On a new variant of Arnoldi method for approximation of eigenpairs

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Abstract

The Arnoldi method is a commonly used technique for finding a few eigenpairs of large, sparse and nonsymmetric matrices. Recently, a new variant of Arnoldi method (NVRA) was proposed. In NVRA, the modified Ritz vector is used to take the place of the Ritz vector by solving a minimization problem. Moreover, it was shown that if the refined Arnoldi method converges, then the NVRA method also converges. The contribution of this work is as follows. First, we point out that the convergence theory of the NVRA method is incomplete. More precisely, the cosine of the angle between the refined Ritz vector and the Ritz vector may not be uniformly lower-bounded, and it can be arbitrarily close to zero in theory. Consequently, the modified Ritz vector may fail to converge even when the search subspace is good enough. A remedy to the convergence of the NVRA method is given. Second, we show that the linear system for solving the modified Ritz vector in the NVRA method will become more and more ill-conditioned as the refined Ritz vector converges. If the Ritz vector also tends to converge as the refined Ritz vector does so, the ill-conditioning of the linear system will have little influence on the convergence of the modified Ritz vector, and the modified Ritz vector can improve the Ritz vector substantially. Otherwise, the ill-conditioning may have significant influence on the convergence of the modified Ritz vector. Third, to fix the NVRA method, we propose an improved refined Arnoldi method that uses improved refined Ritz vector to take the place of the modified Ritz vector. Theoretical results indicate that the improved refined Ritz method is often better than the refined Ritz method. Numerical experiments illustrate the numerical behavior of the improved refined Ritz method, and demonstrate the effectiveness of our theoretical analysis.

Introduction

We are interested in computing a few eigenpairs of the following large-scale eigenvalue problem Ax=λx,where ARn×n is large, sparse and nonsymmetric, with x=1. In this paper, all the norm used is the Euclidean norm (or the 2-norm) unless otherwise stated. A major class of methods for this type of problem are the orthogonal projection methods [1], [2], [3], such as the Arnoldi method [4], [5].

It is well known that orthogonal projection methods are effective for computing exterior eigenpairs, but are not for computing interior ones. The harmonic Arnoldi method is an efficient approach for computing interior eigenpairs of large matrices [6], [7], [8]. However, both the Arnoldi method and the harmonic Arnoldi method may fail to converge even if the search subspace is good enough [9], [10], [11]. In order to deal with this problem, refined projection methods such as the refined Arnoldi method and the refined harmonic Arnoldi method were proposed in [12], [13]. The refined Arnoldi method and the refined harmonic Arnoldi method converge provided that there is sufficiently good information in the search subspace [9], [10], [11].

Recently, a new variant of Arnoldi method for approximation of eigenpairs (NVRA) was proposed in [14]. This method makes use of the so-called modified Ritz vector to take the place of the Ritz vector, by solving some minimization problem. It was proven that if the refined Arnoldi method converges, then the NVRA method also converges [14, Theorem 6].

In this work, we first show that the convergence theorem for this method (i.e., Theorem 6 in [14]) is incomplete. The key is that the cosine of the angle between the refined Ritz vector and the Ritz vector is not uniformly lower-bounded and can be arbitrarily close to zero. Consequently, the modified Ritz vector may fail to converge even if the search subspace contains sufficiently good information on the desired eigenvector. A remedy to the convergence theorem for NVRA is given. Second, we show that the linear system for solving the modified Ritz vector will become more and more ill-conditioned as the refined Ritz vector converges. In this situation, if the Ritz vector also tends to converge, the ill-conditioning of the linear system will have little influence on the convergence of the modified Ritz vector. Otherwise, the ill-conditioning may have significant influence. Consequently, NVRA may even converge slower than the classical Arnoldi method. Third, to fix the NVRA method, we propose an improved refined Arnoldi method, in which improved refined Ritz vector is used to take the place of the modified Ritz vector. Theoretical results show that the improved refined Ritz vector converges as the distance between the desired eigenvector and the search subspace approaches to zero.

This paper is organized as follows. In Section 2, we briefly introduce the Arnoldi method, the refined Arnoldi method, and the new variant of Arnoldi method (NVRA) for computing a few selected eigenpairs of large matrices. In Section 3, we focus on the convergence of the NVRA method. We point out that the convergence of the modified Ritz vector strongly relies on that of the Ritz vector. Similar to the Ritz vector, the modified Ritz vector may fail to converge even if the search subspace is good enough. As a remedy, we propose an improved refined Arnoldi method in Section 4, and the convergence of the proposed method is given. Some numerical experiments are reported in Section 5, which illustrate the numerical behavior of the proposed algorithm, and demonstrate the effectiveness of our theoretical results. Some concluding remarks are given in Section 6.

Section snippets

Preliminaries

In this section, we briefly introduce the Arnoldi method [4], the refined Arnoldi method [12], and the new variant of the Arnoldi method proposed in [14]. Given a matrix ARn×n and a unit vector v1, the m-step Arnoldi process generates an orthonormal basis Vm+1 for the following Krylov subspace [1], [2], [3] Km+1(A,v1)=span{v1,Av1,,Amv1}.The essential details of the Arnoldi process are given as follows; for more details, refer to [1], [2], [3].

Algorithm 1

The m-step Arnoldi process

1. Start:  Choose a

On the convergence of the new variant of Arnoldi method

In this section, we revisit the convergence of the new variant of Arnoldi method. The framework of this section is as follows. First, we indicate that Theorem 2.1 is incorrect. Second, we give new insight into the convergence of the NVRA method, and present a modification to Theorem 2.1. Third, the linear system (2.6) for solving the modified Ritz vector in the NVRA method will become more and more ill-conditioned as the refined Ritz vector converges. We stress that if the Ritz vector also

An improved refined Arnoldi method

Given a search subspace Km(A,v1) and a Ritz value μ, it follows from (2.3) that the refined Ritz vector is the best approximate eigenvector in terms of residual 2-norm. Thus, for the same search subspace and Ritz value, the modified Ritz vector is no better than the refined Ritz vector. With the m-step Arnoldi process at hand, an interesting question is “whether we can find a better approximate eigenvector than the refined Ritz vector”? The answer is yes. In this section, we propose an improved

Numerical examples

In this section, we perform some numerical experiments to illustrate the numerical performance of our proposed algorithm, and to demonstrate the effectiveness of our theoretical results. In the experiments, we run the restarted Arnoldi algorithm (Arnoldi) [5], the restarted refined Arnoldi algorithm (Refined Arnoldi) [12], the new variant of restarting Arnoldi method (NVRA) [14], as well as our proposed improved refined Arnoldi algorithm (Algorithm 3). All the numerical experiments were run on

Conclusions

Recently, a new variant of Arnoldi method for approximation of eigenpairs (NVRA) was proposed [14], in which the modified Ritz vector is used instead of the Ritz vector. It was pointed out that the convergence property of the modified Ritz vector is comparable with that of the refined Ritz vector [14]. In this work, we indicate that this assertion is incomplete, and the modified Ritz vector may fail to converge even if the search subspace contains sufficiently good information on the desired

Acknowledgments

We would like to express our sincere thanks to the anonymous referees for insightful comments and suggestions that greatly improved the representation of this paper.

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    This work is supported by the Fundamental Research Funds for the Central Universities of China under grant 2019XKQYMS89.

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