Refined Ritz Vectors Research Articles

On a new variant of Arnoldi method for approximation of eigenpairs

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.

On expansion of search subspaces for large non-Hermitian eigenproblems

How to expand a given subspace efficiently is an important task for large sparse eigenvalue problems. In this paper, we are interested in the following question: which vector from a given subspace, after multiplied by the matrix A, will give a better expanded search subspace. In (2008) [33], Ye investigated this problem and determined how to choose the optimal expansion vector theoretically. Unfortunately, the result is not computable since it involves some unknown information on the desired eigenvector. When A is symmetric, it was suggested that the Ritz vector may lead to a good candidate for subspace expansion. However, when A is non-Hermitian, the Ritz vector may not be a satisfactory choice. The contribution of this paper is twofold. First, we suggest to use the refined Ritz vector to expand the search subspace, and propose a residual expansion Arnoldi method for subspace expansion. Theoretical results justify the use of the refined Ritz vector. Second, we prove that the elements of the primitive refined Ritz vector have a decreasing pattern going to zero. We then show that the decreasing pattern exists in an arbitrary primitive approximate eigenvector, and derive an inexact residual expansion Arnoldi method for subspace expansion. Numerical examples show the effectiveness of our theoretical results and illustrate the numerical behavior of the new method for subspace expansion.

On refined Ritz vectors and polynomial characterization

It is generally believed that the Ritz vectors do not coincide with the refined Ritz vectors in the Arnoldi method for computing eigenvalues of matrices. We show that this coincidence is theoretically possible. We provide a necessary and sufficient condition for this coincidence to happen and give examples to illustrate the same. Using Lanczos polynomials, we give a polynomial characterization of refined Ritz vectors of symmetric matrices that is different from the one available in the literature.

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.

FDFD Method Based on Refined Shift-and-Invert Arnoldi Technique for Periodic Leaky-Wave Antennas

The finite difference frequency domain (FDFD) method can be used to analyze the properties of arbitrary complex close/open periodic guided-wave structures. For the open problem, it probably results in a large scale nonsymmetrical generalized eigenvalue problem. In the analysis of the leaky-wave antennas, because the leakage constant (attenuation constant) increases a lot in the working frequency range, the extraction process of the eigenvalues (complex propagation constants) is difficult to converge when the shift-and-invert Arnoldi technique is used to resolve this large scale nonsymmetrical generalized eigenvalue problem. In this paper, a refined shift-and-invert Arnoldi algorithm is adopted to speed the convergence rate by using refined Ritz vectors to approximate the desired eigenvectors. Since the refined method always converges, a simple FDFD method with six field components can be used. The difference equations of this method are much simpler than those of the previous method because the longitudinal field components don't have to be eliminated in order to approximately transform the generalized eigenvalue problem to a standard eigenvalue problem. Consequently, the accuracy of the method can be improved and the boundary conditions can be implemented easily.

A refined variant of the inverse-free Krylov subspace method for symmetric generalized eigenvalue problems

A refined variant of the inverse-free Krylov subspace method is proposed in this paper. The new method retains the original Ritz value, and replaces the Ritz vector by a refined Ritz vector in each cycle of the iteration. Each refined Ritz vector is chosen in such a way that the norm of the residual vector formed with the Ritz value is minimized over the subspace involved, and it can be computed cheaply by solving a small sized SVD problem. The refined variant can overcome the irregular convergence behavior of the Ritz vectors which may happen in the inverse-free Krylov subspace method. An a priori error estimate for the refined Ritz vector is given, which shows that the refined Ritz vector converges once the deviation of the eigenvector from the trial Krylov subspace converges to zero. By using spectral transformation, this new method can be applied to compute an interior eigenvalue pair. Numerical experiments are given to show the efficiency of the new methods.

A semiorthogonal generalized Arnoldi method and its variations for quadratic eigenvalue problems

SUMMARYIn this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction for a generalized eigenvalue problem. The method applies the Rayleigh–Ritz orthogonal projection technique on the quadratic eigenvalue problem. Consequently, it preserves the spectral properties of the original quadratic eigenvalue problem. Furthermore, we propose a refinement scheme to improve the accuracy of the Ritz vectors for the quadratic eigenvalue problem. Given shifts, we also show how to restart the method by implicitly updating the starting vector and constructing better projection subspace. We combine the ideas of the refinement and the restart by selecting shifts upon the information of refined Ritz vectors. Finally, an implicitly restarted refined semiorthogonal generalized Arnoldi method is developed. Numerical examples demonstrate that the implicitly restarted semiorthogonal generalized Arnoldi method with or without refinement has superior convergence behaviors than the implicitly restarted Arnoldi method applied to the linearized quadratic eigenvalue problem. Copyright © 2012 John Wiley & Sons, Ltd.

A Novel Arnoldi Method to Calculate the Critical Eigenvalues in Power System

Hopf bifurcation is regarded as one of the three kinds of bifurcations which can cause voltage instability. The Hopf bifurcation was obtained if a pair of complex conjugate eigenvalues of Jacobian matrix crossing the imaginary axis. To calculate the critical eigenvalues of bulk power system, a restarted-refined Arnoldi method is introduced. The refined Ritz vectors are used as the approximations based on the refined projection theory in order to enrich the information of the eigenvectors in the projective subspace. Two examples indicate that the method can work out a set of conjugate eigenvalues which will show whether the Hopf bifurcation appears and provide the datum for the dynamic voltage stability analysis ulteriorly.

Using cross-product matrices to compute the SVD

This paper concerns accurate computation of the singular value decomposition (SVD) of an $m\times n$ matrix $A$ . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of $A^{\rm T}A$ and take the square roots of the large eigenvalues of it as the large singular values of $A$ ; (b) form the Rayleigh quotient of $A^{\rm T}A$ with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of $A^{\rm T}A$ ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of $A$ . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff $\epsilon$ . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.

Some theoretical comparisons of refined Ritz vectors and Ritz vectors

Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (‚;x) of a large matrix A. Given a subspace W that contains an approximation to x, these two methods compute approximations ("; ˜) and ("; ˆ x) to (‚;x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vectoris unique as the deviation of x from W approaches zero if ‚ is simple. Second, in terms of residual norm of the refined approximate eigenpair ("; ˆ), we derive lower and upper bounds for the sine of the angle between the Ritz vector ˜ x and the refined eigenvector approximation ˆ x, and we prove that ˜ x 6 ˆ unless ˆ x = x. Third, we establish relationships between the residual norm kA˜ x i "˜ xk of the conventional methods and the residual norm kAˆ i "ˆ xk of the refined methods, and we show that the latter is always smaller than the former if ("; ˆ) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.

A refined shift-and-invert arnoldi algorithm for large unsymmetric generalized eigenproblems

The shift-and-invert Arnoldi method has been popularly used for computing a number of eigenvalues close to a given shift and/or the associated eigenvectors of a large unsymmetric matrix pair, but there is no guarantee for the approximate eigenvectors, Ritz vectors, obtained by this method to converge even though the subspace is good enough. In order to correct this problem, a refined shift-and-invert Arnoldi method is proposed that uses certain refined Ritz vectors to approximate the desired eigenvectors. The refined Ritz vectors can be computed cheaply and reliably by small-sized singular value decompositions. It is shown that the refined method converges. A refined shift-and-invert Arnoldi algorithm is developed, and several numerical examples are reported. Comparisons are drawn on the refined algorithm and the shift-and-invert Arnoldi algorithm, indicating that the former is considerably more efficient than the latter.

Residuals of refined projection methods for large matrix eigenproblems

In terms of the deviation of the eigenvector ϕ from a projection subspace E, a priori error bounds are established for the residual norm of an approximate eigenpair obtained by a refined projection method. It is shown that the residual converges to zero as the deviation tends to zero. Finally, how to efficiently compute refined Ritz vectors is discussed in the refined symmetric Lanczos method.

An analysis of the Rayleigh--Ritz method for approximating eigenspaces

This paper concerns the Rayleigh- Ritz method for computing an approximation to an eigenspace X of a general matrix A from a subspace W that contains an approximation to X. The method produces a pair (N,X) that purports to approximate a pair (L,X), where X is a basis for X and AX = XL. In this paper we consider the convergence of (N,X) as the sine e of the angle between X and W approaches zero. It is shown that under a natural hypothesis - called the uniform separation condition - the Ritz pairs (N,X) converge to the eigenpair (L,X). When one is concerned with eigenvalues and eigenvectors, one can compute certain refined Ritz vectors whose convergence is guaranteed, even when the uniform separation condition is not satisfied. An attractive feature of the analysis is that it does not assume that A has distinct eigenvalues or is diagonalizable.

A refined subspace iteration algorithm for large sparse eigenproblems

Two innovations are presented to improve the orthogonal subspace iteration. Firstly, a refined strategy is proposed that replaces conventional Ritz vectors by refined Ritz vectors in subspace iteration. Each refined Ritz vector is chosen such that the norm of the residual formed with the Ritz value is minimized over the subspace involved, and it can be cheaply computed by solving a small problem. Secondly, a standard updating matrix, Ritz vector or Schur vector matrix, is replaced by a new one that is generated only by the refined Ritz vectors used to approximate the wanted eigenvectors. A combination of them yields a refined subspace iteration algorithm. A qualitative analysis is given for the refined algorithm, showing that it may be considerably more efficient than a conventional subspace iteration algorithm. When the subspace dimension is bigger than the number of the wanted eigenpairs, numerical examples show that the refined algorithm has a sharp superiority. Numerical comparisons are also drawn for the refined algorithm with the popular package ARPACK and an implicitly restarted version of the refined Arnoldi method. The algorithm is suited for computing a few dominant eigenpairs of large sparse matrices since the only action of a matrix A in question into the program is a subroutine to form the product AQ, where Q is a matrix having much less columns than rows.