### Abstract

The main purpose of this paper is the study of numerical methods for the maximal solution of the matrix equation X + A ^{*}X ^{-1}A = Q, where Q is Hermitian positive definite. We construct a smooth curve parameterized by t ≥ 1 of symplectic pairs with a special structure, in which the curve passes through all iteration points generated by the known numerical methods, including the fixed-point iteration, the structure-preserving doubling algorithm (SDA), and Newton's method provided that A ^{*}Q ^{-1}A = AQ ^{-1}A ^{*}. In the theoretical section, we give a necessary and sufficient condition for the existence of this structure-preserving curve for each parameter t ≥ 1. We also study the monotonicity and boundedness properties of this curve. In the application section, we use this curve to measure the convergence rates of those numerical methods. Numerical results illustrating these solutions are also presented.

Original language | English |
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Pages (from-to) | 597-616 |

Number of pages | 20 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 33 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Sep 3 |

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### Keywords

- Fixed-point iteration
- Newton's method
- Nonlinear matrix equation
- Structure-preserving curve
- Structure-preserving doubling algorithm
- Symplectic pair

### ASJC Scopus subject areas

- Analysis

### Cite this

**A structure-preserving curve for symplectic pairs and its applications.** / Kuo, Yueh Cheng; Shieh, Shih-Feng.

Research output: Contribution to journal › Article

*SIAM Journal on Matrix Analysis and Applications*, vol. 33, no. 2, pp. 597-616. https://doi.org/10.1137/110843137

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TY - JOUR

T1 - A structure-preserving curve for symplectic pairs and its applications

AU - Kuo, Yueh Cheng

AU - Shieh, Shih-Feng

PY - 2012/9/3

Y1 - 2012/9/3

N2 - The main purpose of this paper is the study of numerical methods for the maximal solution of the matrix equation X + A *X -1A = Q, where Q is Hermitian positive definite. We construct a smooth curve parameterized by t ≥ 1 of symplectic pairs with a special structure, in which the curve passes through all iteration points generated by the known numerical methods, including the fixed-point iteration, the structure-preserving doubling algorithm (SDA), and Newton's method provided that A *Q -1A = AQ -1A *. In the theoretical section, we give a necessary and sufficient condition for the existence of this structure-preserving curve for each parameter t ≥ 1. We also study the monotonicity and boundedness properties of this curve. In the application section, we use this curve to measure the convergence rates of those numerical methods. Numerical results illustrating these solutions are also presented.

AB - The main purpose of this paper is the study of numerical methods for the maximal solution of the matrix equation X + A *X -1A = Q, where Q is Hermitian positive definite. We construct a smooth curve parameterized by t ≥ 1 of symplectic pairs with a special structure, in which the curve passes through all iteration points generated by the known numerical methods, including the fixed-point iteration, the structure-preserving doubling algorithm (SDA), and Newton's method provided that A *Q -1A = AQ -1A *. In the theoretical section, we give a necessary and sufficient condition for the existence of this structure-preserving curve for each parameter t ≥ 1. We also study the monotonicity and boundedness properties of this curve. In the application section, we use this curve to measure the convergence rates of those numerical methods. Numerical results illustrating these solutions are also presented.

KW - Fixed-point iteration

KW - Newton's method

KW - Nonlinear matrix equation

KW - Structure-preserving curve

KW - Structure-preserving doubling algorithm

KW - Symplectic pair

UR - http://www.scopus.com/inward/record.url?scp=84865489145&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865489145&partnerID=8YFLogxK

U2 - 10.1137/110843137

DO - 10.1137/110843137

M3 - Article

VL - 33

SP - 597

EP - 616

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 2

ER -