Iterative Solution of a Nonsymmetric Algebraic Riccati Equation

@article{Guo2007IterativeSO,
  title={Iterative Solution of a Nonsymmetric Algebraic Riccati Equation},
  author={Chun-Hua Guo and Nicholas John Higham},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2007},
  volume={29},
  pages={396-412}
}
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… 

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References

SHOWING 1-10 OF 28 REFERENCES

ON THE MINIMAL NONNEGATIVE SOLUTION OFNONSYMMETRIC ALGEBRAIC RICCATI EQUATION

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.

Solution Form and Simple Iteration of a Nonsymmetric Algebraic Riccati Equation Arising in Transport Theory

  • Linzhang Lu
  • Mathematics, Computer Science
    SIAM 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 Iterative Solution of a Class of Nonsymmetric Algebraic Riccati Equations

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.

On a quadratic matrix equation associated with an M‐matrix

We study the quadratic matrix equation X 2 - EX - F = 0, where E is diagonal and F is an M-matrix. Quadratic matrix equations of this type arise in noisy Wiener-Hopf problems for Markov chains. The

Existence of algebraic matrix Riccati equations arising in transport theory

Algebraic Riccati equations

1. Preliminaries from the theory of matrices 2. Indefinite scalar products 3. Skew-symmetric scalar products 4. Matrix theory and control 5. Linear matrix equations 6. Rational matrix functions 7.