Noda iterations for generalized eigenproblems following Perron-Frobenius theory
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In this paper, we investigate the generalized eigenvalue problem Ax = λBx arising from economic models. Under certain conditions, there is a simple generalized eigenvalue ρ(A, B) in the interval (0, 1) with a positive eigenvector. Based on the Noda iteration, a modified Noda iteration (MNI) and a generalized Noda iteration (GNI) are proposed for finding the generalized eigenvalue ρ(A, B) and the associated unit positive eigenvector. It is proved that the GNI method always converges and has a quadratic asymptotic convergence rate. So GNI has a similar convergence behavior as MNI. The efficiency of these algorithms is illustrated by numerical examples.
KeywordsGeneralized eigenproblem Generalized Noda iteration Nonnegative irreducible matrix M-matrix Quadratic convergence Perron-Frobenius theory
Mathematics Subject Classification (2010)65F15 65F99
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We thank Prof. Weizhang Huang of University of Kansas for introducing his work  and providing the data of Example 5.3. We would like to thank an anonymous referee for his/her valuable comments.
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