求解陀螺系统特征值问题的收缩二阶Lanczos方法

本文研究陀螺系统特征值问题的数值解法,利用反对称矩阵Lanczos算法,提出了求解陀螺系统特征值问题的二阶Lanczos方法.基于提出的陀螺系统特征值问题的非等价低秩收缩技术,给出了计算陀螺系统极端特征值的收缩二阶Lanczos方法.数值结果说明了算法的有效性.     

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.     

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 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.     

A semidefinite framework for trust region subproblems with applications to large scale minimization

Primal-dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem (TRS). This latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and, therefore, it is a generalization of the minimum eigenvalue problem. The importance of (TRS) is due to the fact that it provides the step in trust region minimization algorithms. The semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for (TRS). In particular, a dual simplex type method is studied that solves (TRS) as a parametric eigenvalue problem. This method uses the Lanczos algorithm for the smallest eigenvalue as a black box. Therefore, the essential cost of the algorithm is the matrix-vector multiplication and, thus, sparsity can be exploited. A primal simplex type method provides steps for the so-called hard case. Extensive numerical tests for large sparse problems are discussed. These tests show that the cost of the algorithm is 1 +α(n) times the cost of finding a minimum eigenvalue using the Lanczos algorithm, where 0<α(n)<1 is a fraction which decreases as the dimension increases. Research supported by the National Science and Engineering Research Council Canada.     

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.     

On conjugate gradient-like methods for eigen-like problems

Numerical analysts, physicists, and signal processing engineers have proposed algorithms that might be called conjugate gradient for problems associated with the computation of eigenvalues. There are many variations, mostly one eigenvalue at a time though sometimes block algorithms are proposed. Is there a correct “conjugate gradient” algorithm for the eigenvalue problem? How are the algorithms related amongst themselves and with other related algorithms such as Lanczos, the Newton method, and the Rayleigh quotient?     

A Variant of the Inverted Lanczos Method

In this note we study a variant of the inverted Lanczos method which computes eigenvalue approximates of a symmetric matrix A as Ritz values of A from a Krylov space of A ^–1. The method turns out to be slightly faster than the Lanczos method at least as long as reorthogonalization is not required. The method is applied to the problem of determining the smallest eigenvalue of a symmetric Toeplitz matrix. It is accelerated taking advantage of symmetry properties of the correspond ng eigenvector.This revised version was published online in October 2005 with corrections to the Cover Date.     

Rational Lanczos approximations to the matrix square root and related functions

We consider restricted rational Lanczos approximations to matrix functions representable by some integral forms. A convergence analysis that stresses the effectiveness of the proposed method is developed. Error estimates are derived. Numerical experiments are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

ARNOLDI TYPE ALGORITHMS FOR LARGE UNSYMMETRIC MULTIPLE EIGENVALUE PROBLEMS

1.IntroductionTheLanczosalgorithm[Zo]isaverypowerfultoolforextractingafewextremeeigenvaluesandassociatedeigenvectorsoflargesymmetricmatrices[4'5'22].Sincethe1980's,considerableattentionhasbeenpaidtogeneralizingittolargeunsymmetricproblems.Oneofitsgen...     

Strategies for spectrum slicing based on restarted Lanczos methods

In the context of symmetric-definite generalized eigenvalue problems, it is often required to compute all eigenvalues contained in a prescribed interval. For large-scale problems, the method of choice is the so-called spectrum slicing technique: a shift-and-invert Lanczos method combined with a dynamic shift selection that sweeps the interval in a smart way. This kind of strategies were proposed initially in the context of unrestarted Lanczos methods, back in the 1990’s. We propose variations that try to incorporate recent developments in the field of Krylov methods, including thick restarting in the Lanczos solver and a rational Krylov update when moving from one shift to the next. We discuss a parallel implementation in the SLEPc library and provide performance results.     

Zero cancellation for general rational matrix functions

The problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints. Firstly, the class of all factors that dislocate the zeros and feature minimal McMillan degree are derived. Further, necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being J-unitary, or J-inner, either with respect to the imaginary axis or to the unit circle. The main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial, proper or improper, rank deficient, with arbitrary poles and zeros including at infinity. A particular attention is paid to the numerically-sound construction of solutions by employing at each stage unitary transformations, reliable numerical algorithms for eigenvalue assignment and efficient Lyapunov equation solvers.     

The implicit application of a rational filter in the RKS method

The implicitly restarted Arnoldi method implicitly applies a polynomial filter to the Arnoldi vectors by use of orthogonal transformations. In this paper, an implicit filtering by rational functions is proposed for the rational Krylov method. This filtering is performed in an efficient way. Two applications are considered. The first one is the filtering of unwanted eigenvalues using exact shifts. This approach is related to the use of exact shifts in the implicitly restarted Arnoldi method. Second, eigenvalue problems can have an infinite eigenvalue without physical relevance. This infinite eigenvalue can corrupt the eigensolution. An implicit filtering is proposed for avoiding such corruptions. The work of Gorik De Samblanx and Adhemar Bultheel was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93 and by the Human Capital and Mobility project ROLLS of the European Community under contract ERBCHRXCT930416. The research by Karl Meerbergen was supported by the Belgian programme on Interuniversity Poles of Attraction (IUAP 17), initiated by the Belgian State—Prime Minister's Service—Federal Office for Scientific, Technical and Cultural Affairs and the project Iterative Methods in Scientific Computing, contract number HCM network CHRCCT93-0420, coordinated by CERFACS, Toulouse, France.     

Error bounds of Lanczos approach for trust-region subproblem

Because of its vital role of the trust-region subproblem (TRS) in various applications, for example, in optimization and in ill-posed problems, there are several factorization-free algorithms for solving the large-scale sparse TRS. The truncated Lanczos approach proposed by N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint [SIAM J. Optim., 1999, 9: 504–525] is a natural extension of the classical Lanczos method for the symmetric linear system and eigenvalue problem and, indeed follows the classical Rayleigh-Ritz procedure for eigenvalue computations. It consists of 1) projecting the original TRS to the Krylov subspaces to yield smaller size TRS’s and then 2) solving the resulted TRS’s to get the approximates of the original TRS. This paper presents a posterior error bounds for both the global optimal value and the optimal solution between the original TRS and their projected counterparts. Our error bounds mainly rely on the factors from the Lanczos process as well as the data of the original TRS and, could be helpful in designing certain stopping criteria for the truncated Lanczos approach.     

Analysis of the symmetric Lanczos algorithm with reorthogonalization methods

We present an error analysis of the symmetric Lanczos algorithm in finite precision arithmetic. The loss of orthogonality among the computed Lanczos vectors is explained with the help of a recurrence formula. A backward error analysis shows that semiorthogonality among the Lanczos vectors is enough to guarantee the accuracy of the computed quantities up to machine precision. The results of this analysis are then extended to the more general case of the Lanczos algorithm with a semiorthogonalization strategy. Based on the recurrence formula, a new reorthogonalization method called partial reorthogonalization is introduced. We show that both partial reorthogonalization and selective orthogonalization as introduced by Parlett and Scott [15] are semiorthogonalization strategies. Finally we discuss the application of our results to the solution of linear systems of equations and to the eigenvalue problem.     

Chebyshev rational interpolation

The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we present a suitable fast modification of a general look-ahead version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for approximating analytic functions by means of rational interpolation at certain nodes located on the boundary of an elliptical region of the complex plane. In fact, in this case it overcomes some of the numerical difficulties which limited the applicability of the look-ahead Lanczos process for determining the coefficients both of the numerators and of the denominators with respect to the standard power basis. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Lanczos phenomenon—An interpretation based upon conjugate gradient optimization

The equivalence in exact arithmetic of the Lanczos tridiagonalization procedure and the conjugate gradient optimization procedure for solving Ax = b, where A is a real symmetric, positive definite matrix, is well known. We demonstrate that a relaxed equivalence is valid in the presence of errors. Specifically we demonstrate that local ε-orthonormality of the Lanczos vectors guarantees local ε-A-conjugacy of the direction vectors in the associated conjugate gradient procedure. Moreover we demonstrate that all the conjugate gradient relationships are satisfied approximately. Therefore, any statements valid for the conjugate gradient optimization procedure, which we show converges under very weak conditions, apply directly to the Lanczos procedure. We then use this equivalence to obtain an explanation of the Lanczos phenomenon: the empirically observed “convergence” of Lanczos eigenvalue procedures despite total loss of the global orthogonality of the Lanczos vectors.     

A rational Lanczos algorithm for model reduction

This paper presents a model reduction method for large-scale linear systems that is based on a Lanczos-type approach. A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padé approximant) for the large-scale system. An exact expression for the error in the interpolant is derived. Examples are utilized to demonstrate that the rational Lanczos method provides opportunities for significant improvements in the rate of convergence over single-point Lanczos approaches.     

Expansion of random field gradients using hierarchical matrices

We present two expansions for the gradient of a random field. In the first approach, we differentiate its truncated Karhunen-Loève expansion. In the second approach, the Karhunen-Loève expansion of the random field gradient is computed directly. Both strategies require the solution of dense, symmetric matrix eigenvalue problems which can be handled efficiently by combining hierachical matrix techniques with a thick-restart Lanczos method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)