Newton’s iteration for the extinction probability of a Markovian binary tree

Continuous-time multi-type branching processes have applications in a large number of fields such as biology and telecommunication systems. A basic problem for this kind of processes is to determine the extinction probability and, in order to compute it, it is necessary to find the minimal nonnegative solution of a non-linear matrix equation. We consider here a particular family of branching processes called Markovian binary trees. These give rise to second-order equations and we apply Newton’s method for fixed-point equations. We show that this algorithm is well defined and converges quadratically in the domain of interest. We also give it a probabilistic interpretation in terms of the branching process itself.