The use of a refined error bound when updating eigenvalues of tridiagonals

The Lanczos algorithm is used to compute some eigenvalues of a given symmetric matrix of large order. At each step of the Lanczos algorithm it is valuable to know which eigenvalues of the associated tridiagonal matrix have stabilized at eigenvalues of the given symmetric matrix. We present a robust algorithm which is fast (20j to 40j operations at the jth Lanczos step), uses about 30 words of extra storage, and has a fairly short program (approxiamately 200 executable statements)     

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     

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.     

Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem

Eigenvalues and eigenvectors of a large sparse symmetric matrix A can be found accurately and often very quickly using the Lanczos algorithm without reorthogonalization. The algorithm gives essentially correct information on the eigensystem of A, although it does not necessarily give the correct multiplicity of multiple, or even single, eigenvalues. It is straightforward to determine a useful bound on the accuracy of every eigenvalue given by the algorithm. The initial behavior of the algorithm is surprisingly good: it produces vectors spanning the Krylov subspace of a matrix very close to A until this subspace contains an exact eigenvector of a matrix very close to A, and up to this point the effective behavior of the algorithm for the eigenproblem is very like that of the Lanczos algorithm using full reorthogonalization. This helps to explain the remarkable behavior of the basic Lanczos algorithm.     

A numerical solution of the constrained weighted energy problem

A numerical algorithm is presented to solve the constrained weighted energy problem from potential theory. As one of the possible applications of this algorithm, we study the convergence properties of the rational Lanczos iteration method for the symmetric eigenvalue problem. The constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of the eigenvalues, on the distribution of the poles, and on the ratio between the size of the matrix and the number of iterations. Our algorithm gives the possibility of finding the boundary of this region in an effective way.We give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational Lanczos algorithm.     

The block conjugate gradient algorithm and related methods

The development of the Lanczos algorithm for finding eigenvalues of large sparse symmetric matrices was followed by that of block forms of the algorithm. In this paper, similar extensions are carried out for a relative of the Lanczos method, the conjugate gradient algorithm. The resulting block algorithms are useful for simultaneously solving multiple linear systems or for solving a single linear system in which the matrix has several separated eigenvalues or is not easily accessed on a computer. We develop a block biconjugate gradient algorithm for general matrices, and develop block conjugate gradient, minimum residual, and minimum error algorithms for symmetric semidefinite matrices. Bounds on the rate of convergence of the block conjugate gradient algorithm are presented, and issues related to computational implementation are discussed. Variants of the block conjugate gradient algorithm applicable to symmetric indefinite matrices are also developed.     

On some structured inverse eigenvalue problems

This work deals with various finite algorithms that solve two special Structured Inverse Eigenvalue Problems (SIEP). The first problem we consider is the Jacobi Inverse Eigenvalue Problem (JIEP): given some constraints on two sets of reals, find a Jacobi matrix J (real, symmetric, tridiagonal, with positive off-diagonal entries) that admits as spectrum and principal subspectrum the two given sets. Two classes of finite algorithms are considered. The polynomial algorithm which is based on a special Euclid–Sturm algorithm (Householder's terminology) and has been rediscovered several times. The matrix algorithm which is a symmetric Lanczos algorithm with a special initial vector. Some characterization of the matrix ensures the equivalence of the two algorithms in exact arithmetic. The results of the symmetric situation are extended to the nonsymmetric case. This is the second SIEP to be considered: the Tridiagonal Inverse Eigenvalue Problem (TIEP). Possible breakdowns may occur in the polynomial algorithm as it may happen with the nonsymmetric Lanczos algorithm. The connection between the two algorithms exhibits a similarity transformation from the classical Frobenius companion matrix to the tridiagonal matrix. This result is used to illustrate the fact that, when computing the eigenvalues of a matrix, the nonsymmetric Lanczos algorithm may lead to a slow convergence, even for a symmetric matrix, since an outer eigenvalue of the tridiagonal matrix of order n − 1 can be arbitrarily far from the spectrum of the original matrix. This revised version was published online in August 2006 with corrections to the Cover Date.     

Estimating a largest eigenvector by Lanczos and polynomial algorithms with a random start

We study Lanczos and polynomial algorithms with random start for estimating an eigenvector corresponding to the largest eigenvalue of an n × n large symmetric positive definite matrix. We analyze the two error criteria: the randomized error and the randomized residual error. For the randomized error, we prove that it is not possible to get distribution-free bounds, i.e., the bounds must depend on the distribution of eigenvalues of the matrix. We supply such bounds and show that they depend on the ratio of the two largest eigenvalues. For the randomized residual error, distribution-free bounds exist and are provided in the paper. We also provide asymptotic bounds, as well as bounds depending on the ratio of the two largest eigenvalues. The bounds for the Lanczos algorithm may be helpful in a practical implementation and termination of this algorithm. © 1998 John Wiley & Sons, Ltd.     

Computing the field of values and pseudospectra using the Lanczos method with continuation

The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes. To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm. We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices. The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type.This work was supported by Engineering and Physical Sciences Research Council grants GR/H/52139 and GR/H/94528.     

Use of the Lanczos Method for Finding Complete Sets of Eigenvalues of Large Sparse Symmetric Matrices

A way of using the Lanczos method to find all the eigenvaluesof a large sparse symmetric matrix is described, and some empiricalobservations on the manner in which the method works in practiceare given. The method seems to be accurate, fast, and not verydemanding on storage. A formula for the number of iterationsnecessary to get the eigenvalues is proposed.     

A symmetry exploiting Lanczos method for symmetric Toeplitz matrices

Several methods for computing the smallest eigenvalues of a symmetric and positive definite Toeplitz matrix T have been studied in the literature. Most of them share the disadvantage that they do not reflect symmetry properties of the corresponding eigenvector. In this note we present a Lanczos method which approximates simultaneously the odd and the even spectrum of T at the same cost as the classical Lanczos approach.     

Inertia‐based spectrum slicing for symmetric quadratic eigenvalue problems

In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory‐efficient by representing the computed pseudo‐Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.     

Dealing with linear dependence during the iterations of the restarted block Lanczos methods

The Lanczos method can be generalized to block form to compute multiple eigenvalues without the need of any deflation techniques. The block Lanczos method reduces a general sparse symmetric matrix to a block tridiagonal matrix via a Gram–Schmidt process. During the iterations of the block Lanczos method an off-diagonal block of the block tridiagonal matrix may become singular, implying that the new set of Lanczos vectors are linearly dependent on the previously generated vectors. Unlike the single vector Lanczos method, this occurrence of linearly dependent vectors may not imply an invariant subspace has been computed. This difficulty of a singular off-diagonal block is easily overcome in non-restarted block Lanczos methods, see [12,30]. The same schemes applied in non-restarted block Lanczos methods can also be applied in restarted block Lanczos methods. This allows the largest possible subspace to be built before restarting. However, in some cases a modification of the restart vectors is required or a singular block will continue to reoccur. In this paper we examine the different schemes mentioned in [12,30] for overcoming a singular block for the restarted block Lanczos methods, namely the restarted method reported in [12] and the Implicitly Restarted Block Lanczos (IRBL) method developed by Baglama et al. [3]. Numerical examples are presented to illustrate the different strategies discussed.     

隐式重启精化全局Lanczos方法

The numerical methods for solving large symmetric eigenvalue problems are considered in this paper.Based on the global Lanczos process,a global Lanczos method for solving large symmetric eigenvalue problems is presented.In order to accelerate the convergence of the F-Ritz vectors,the refined global Lanczos method is developed.Combining the implicitly restarted strategy with the deflation technique,an implicitly restarted and refined global Lanczos method for computing some eigenvalues of large symmetric matrices is proposed.Numerical results show that the proposed methods are efficient.     

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…     

The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices

A new algorithm for the computation of eigenvalues of a nonsymmetric matrix pencil is described. It is a generalization of the shifted and inverted Lanczos (or Arnoldi) algorithm, in which several shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil, gives Ritz approximations to the solution of the original pencil. It is shown how complex shifts can be used to compute a real block Hessenberg pencil to a real matrix pair.Two applicationx, one coming from an aircraft stability problem and the other from a hydrodynamic bifurcation, have been tested and results are reported.Dedicated to Carl-Erik Fröberg on the occasion of his 75th birthday.     

广义块Toeplitz特征值问题的基于sine变换的预处理子

在求块Toeplitz矩阵束（Amn，Bmn）特征值的Lanczos过程中，通过对移位块Toepltz矩阵Amn-ρBmn进行基于sine变换的块预处理，从而改进了位移块Toeplitz矩阵的谱分布，加速了Lanczos过程的收敛速度．该块预处理方法能通过快速算法有效快速执行．本文证明了预处理后Lanczos过程收敛迅速，并通过实验证明该算法求解大规模矩阵问题尤其有效．     

A QUASI-REFINED ITERATIVE ALGORITHM BASED ON THE LANCZOS BIORTHOGONALIZATION PROCEDURE FOR LARGE EIGENPROBLEMS

The two-sided Lanczos method is popular for computing a few selected eigentriplets of large non-Hermitian matrices. However, it has been revealed that theRitz vectors gained by this method may not converge even if the subspaces are good enough and the associated eigenvalues converge. In order to remedy this drawback, a novel method is proposed which is based on the refined strategy, the quasi-refined ideaand the Lanczos biothogonalization procedure, the resulting algorithm is presented. Therelationship between the new method and the classical oblique projection technique isalso established. We report some numericalwith the conventional one, the results showthe latter.experiments and compare the new algorithmthat the former is often more powerful than     

Convergence estimates of nonrestarted and restarted block‐Lanczos methods

The block‐Lanczos method serves to compute a moderate number of eigenvalues and the corresponding invariant subspace of a symmetric matrix. In this paper, the convergence behavior of nonrestarted and restarted versions of the block‐Lanczos method is analyzed. For the nonrestarted version, we improve an estimate by Saad by means of a change of the auxiliary vector so that the new estimate is much more accurate in the case of clustered or multiple eigenvalues. For the restarted version, an estimate by Knyazev is generalized by extending our previous results on block steepest descent iterations and single‐vector restarted Krylov subspace iterations. The new estimates can also be reformulated and applied to invert‐block‐Lanczos methods for solving generalized matrix eigenvalue problems.     

An adaptive block Lanczos algorithm

A generalization of the block Lanczos algorithm will be given, which allows the block size to be increased during the iteration process. In particular, the algorithm can be implemented with the block size chosen adaptively according to clustering of Ritz values. In this way, multiple and clustered eigenvalues can be found and the difficulty of choosing the block size is eased. Residual bounds for clustered eigenvalues are given. Numerical examples are presented to illustrate the adaptive algorithm.Research supported by a grant from Natural Sciences and Engineering Research Council of Canada.