On the Doubling Algorithm for a (Shifted) Nonsymmetric Algebraic Riccati Equation
@article{Guo2007OnTD,
title={On the Doubling Algorithm for a (Shifted) Nonsymmetric Algebraic Riccati Equation},
author={Chun-Hua Guo and Bruno Iannazzo and Beatrice Meini},
journal={SIAM J. Matrix Anal. Appl.},
year={2007},
volume={29},
pages={1083-1100}
}Nonsymmetric algebraic Riccati equations for which the four coefficient matrices form an irreducible $M$-matrix $M$ are considered. The emphasis is on the case where $M$ is an irreducible singular $M$-matrix, which arises in the study of Markov models. The doubling algorithm is considered for finding the minimal nonnegative solution, the one of practical interest. The algorithm has been recently studied by others for the case where $M$ is a nonsingular $M$-matrix. A shift technique is proposed…
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References
SHOWING 1-10 OF 23 REFERENCES
Iterative Solution of a Nonsymmetric Algebraic Riccati Equation
- MathematicsSIAM J. Matrix Anal. Appl.
- 2007
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.
A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation
- Mathematics, Computer ScienceNumerische Mathematik
- 2006
A structure-preserving doubling algorithm for the computation of the minimal nonnegative solution to the nonsymmetric algebraic Riccati equation (NARE), based on the techniques developed for the symmetric cases, which outperforms Newton's iteration and the fixed-point iteration methods.
On the Iterative Solution of a Class of Nonsymmetric Algebraic Riccati Equations
- MathematicsSIAM J. Matrix Anal. Appl.
- 2000
Newton's method and a class of basic fixed-point iterations can be used to find its minimal positive solution whenever it has a positive solution of any equation in this class of nonsymmetric algebraic Riccati equations.
Solution Form and Simple Iteration of a Nonsymmetric Algebraic Riccati Equation Arising in Transport Theory
- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2005
It is shown that this computation can be done via computing only the minimal positive solution of a vector equation, which is derived from the special form of solutions of the Riccati equation.
ON THE MINIMAL NONNEGATIVE SOLUTION OFNONSYMMETRIC ALGEBRAIC RICCATI EQUATION
- Mathematics, Computer Science
- 2005
Using the matrix sign function method, it is shown how to compute the desired M-matrix solution of the quadratic matrix equation X^2 - EX - F = 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M- matrix.
Efficient methods for solving a nonsymmetric algebraic Riccati equation arising in stochastic fluid models
- Mathematics
- 2006
A note on the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation
- Mathematics
- 2002
On the solution of algebraic Riccati equations arising in fluid queues
- Mathematics, Computer Science
- 2006