Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer...

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Bibliographic Details
Main Authors: Anna Concas, Lothar Reichel, Giuseppe Rodriguez, Yunzi Zhang
Format: Article
Language:English
Published: MDPI AG 2021-06-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/9/13/1522
Description
Summary:The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.
ISSN:2227-7390