Quadratic alternating direction implicit iteration for the fast solution of algebraic Riccati equations

Wong, Ngai; Balakrishnan, V.

doi:10.1109/ispacs.2005.1595424

Cited by 5 publications

(1 citation statement)

References 14 publications

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“…However, determining the optimal ADI parameter and finding approximations for the Lyapunov equation increase the burden on memory requirements and computational time. Recently, Wong and Balakrishnan [16] introduced an algorithm called Quadratic ADI (qADI) method to solve the algebraic Riccati equation (1). Their method is a direct extension of the Lyapunov ADI method.…”

Section: Introductionmentioning

confidence: 99%

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

Zhang,

Xun

2023

Preprint

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This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The method is based on generalized alternating direction implicit iteration (GADI), which exploits the low-rank property of matrices and utilizes the Cholesky factorization approach for solving. The advantage of the new algorithm lies in its direct and efficient low-rank formulation, which is a variant of the Cholesky decomposition in the Lyapunov GADI method, saving storage space and making it computationally effective. When solving the continuous-time algebraic Riccati matrix equation, the Riccati equation is first simplified to a Lyapunov equation using the Newton method, and then the R-GADI method is employed for computation. Additionally, we analyze the convergence of the R-GADI method and prove its consistency with the convergence of the GADI method. Finally, the effectiveness of the new algorithm is demonstrated through corresponding numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

Zhang,

Xun

2023

Preprint

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RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations

et al. 2017

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This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature—all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.

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The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

Guo

Liang

2022

Numer Algor

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Quadratic alternating direction implicit iteration for the fast solution of algebraic Riccati equations

Cited by 5 publications

References 14 publications

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations

The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

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