Abstract
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.
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Acknowledgements
We thank Prof. Weizhang Huang of University of Kansas for introducing his work [7] 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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This project was supported by National Natural Science Foundations of China (11771159, 11671158) and grant MYRG2017-00098-FST from University of Macau, and Major Project (2016KZDXM025) and Innovation Team Project (2015KCXTD007) of Guangdong Provincial Universities
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Chen, X.S., Vong, SW., Li, W. et al. Noda iterations for generalized eigenproblems following Perron-Frobenius theory. Numer Algor 80, 937–955 (2019). https://doi.org/10.1007/s11075-018-0512-4
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DOI: https://doi.org/10.1007/s11075-018-0512-4
Keywords
- Generalized eigenproblem
- Generalized Noda iteration
- Nonnegative irreducible matrix
- M-matrix
- Quadratic convergence
- Perron-Frobenius theory