Iterative Solution of a Nonsymmetric Algebraic Riccati Equation

Guo, Chun-Hua and Higham, Nicholas J. (2005) Iterative Solution of a Nonsymmetric Algebraic Riccati Equation. [MIMS Preprint]

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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 tranformed 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 Latouche--Ramaswami algorithm. We show that the Latouche--Ramawami algorithm, combined with a shift technique suggested by He, Mini, and Rhee, is breakdown-free in all cases and is able to find the minimal solution more efficiently and more accurately than the algorithm without a shift. Our approach is to find a proper stochastic solution using the shift technique even if it is not the minimal solution. We show how we can easily recover the minimal solution when it is not the computed stochastic solution.

Item Type: MIMS Preprint
Uncontrolled Keywords: nonsymmetric algebraic Riccati equation, $M$-matrix, minimal nonnegative solution, perturbation analysis, Newton's method, Latouche--Ramaswami algorithm, shifts
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 16 Dec 2005
Last Modified: 08 Nov 2017 18:18

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