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On the solution of a quadratic vector equation arising in Markovian Binary Trees

Authors


Federico Poloni, Scuola Normale Superiore. Piazza dei Cavalieri 7, 56126 Pisa, Italy.

E-mail: f.poloni@sns.it

SUMMARY

We present some advances, both from a theoretical and a computational point of view, on a quadratic vector equation (QVE) arising in Markovian Binary Trees. Concerning the theoretical advances, some irreducibility assumptions are relaxed, and the minimality of the solution of the QVE is expressed in terms of properties of the Jacobian of a suitable function. From the computational point of view, we elaborate on the Perron vector-based iteration proposed in [Meini and Poloni, A Perron iteration for the solution of a quadratic vector equation arising in Markovian binary trees]. In particular, we provide a condition that ensures that the Perron iteration converges to the sought solution of the QVE. Moreover we introduce a variant of the algorithm that consists of the application of the Newton method instead of a fixed-point iteration. This method has the same convergence behavior as the Perron iteration, namely it tends to converge faster for close-to-critical problems. Moreover, unlike the Perron iteration, the method has a quadratic convergence. Finally, we show that it is possible to alter the bilinear form defining the QVE in several ways without changing the solution. This modification has an impact on convergence speed of the algorithms. Copyright © 2011 John Wiley & Sons, Ltd.

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