Noda iterations for generalized eigenproblems following Perron-Frobenius theory
- 26 Downloads
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
Unable to display preview. Download preview PDF.
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.
- 5.Fujimoto, T.: A generalization of the Frobenius theorem. Toyama University Econ. Rev. 23(2), 269–274 (1979)Google Scholar
- 6.Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)Google Scholar
- 14.Sraffa, P.: Production of commodities by means of commodities. Cambridge University Press, Cambridge (1960)Google Scholar
- 16.Steward, G.W., Sun, J.G.: Matrix perturbation theory. Academic Press, Boston (1990)Google Scholar
- 17.Varga, R.S.: Matrix iterative analysis. Springer, Berlin (1962)Google Scholar