMMETRIC MATRIX EIGENPROBLEMS

1.IntroductionLarge-scalematrixeigenproblemsariseinappliedsciencesandmanyengineeringapplications.Arnoldi'smethod[1'2]anditsblockversion[3--6]areverypopularforsolvingthem.Thesemethodshavebeenintensivelyinvestigatedsincethe1980s,bothintheoryandinalgorithms;wereferto[7--17]fordetails.WhenmstepsoftheblockArnoldiprocessareperformed,anorthonormalbasis{K}7=1oftheblockKrylovsubspaceK.(VI,A)spannedbyVI5AVI,'IAm--1VIisgenerated,whereVIisaninitialNxporthogonalmatrix,andtherestrictionofAtoKm(V…     

求解大规模非对称矩阵特征问题的精化双正交Lanczos算法

We combine Lanczos algorithm with the thought of the refined projection method using QR factorization and propose the refined biothogonalization Lanczos method for computing the desired eigenvalues of large unsymmetric matrix. With low cost of work space and flops the algorithm cures the disease that the Ritz vectors may not converge when the Ritz values converge usingthe Lanczos method. Numerical experiments show our algorithm is considerably more stable and efficient than its counterpart.     

A modified harmonic block Arnoldi algorithm with adaptive shifts for large interior eigenproblems

The harmonic block Arnoldi method can be used to find interior eigenpairs of large matrices. Given a target point or shift τ$\tau$ to which the needed interior eigenvalues are close, the desired interior eigenpairs are the eigenvalues nearest τ$\tau$ and the associated eigenvectors. However, it has been shown that the harmonic Ritz vectors may converge erratically and even may fail to do so. To do a better job, a modified harmonic block Arnoldi method is coined that replaces the harmonic Ritz vectors by some modified harmonic Ritz vectors. The relationships between the modified harmonic block Arnoldi method and the original one are analyzed. Moreover, how to adaptively adjust shifts during iterations so as to improve convergence is also discussed. Numerical results on the efficiency of the new algorithm are reported.     

Block tensor conjugate gradient-type method for Rayleigh quotient minimization in two-dimensional case

A method for solving a partial algebraic eigenvalues problem is constructed. It exploits tensor structure of eigenvectors in two-dimensional case. For a symmetric matrix represented in tensor format, the method finds low-rank approximations to the eigenvectors corresponding to the smallest eigenvalues. For sparse matrices, execution time and required memory for the proposed method are proportional to the square root of miscellaneous overall number of unknowns, whereas this dependence is usually linear. To maintain tensor structure of vectors at each iteration step, low-rank approximations are performed, which introduces errors into the original method. Nevertheless, the new method was proved to converge. Convergence rate estimates are obtained for various tensor modifications of the abstract one-step method. It is shown how the convergence of a multistep method can be derived from the convergence of the corresponding one-step method. Several modifications of the method with an low-rank approximation techniques were implemented on the basis of the block conjugate gradient method. Their performance is compared on numerical examples.     

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.     

A PROCESS FOR SOLVING A FEW EXTREME EIGENPAIRS OF LARGE SPARSE POSITIVE DEFINITE GENERALIZED EIGENVALUE PROBLEM

1. IntroductionWe are concerned in this work with finding a few extreme eigenvalues and theircorresponding eigenvectors of a generalized large scale eigenvalue problem in which thematrices are sparse and symmetric positive definite.Although finding a few extreme eigenpairs is of interest both in theory and practice,there are only few usable and efficient methods up to now. Reinsch and Baner ([12]),suggested a oR algorithm with Newton shift for the standard eigenproblem which included an ingen…     

On proximity of Rayleigh quotients for different vectors and Ritz values generated by different trial subspaces

The Rayleigh quotient is unarguably the most important function used in the analysis and computation of eigenvalues of symmetric matrices. The Rayleigh-Ritz method finds the stationary values of the Rayleigh quotient, called Ritz values, on a given trial subspace as optimal, in some sense, approximations to eigenvalues.In the present paper, we derive upper bounds for proximity of the Ritz values in terms of the proximity of the trial subspaces without making an assumption that the trial subspace is close to an invariant subspace. The main result is that the absolute value of the perturbations in the Ritz values is bounded by a constant times the gap between the original trial subspace and its perturbation. The constant is the spread in the matrix spectrum, i.e. the difference between the largest and the smallest eigenvalues of the matrix. It’s shown that the constant cannot be improved. We then generalize this result to arbitrary unitarily invariant norms, but we have to increase the constant by a factor of .Our results demonstrate, in particular, the stability of the Ritz values with respect to a perturbation in the trial subspace.     

An augmented LSQR method

The LSQR iterative method for solving least-squares problems may require many iterations to determine an approximate solution with desired accuracy. This often depends on the fact that singular vector components of the solution associated with small singular values of the matrix require many iterations to be determined. Augmentation of Krylov subspaces with harmonic Ritz vectors often makes it possible to determine the singular vectors associated with small singular values with fewer iterations than without augmentation. This paper describes how Krylov subspaces generated by the LSQR iterative method can be conveniently augmented with harmonic Ritz vectors. Computed examples illustrate the competitiveness of the augmented LSQR method proposed.     

Error bounds of Rayleigh–Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems

We investigate contour integral-based eigensolvers for computing all eigenvalues located in a certain region and their corresponding eigenvectors. In this paper, we focus on a Rayleigh–Ritz type method and analyze its error bounds. From the results of our analysis, we conclude that the Rayleigh–Ritz type contour integral-based eigensolver with sufficient subspace size can achieve high accuracy for target eigenpairs even if some eigenvalues exist outside but near the region.     

Angles between infinite dimensional subspaces with applications to the Rayleigh-Ritz and alternating projectors methods

We define angles for infinite dimensional subspaces of Hilbert spaces, inspired by the work of E.J. Hannan, 1961/1962. The angles of Dixmier and Friedrichs, and the gaps are characterized. We establish connections between the angles corresponding to orthogonal complements. The sensitivity of angles with respect to subspaces is estimated. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. An ultimate acceleration of the classical alternating projectors method is proposed. Its convergence rate is estimated in terms of the angles. We illustrate the acceleration for a domain decomposition method with a small overlap for the 1D diffusion equation.     

求解大规模Hamilton矩阵特征问题的辛Lanczos算法的误差分析

对求解大规模稀疏Hamilton矩阵特征问题的辛Lanczos算法给出了舍入误差分析.分析表明辛Lanczos算法在无中断时,保Hamilton结构的限制没有破坏非对称Lanczos算法的本质特性.本文还讨论了辛Lanczos算法计算出的辛Lanczos向量的J一正交性的损失与Ritz值收敛的关系.结论正如所料,当某些Ritz值开始收敛时.计算出的辛Lanczos向量的J-正交性损失是必然的.以上结果对辛Lanczos算法的改进具有理论指导意义.     

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     

Error analysis of symplectic Lanczos method for Hamiltonian eigenvalue problem

A rounding error analysis for the symplectic Lanczos method is given to solve the large-scale sparse Hamiltonian eigenvalue problem. If no breakdown occurs in the method, then it can be shown that the Hamiltonian structure preserving requirement does not destroy the essential feature of the nonsymmetric Lanczos algorithm. The relationship between the loss of J-orthogonality among the symplectic Lanczos vectors and the convergence of the Ritz values in the symmetric Lanczos algorithm is discussed. It is demonstrated that under certain assumptions the computed J-orthogonal Lanczos vectors lose the J-orthogonality when some Ritz values begin to converge. Our analysis closely follows the recent works of Bai and Fabbender. Selected from Journal of Mathematical Research and Exposition, 2004, 24(1): 91–106     

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.     

Error Analysis of the Symplectic Lanczos Method for the Symplectic Eigenvalue Problem

A rounding error analysis of the symplectic Lanczos algorithm for the symplectic eigenvalue problem is given. It is applicable when no break down occurs and shows that the restriction of preserving the symplectic structure does not destroy the characteristic feature of nonsymmetric Lanczos processes. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. As to be expected, it follows that (under certain assumptions) the computed J-orthogonal Lanczos vectors loose J-orthogonality when some Ritz values begin to converge.