### Abstract

In this paper, we propose structured doubling algorithms for the computation of the weakly stabilizing Hermitian solutions of the continuous- and discrete-time algebraic Riccati equations, respectively. Assume that the partial multiplicities of purely imaginary and unimodular eigenvalues (if any) of the associated Hamiltonian and symplectic pencil, respectively, are all even and the C/DARE and the dual C/DARE have weakly stabilizing Hermitian solutions with property (P). Under these assumptions, we prove that if these structured doubling algorithms do not break down, then they converge to the desired Hermitian solutions globally and linearly. Numerical experiments show that the structured doubling algorithms perform efficiently and reliably.

Original language | English |
---|---|

Pages (from-to) | 1452-1478 |

Number of pages | 27 |

Journal | Linear Algebra and Its Applications |

Volume | 430 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - 2009 Mar 1 |

### Fingerprint

### Keywords

- Algebraic Riccati equation
- Global and linear convergence
- Hermitian solution
- Purely imaginary eigenvalue
- Structured doubling algorithm
- Unimodular eigenvalue

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*430*(5-6), 1452-1478. https://doi.org/10.1016/j.laa.2007.08.043

**Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations.** / Hwang, Tsung-Min; Lin, Wen Wei.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 430, no. 5-6, pp. 1452-1478. https://doi.org/10.1016/j.laa.2007.08.043

}

TY - JOUR

T1 - Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations

AU - Hwang, Tsung-Min

AU - Lin, Wen Wei

PY - 2009/3/1

Y1 - 2009/3/1

N2 - In this paper, we propose structured doubling algorithms for the computation of the weakly stabilizing Hermitian solutions of the continuous- and discrete-time algebraic Riccati equations, respectively. Assume that the partial multiplicities of purely imaginary and unimodular eigenvalues (if any) of the associated Hamiltonian and symplectic pencil, respectively, are all even and the C/DARE and the dual C/DARE have weakly stabilizing Hermitian solutions with property (P). Under these assumptions, we prove that if these structured doubling algorithms do not break down, then they converge to the desired Hermitian solutions globally and linearly. Numerical experiments show that the structured doubling algorithms perform efficiently and reliably.

AB - In this paper, we propose structured doubling algorithms for the computation of the weakly stabilizing Hermitian solutions of the continuous- and discrete-time algebraic Riccati equations, respectively. Assume that the partial multiplicities of purely imaginary and unimodular eigenvalues (if any) of the associated Hamiltonian and symplectic pencil, respectively, are all even and the C/DARE and the dual C/DARE have weakly stabilizing Hermitian solutions with property (P). Under these assumptions, we prove that if these structured doubling algorithms do not break down, then they converge to the desired Hermitian solutions globally and linearly. Numerical experiments show that the structured doubling algorithms perform efficiently and reliably.

KW - Algebraic Riccati equation

KW - Global and linear convergence

KW - Hermitian solution

KW - Purely imaginary eigenvalue

KW - Structured doubling algorithm

KW - Unimodular eigenvalue

UR - http://www.scopus.com/inward/record.url?scp=58349112763&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58349112763&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2007.08.043

DO - 10.1016/j.laa.2007.08.043

M3 - Article

AN - SCOPUS:58349112763

VL - 430

SP - 1452

EP - 1478

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 5-6

ER -