On the Randomized Error of Polynomial Methods for Eigenvector and Eigenvalue Estimates |
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Authors: | Gianna M Del Corso Giovanni Manzini |
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Institution: | aDipartimento di Matematica, Università di Milano, Via Saldini 52, I-20133, Milano;bIMC-CNR, Via Santa Maria 46, I-56100, Pisa, Italy;cDipartimento Scienze e Tecnologie Avanzate, Via Cavour 84, I-15100, Alessandria |
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Abstract: | In this paper we consider the problem of estimating the largest eigenvalue and the corresponding eigenvector of a symmetric matrix. In particular, we consider iterative methods, such as the power method and the Lanczos method. These methods need a starting vector which is usually chosen randomly. We analyze the behavior of these methods when the initial vector is chosen with uniform distribution over the unitn-dimensional sphere. We extend and generalize the results reported earlier. In particular, we give upper and lower bounds on the pnorm of the randomized error, and we improve previously known bounds with a detailed analysis of the role of the multiplicity of the largest eigenvalue. |
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Keywords: | power and Lanczos methods eigenvalues and eigenvectors random start randomized error |
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