Improved global algorithms for maximal eigenpair

Chen, Mu-Fa; Li, Yue-Shuang

doi:10.1007/s11464-019-0799-z

Cited by 2 publications

(11 citation statements)

References 12 publications

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“…This paper is motivated by the study on the global algorithms given in [3,7], where some effective algorithms were presented for computing the maximal eigenpair of a rather larger class of matrices. Roughly speaking, two approaches are adopted there: the power iteration (abbrev.…”

Section: Introduction: Extended Global Algorithmmentioning

confidence: 99%

“…It is also economical (having lower computational complexity), but has a quite slow convergence speed, especially near the target eigenvalue. The fast convergence speed of the algorithms given in [3,7] is mainly due to the use of IPI v (having higher computational complexity). It is however quite dangerous if the initial is not close enough (from above) to the target eigenvalue.…”

Section: Introduction: Extended Global Algorithmmentioning

confidence: 99%

“…It is however quite dangerous if the initial is not close enough (from above) to the target eigenvalue. The last problem was avoided in [3,7] mainly due to the assumption: the off-diagonal elements of the matrix are all nonnegative. This is essential: it implies the existence of the maximal eigenpair (as an application of the Perron-Frobenius theorem, by a shift if necessary).…”

Section: Introduction: Extended Global Algorithmmentioning

confidence: 99%

“…Starting from 1ß 1 , where 1 is the constant column vector having its component 1 everywhere and x is the L 2 -norm of x. The resulting θ defined by ( 8) is what we need for (7). This method is especially good for PI, it converges economically to λ ¦ , the maximal eigenvalue in modulus, but not the real maximum, effective enough unless it is too close to λ ¦ .…”

Section: Introduction: Extended Global Algorithmmentioning

confidence: 99%

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Top Eigenpairs of Large Scale Matrices

null¹,

Chen²

2022

CSIAM-AM

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This paper is devoted to the study of an extended global algorithm on computing the top eigenpairs of a large class of matrices. Three versions of the algorithm are presented that includes a preliminary version for real matrices, one for complex matrices, and one for large scale sparse real matrix. Some examples are illustrated as powerful applications of the algorithms. The main contributions of the paper are two localized estimation techniques, plus the use of a machine learning inspired approach in terms of a modified power iteration. Based on these new tools, the proposed algorithm successfully employs the inverse iteration with varying shifts (a very fast "cubic algorithm") to achieve a superior estimation accuracy and computation efficiency to existing approaches under the general setup considered in this work.

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Section: Introduction: Extended Global Algorithmmentioning

confidence: 99%

Section: Introduction: Extended Global Algorithmmentioning

confidence: 99%

Section: Introduction: Extended Global Algorithmmentioning

confidence: 99%

Section: Introduction: Extended Global Algorithmmentioning

confidence: 99%

See 2 more Smart Citations

Top Eigenpairs of Large Scale Matrices

null¹,

Chen²

2022

CSIAM-AM

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“…Finding the largest eigenpairs has many applications in signal processing, control, and recent development of Google's PageRank algorithm. In a series of papers, Mufa Chen [3][4][5] developed some efficient algorithms for computing the maximal eigenpairs for tridiagonal matrices. The key idea is to explicitly construct effective initials for the maximal eigenpairs and also employ a self-closed iterative algorithm.…”

Section: Introductionmentioning

confidence: 99%

Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm

Yang¹

2020

CMR

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A hybrid method is presented for determining maximal eigenvalue and its eigenvector (called eigenpair) of a large, dense, symmetric matrix. Many problems require finding only a small part of the eigenpairs, and some require only the maximal one. In a series of papers, efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements. The key idea is to explicitly construct effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm. In this paper, we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale, dense, symmetric matrix. Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations. We then handle the cases that prevent us from applying Chen's algorithm directly, e.g., the cases with zero or negative super-or sub-diagonal elements. Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.

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Improved global algorithms for maximal eigenpair

Cited by 2 publications

References 12 publications

Top Eigenpairs of Large Scale Matrices

Top Eigenpairs of Large Scale Matrices

Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm

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