Iterative Solution of a Nonsymmetric Algebraic Riccati Equation

Abstract

Abstract. We study the nonsymmetric algebraic Riccati equation whose four coefficient matrices are the blocks of a nonsingular M -matrix or an irreducible singular M -matrix M . The solution of practical interest is the minimal nonnegative solution. We show that Newton's method with zero initial guess can be used to find this solution without any further assumptions. We also present a qualitative perturbation analysis for the minimal solution, which is instructive in designing algorithms for finding more accurate approximations. For the most practically important case, in which M is an irreducible singular M -matrix with zero row sums, the minimal solution is either stochastic or substochastic and the Riccati equation can be transformed into a unilateral matrix equation by a procedure of Ramaswami. The minimal solution of the Riccati equation can then be found by computing the minimal nonnegative solution of the unilateral equation using the LatoucheRamaswami algorithm. When the minimal solution of the Riccati equation is stochastic, we show that the Latouche-Ramaswami algorithm, combined with a shift technique suggested by He, Meini, and Rhee, is breakdown-free and is able to find the minimal solution more efficiently and more accurately than the algorithm without a shift. When the minimal solution of the Riccati equation is substochastic, we show how the substochastic minimal solution can be found by computing the stochastic minimal solution of a related Riccati equation of the same type.

Explicit Solutions for a Riccati Equation from Transport Theory

“…which is (12). Clearly, the eigenvalues ofR 2 are the n leftmost eigenvalues ofH, which are −ν 1 , .…”

“…The computation of this minimal solution has been investigated in several publications. Various direct and iterative methods [1,10,11,12,13,14,15,17,16,24] have been proposed by either directly solving the Riccati equation or by computing specific invariant subspaces of the 2n × 2n matrix…”

Newton's Method for Solving a Quadratic Matrix Equation With Special Coefficient Matrices

“…Definition 1.1. [3], [17, p. 42] Let a matrix A ∈ R n×n . A is an Z-matrix if all its off-diagonal elements are nonpositive.…”

Transforming algebraic Riccati equations into unilateral quadratic matrix equations

“…is either a nonsingular M-matrix or a singular irreducible M-matrix, see the papers [7,8,[16][17][18].…”