Guo, ChunHua and Higham, Nicholas J. (2006) Iterative Solution of a Nonsymmetric Algebraic Riccati Equation. [MIMS Preprint]
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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 LatoucheRamaswami algorithm, combined with a shift technique suggested by He, Mini, and Rhee, is breakdownfree 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.
Item Type:  MIMS Preprint 

Uncontrolled Keywords:  nonsymmetric algebraic Riccati equation, $M$matrix, minimal nonnegative solution, perturbation analysis, Newton's method, LatoucheRamaswami 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:  21 Jun 2006 
Last Modified:  08 Nov 2017 18:18 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/309 
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Iterative Solution of a Nonsymmetric Algebraic Riccati Equation. (deposited 16 Dec 2005)
 Iterative Solution of a Nonsymmetric Algebraic Riccati Equation. (deposited 21 Jun 2006) [Currently Displayed]
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