A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations

Chu, Eric King-wah; Fan, Hung-Yuan; Lin, Wen‐Wei

doi:10.1016/j.laa.2004.10.010

Cited by 90 publications

(91 citation statements)

References 31 publications

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“…The nonlinear matrix equations CARE and DARE have been studied extensively (see [1]- [2], [4]- [8], [13]- [14], [17]- [25], [27]- [29], and [31]); and the nonlinear matrix equations NME-P and NME-M has been studied recently by several authors (see [3], [9]- [11], [15]- [16], [26], [30], and [32]). …”

Section: Introductionmentioning

confidence: 99%

“…Doubling algorithms were largely forgotten in the past decade. Recently, they have been revived for DAREs and CAREs, because their nice numerical behavior: quadratical convergence rate, low cost computational cost per step, and good numerical stability (see [6,7,8]). Concerning the matrix equations NME-Ps and NME-Ms, in 2001 B. Meini proposed an iterative method with the same numerical behavior as the SDA algorithms for solving the DAREs and CAREs (see [26,16]).…”

Section: Introductionmentioning

confidence: 99%

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Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations

Lin¹,

Xu²

2006

SIAM J. Matrix Anal. & Appl.

102

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In this paper, a structure-preserving transformation of a symplectic pencil is introduced, referred to as the doubling transformation, and its some basic properties are presented. Based on the nice properties of this kind of transformations, a unified convergence theory for the structure-preserving doubling algorithms for solving a class of Riccati-type matrix equations is established by using only the knowledge from elementary matrix theory.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations

Lin¹,

Xu²

2006

SIAM J. Matrix Anal. & Appl.

102

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“…The relation L 1 R 0 = L 0 R 1 has been studied extensively for regular pencils in the context of pencil arithmetic and inverse-free matrix iterative algorithms [2,3,9,16,31]. Two main techniques exist for constructing L 0 , L 1 starting from R 0 , R 1 (or vice versa).…”

Section: Constructing Dualsmentioning

confidence: 99%

“…Two main techniques exist for constructing L 0 , L 1 starting from R 0 , R 1 (or vice versa). Enforcing an identity block [9,32] Suppose that the identity matrix is a submatrix of col(R), for a pencil R ∈ C[x] n×p…”

Section: Constructing Dualsmentioning

confidence: 99%

“…Dual pencils appear in [26,Section 1.3], where they are given the name of consistent pencils and some of their theoretical properties are stated. Moreover, one can recognize the use of duality (of regular pencils only, which is a less interesting case) in the study of doubling and inverse-free methods [2,9,31], as well as in the work [3], which gives an elegant algebraic theory of operations on matrix pencils.…”

Section: Introductionmentioning

confidence: 99%

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Duality of matrix pencils, Wong chains and linearizations

Noferini

Poloni

2015

Linear Algebra and its Applications

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We consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils. One is dual pencils, a pair of pencils with the same regular part and related singular structures. They were introduced by V. Kublanovskaya in the 1980s. The other is Wong chains, families of subspaces, associated with (possibly singular) matrix pencils, that generalize Jordan chains. They were introduced by K.T. Wong in the 1970s. Together, dual pencils and Wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications, especially in the context of linearizations of matrix polynomials.We first give a self-contained introduction to these two concepts, using modern language and extending them to a more general form; we describe the relation between them and show how they act on the Kronecker form of a pencil and on spectral and singular structures (eigenvalues, eigenvectors and minimal bases). Then we present several new applications of these results to more recent topics in matrix pencil theory, including: constraints on the minimal indices of singular Hamiltonian and symplectic pencils, new sufficient conditions under which pencils in L 1 , L 2 linearization spaces are strong linearizations, a new perspective on Fiedler pencils, and a link between the Möller-Stetter theorem and some linearizations of matrix polynomials.

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A GPU‐aware mixed‐precision solver for low‐rank algebraic Riccati equations

Benner

Dufrechou

Ezzatti

et al. 2018

Concurrency and Computation

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Summary We investigate different alternatives for the solution of algebraic Riccati equations on hybrid hardware platforms (ie, CPUs+GPUs). We evaluate a mixed‐precision approach that uses single‐precision arithmetic to obtain an approximation to the solution and later improve it to the desired precision, applying some steps of an economic iterative refinement. This method exploits the higher performance of the hardware to accelerate the solver when single‐precision arithmetic is employed and simultaneously obtains a high‐accuracy solution with the iterative refinement. We extend this approach to exploit the low‐rank property of the equation, when possible, to further improve its efficiency. The experimental evaluation shows that the mixed‐precision approach reports time and energy savings and provides similar or even more accurate solutions than well‐known methods like the sign function iteration or the structure‐preserving doubling algorithm.

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A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations

Cited by 90 publications

References 31 publications

Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations

Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations

Duality of matrix pencils, Wong chains and linearizations

A GPU‐aware mixed‐precision solver for low‐rank algebraic Riccati equations

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