A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity

A restarted Arnoldi algorithm is given that computes eigenvalues and eigenvectors. It is related to implicitly restarted Arnoldi, but has a simpler restarting approach. Harmonic and regular Rayleigh-Ritz versions are possible.For multiple eigenvalues, an approach is proposed that first computes eigenvalues with the new harmonic restarted Arnoldi algorithm, then uses random restarts to determine multiplicity. This avoids the need for a block method or for relying on roundoff error to produce the multiple copies.     

A restarted Krylov method with inexact inversions

In this paper, we present a new type of restarted Krylov method for calculating the smallest eigenvalues of a symmetric positive definite matrix G. The new framework avoids the Lanczos tridiagonalization process and the use of polynomial filtering. This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the eigenvalues toward their limits. Another innovation is the use of inexact inversions of G to generate the Krylov matrices. In this approach, the inverse of G is approximated by using an iterative method to solve the related linear system. Numerical experiments illustrate the usefulness of the proposed approach.     

A new framework for implicit restarting of the Krylov–Schur algorithm

This paper introduces a new framework for implicit restarting of the Krylov–Schur algorithm. It is shown that restarting with arbitrary polynomial filter is possible by reassigning some of the eigenvalues of the Rayleigh quotient through a rank‐one correction, implemented using only the elementary transformations (translation and similarity) of the Krylov decomposition. This framework includes the implicitly restarted Arnoldi (IRA) algorithm and the Krylov–Schur algorithm with implicit harmonic restart as special cases. Further, it reveals that the IRA algorithm can be turned into an eigenvalue assignment method. Copyright © 2014 John Wiley & Sons, Ltd.     

A framework of the harmonic Arnoldi method for evaluating <Emphasis Type="Italic">φ</Emphasis>-functions with applications to exponential integrators

In recent years, a great deal of attention has been focused on exponential integrators. The important ingredient to the implementation of exponential integrators is the efficient and accurate evaluation of the so called φ-functions on a given vector. The Krylov subspace method is an important technique for this problem. For this type of method, however, restarts become essential for the sake of storage requirements or due to computational complexities of evaluating matrix function on a reduced matrix of growing size. Another problem in computing φ-functions is the lack of a clear residual notion. The contribution of this work is threefold. First, we introduce a framework of the harmonic Arnoldi method for φ-functions, which is based on the residual and the oblique projection technique. Second, we establish the relationship between the harmonic Arnoldi approximation and the Arnoldi approximation, and compare the harmonic Arnoldi method and the Arnoldi method from a theoretical point of view. Third, we apply the thick-restarting strategy to the harmonic Arnoldi method, and propose a thick-restarted harmonic Arnoldi algorithm for evaluating φ-functions. An advantage of the new algorithm is that we can compute several φ-functions simultaneously in the same search subspace after restarting. The relationship between the error and the residual of the harmonic Arnoldi approximation is also investigated. Numerical experiments show the superiority of our new algorithm over many state-of-the-art algorithms for computing φ-functions.     

A rank‐exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter‐dependent matrix. In several applications, M(λ) has a structure where the higher‐order terms of its Taylor expansion have a particular low‐rank structure. We propose a new Arnoldi‐based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector‐valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169–195]. In this paper, the low‐rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large‐scale problems. Copyright © 2016 John Wiley & Sons, Ltd.     

A reflection on the implicitly restarted Arnoldi method for computing eigenvalues near a vertical line

In this article, we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted Arnoldi method. The extension to computing eigenvalues closest to a vertical line is straightforward, by incorporating a shift. Without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis.In a recent publication, Meerbergen and Spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to Hopf bifurcations, which is of interest for the stability of dynamical systems. The novel method is based on inverse iteration (inverse power method) applied on a Lyapunov-like eigenvalue problem. To reduce the computational overhead significantly a projection was added.This method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane. We will prove in this paper that the combination of inverse iteration with the projection step is equivalent to Sorensen’s implicitly restarted Arnoldi method utilizing well-chosen shifts.     

Krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices

In this paper we propose a general approach by which eigenvalues with a special property of a given matrix A can be obtained. In this approach we first determine a scalar function ψ: C → C whose modulus is maximized by the eigenvalues that have the special property. Next, we compute the generalized power iterations uinj + 1 = ψ(A)u_j, j = 0, 1,…, where u₀ is an arbitrary initial vector. Finally, we apply known Krylov subspace methods, such as the Arnoldi and Lanczos methods, to the vector u_n for some sufficiently large n. We can also apply the simultaneous iteration method to the subspace span{x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_k} with some sufficiently large n, where x^(j+1)_m = ψ(A)x^(j)_m, j = 0, 1,…, m = 1,…, k. In all cases the resulting Ritz pairs are approximations to the eigenpairs of A with the special property. We provide a rather thorough convergence analysis of the approach involving all three methods as n → ∞ for the case in which A is a normal matrix. We also discuss the connections and similarities of our approach with the existing methods and approaches in the literature.     

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…     

A global harmonic Arnoldi method for large non-Hermitian eigenproblems with an application to multiple eigenvalue problems

The global Arnoldi method can be used to compute exterior eigenpairs of a large non-Hermitian matrix A, but it does not work well for interior eigenvalue problems. Based on the global Arnoldi process that generates an F-orthonormal basis of a matrix Krylov subspace, we propose a global harmonic Arnoldi method for computing certain harmonic F-Ritz pairs that are used to approximate some interior eigenpairs. We propose computing the F-Rayleigh quotients of the large non-Hermitian matrix with respect to harmonic F-Ritz vectors and taking them as new approximate eigenvalues. They are better and more reliable than the harmonic F-Ritz values. The global harmonic Arnoldi method inherits convergence properties of the harmonic Arnoldi method applied to a larger matrix whose distinct eigenvalues are the same as those of the original given matrix. Some properties of the harmonic F-Ritz vectors are presented. As an application, assuming that A is diagonalizable, we show that the global harmonic Arnoldi method is able to solve multiple eigenvalue problems both in theory and in practice. To be practical, we develop an implicitly restarted global harmonic Arnoldi algorithm with certain harmonic F-shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.     

An Investigation of Restarted GMRES Method by Using Flexible Starting Vectors

<正>We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations.The merit of the variant is that previously generated information can be utilized to select a new starting vector,such that the occurrence of stagnation be mitigated or the convergence be accelerated.The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector,which is discussed in detail.Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.     

Pseudospectra of rectangular matrices

Pseudospectra of rectangular matrices vary continuously withthe matrix entries, a feature that eigenvalues of these matricesdo not have. Some properties of eigenvalues and pseudospectraof rectangular matrices are explored, and an efficient algorithmfor the computation of pseudospectra is proposed. Applicationsare given in (square) eigenvalue computation (Lanczos iteration),square pseudospectra approximation (Arnoldi iteration), controltheory (nearest uncontrollable system) and game theory.     

Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices

It is shown that the method of Arnoldi can be successfully used for solvinglarge unsymmetric eigenproblems. Like the symmetric Lanczos method, Arnoldi's algorithm realizes a projection process onto the Krylov subspace K_m spanned by v₁,Av₁,...,A^m?1v₁, where v₁ is the initial vector. We therefore study the convergence of the approximate eigenelements obtained by such a process. In particular, when the eigenvalues of A are real, we obtain bounds for the rates of convergence similar to those for the symmetric Lanczos algorithm. Some practical methods are presented in addition to that of Arnoldi, and several numerical experiments are described.     

Numerical experiments on a flexible variant of GMRES-DR

The Flexible GMRES (FGMRES [1]) and the GMRES with deflated restarting (GMRES-DR [2]) methods are two algorithms derived from GMRES [3], that are considered as powerful when solving large non hermitian systems of linear equations. GMRES-DR is a variant of GMRES with an improved restarting technique that maintains in the Krylov subspace harmonic Ritz vector from the previous restart. In situations where the convergence of restarted GMRES is slow and where the matrix has few eigenvalues close to the origin, this technique has proved very efficient. The new method that we propose is the Flexible GMRES with deflated restarting (FGMRES-DR [6]), which combines the two above mentioned algorithms in order to yield better performance. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A linear eigenvalue algorithm for the nonlinear eigenvalue problem

The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted ${\mathcal {B}}$ . We consider the Arnoldi method for the operator ${\mathcal {B}}$ and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator ${\mathcal {B}}$ in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.     

A rational Arnoldi process with applications

The rational Arnoldi process is a popular method for the computation of a few eigenvalues of a large non‐Hermitian matrix and for the approximation of matrix functions. The method is particularly attractive when the rational functions that determine the process have only few distinct poles , because then few factorizations of matrices of the form A ? z_jI have to be computed. We discuss recursion relations for orthogonal bases of rational Krylov subspaces determined by rational functions with few distinct poles. These recursion formulas yield a new implementation of the rational Arnoldi process. Applications of the rational Arnoldi process to the approximation of matrix functions as well as to the computation of eigenvalues and pseudospectra of A are described. The new implementation is compared to several available implementations.     

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.     

Memory‐efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Novel memory‐efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree d. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor d. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so‐called quadratic Arnoldi method and two‐level orthogonal Arnoldi procedure methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift‐and‐invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree 30 arising from the interpolation of nonlinear eigenvalue problems, which stem from boundary element discretizations of PDE eigenvalue problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems

Convergence of the implicitly restarted Arnoldi (IRA) method for nonsymmetric eigenvalue problems has often been studied by deriving bounds for the angle between a desired eigenvector and the Krylov projection subspace. Bounds for residual norms of approximate eigenvectors have been less studied and this paper derives a new a-posteriori residual bound for nonsymmetric matrices with simple eigenvalues. The residual vector is shown to be a linear combination of exact eigenvectors and a residual bound is obtained as the sum of the magnitudes of the coefficients of the eigenvectors. We numerically illustrate that the convergence of the residual norm to zero is governed by a scalar term, namely the last element of the wanted eigenvector of the projected matrix. Both cases of convergence and non-convergence are illustrated and this validates our theoretical results. We derive an analogous result for implicitly restarted refined Arnoldi (IRRA) and for this algorithm, we numerically illustrate that convergence is governed by two scalar terms appearing in the linear combination which drives the residual norm to zero. We provide a set of numerical results that validate the residual bounds for both variants of Arnoldi methods.     

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.