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

On the link between Markovian trees and tree-structured Markov chains

“…We further show that the two natural fixed-point iterations for tree-structured QBDs [24], when applied to the associated TLQBD, give rise to the depth and (both) order algorithms for MBTs (see Section 2.1.1). The Newton iteration for TLQBDs however reduces to a different type of Newton iteration for MBTs than the one discussed in Hautphenne et al [6]. We analyze the convergence properties of this novel Newton iteration, and we show that it needs less iterations than the existing Newton algorithm for MBTs.…”

“…It offers some advantages in that it better exploits possible dissymmetries in the structure of the matrix B. Finally, a quadratic algorithm [6], called the Newton algorithm, is obtained using Newton's iteration method on (1).…”

“…An important problem of MBTs exists in determining the extinction probability vector q, where its ith entry holds the probability that an MBT starting in phase i becomes extinct. Several algorithms to compute q with linear convergence [12,2,5] and one with quadratic convergence based on a Newton iteration [6] have been developed.…”

Extinction Probabilities of Supercritical Decomposable Branching Processes

“…Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints, and that it is also equal to the total extinction of a modified branching process. We discuss the algorithmic issues to compute the partial extinction probabilities, in particular for special classes of multitype branching processes called Markovian trees [1], [6], [7].…”

“…This allows for the direct use of any available algorithm. For instance, in some special cases of multitype branching processes called Markovian trees, several linear and quadratic algorithms have been developed to compute the total extinction probability; see [1], [6], and [7].…”