Highly accurate doubling algorithm for quadratic matrix equation from quasi-birth-and-death process

Abstract

A highly accurate doubling algorithm to solve the most fundamental quadratic matrix equation in the quasi-birth-and-death (qbd) process is developed. It follows from the general framework of the doubling algorithm for the first standard form (SF1) but can be implemented to compute the minimal nonnegative solution with high entrywise relative accuracy for all entries, large or tiny. The algorithm is globally and quadratically convergent, except for qbd equations in the critical case where convergence is linear with the linear rate 1/2. Numerical examples are presented to demonstrate and confirm our claims. The development here parallels the recent work of Xue and Li (2017) [3] on the M-matrix algebraic Riccati equation.