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

Anna Concas, Lothar Reichel, Giuseppe Rodriguez and Yunzi Zhang
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Anna Concas: Department of Mathematics and Computer Science, University of Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
Lothar Reichel: Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
Giuseppe Rodriguez: Department of Mathematics and Computer Science, University of Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
Yunzi Zhang: Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

Mathematics, 2021, vol. 9, issue 13, 1-16

Abstract: 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.

Keywords: networks; perron vector; power method; lanczos method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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