Harmonic projection methods for large non-symmetric eigenvalue problems

Morgan, Ronald B.; Zeng, Min

doi:10.1002/(sici)1099-1506(199801/02)5:1<33::aid-nla125>3.0.co;2-1

Cited by 91 publications

(85 citation statements)

References 35 publications

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“…Third, better results for the convergence of µ + τ are derived whenx converges. We show why the harmonic method can produce a spurious harmonic Ritz value and fail to find the desired eigenvalue λ if it is very close to the target τ , a phenomenon observed by some researchers [15,16,21]. We construct a set of examples to illustrate this phenomenon.…”

Section: Introductionmentioning

confidence: 80%

“…However, it is often the case that in practical applications factoring A − τI is too expensive and even impractical. The harmonic projection method is primarily used to settle this problem and to compute interior eigenvalues and eigenvectors of large matrices without factoring A − τI; see [15,16,22,24] for more details.…”

Section: Introductionmentioning

confidence: 99%

“…In the simplest form the harmonic projection method works as follows (see, e.g., [16], where the method is also referred to as the harmonic Rayleigh-Ritz method. Here we drop the iteration subscript).…”

Section: Introductionmentioning

confidence: 99%

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The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

Jia

2004

Math. Comp.

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Abstract. This paper concerns a harmonic projection method for computing an approximation to an eigenpair (λ, x) of a large matrix A. Given a target point τ and a subspace W that contains an approximation to x, the harmonic projection method returns an approximation (µ + τ,x) to (λ, x). Three convergence results are established as the deviation of x from W approaches zero. First, the harmonic Ritz value µ + τ converges to λ if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector x converges to x if the Rayleigh quotient matrix is uniformly nonsingular and µ + τ remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of µ + τ are derived whenx converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue λ-in other words, the method can miss λ if it is very close to τ . To this end, we propose to compute the Rayleigh quotient ρ of A with respect tox and take it as a new approximate eigenvalue. ρ is shown to converge to λ oncex tends to x, no matter how τ is close to λ. Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

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Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

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The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

Jia

2004

Math. Comp.

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“…The harmonic Ritz vectors corresponding to the smallest eigenvalues of A is used in SGMRES-DR because they are better approximate eigenvectors for eigenvalues with small modulus [10]. Note that the term "the smallest eigenvalues" means the smallest eigenvalues in modulus.…”

Section: Sgmres With Deflated Restartingmentioning

confidence: 99%

An Intelligent Method for Accelerating the Convergence of Different Versions of SGMRES Algorithm

Zarch¹,

Fazeli²,

Dowlatshahi³

2016

ACII

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In a wide range of applications, solving the linear system of equations Ax = b is appeared. One of the best methods to solve the large sparse asymmetric linear systems is the simplified generalized minimal residual (SGMRES(m)) method. Also, some improved versions of SGMRES(m) exist: SGMRES-E(m, k) and SGMRES-DR(m, k). In this paper, an intelligent heuristic method for accelerating the convergence of three methods SGMRES(m), SGMRES-E(m, k), and SGMRES-DR(m, k) is proposed. The numerical results obtained from implementation of the proposed approach on several University of Florida standard matrixes confirm the efficiency of the proposed method.

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“…Instead of Lan-DR, a method Minres-DR can be used. It solves the linear equations problem with a minimum residual projection and it computes harmonic Ritz vectors [27,17,42,34,66,35] instead of regular Ritz vectors. Harmonic Ritz approximations are more reliable for interior eigenvalues.…”

Section: Example From Qcdmentioning

confidence: 99%

Deflated and Restarted Symmetric Lanczos Methods for Eigenvalues and Linear Equations with Multiple Right-Hand Sides

Abdel-Rehim¹,

Morgan²,

Nicely³

et al. 2010

SIAM J. Sci. Comput.

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Abstract.A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is necessary to control roundoff error, and several approaches are discussed. The eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides. Experiments are given with large matrices from quantum chromodynamics that have many right-hand sides.Key words. linear equations, deflation, eigenvalues, Lanczos, conjugate gradient, QCD, multiple right-hand sides, symmetric, Hermitian AMS subject classifications. 65F10, 65F15, 15A06, 15A181. Introduction. We consider a large matrix A that is either real symmetric or complex Hermitian. We are interested in solving the system of equations Ax = b, possibly with multiple right-hand sides, and in solving the associated eigenvalue problem. Both eigenvalues and eigenvectors are desired. Symmetric and Hermitian problems can take advantage of fast algorithms such as the conjugate gradient method (CG) [22,52] for linear equations and the related Lanczos algorithm [25,46] for eigenvalues. However, regular CG can be improved upon for the case of multiple right-hand sides, and Lanczos may have storage and accuracy issues while computing eigenvectors. We give new methods for these problems.An approach is presented called Lanczos with deflated restarting or Lan-DR. It simultaneously solves the linear equations and computes the eigenvalues and eigenvectors. A restarted Krylov subspace approach is used for the linear equations, but it also saves approximate eigenvectors at the restart and uses them in the subspace for the next cycle. The restarting of the Lanczos algorithm does not slow down the convergence as it normally would, because once the approximate eigenvectors are accurate enough, they essentially remove or deflate the associated eigenvalues from the problem. The eigenvalue portion of Lan-DR has already been presented in [68], but as mentioned, we add on the solution of linear equations. Also, some reorthogonalization is necessary to control roundoff error. We give some new approaches for this reorthogonalization. We also give a Minres/harmonic version of the algorithm.

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Harmonic projection methods for large non-symmetric eigenvalue problems

Cited by 91 publications

References 35 publications

The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

An Intelligent Method for Accelerating the Convergence of Different Versions of SGMRES Algorithm

Deflated and Restarted Symmetric Lanczos Methods for Eigenvalues and Linear Equations with Multiple Right-Hand Sides

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