A symplectic QR like algorithm for the solution of the real algebraic Riccati equation

Bunse-Gerstner, Angelika; Mehrmann, Volker

doi:10.1109/tac.1986.1104186

Cited by 108 publications

(94 citation statements)

References 9 publications

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“…A well-known fact is that X11 is invertible and Pmin = P Hamiltonian structure. Here we present an effective implementation of the SR algorithm [20] to achieve subspace separation. It is assumed that H has already been transformed into J-tridiagonal form (see remarks):…”

Section: Solving Dual Caresmentioning

confidence: 99%

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Passivity-preserving model reduction via a computationally efficient project-and-balance scheme

Wong

Balakrishnan

Koh

2004

Proceedings of the 41st Annual Design Automation Conference

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This paper presents an efficient two-stage project-and-balance scheme for passivity-preserving model order reduction. Orthogonal dominant eigenspace projection is implemented by integrating the Smith method and Krylov subspace iteration. It is followed by stochastic balanced truncation wherein a novel method, based on the complete separation of stable and unstable invariant subspaces of a Hamiltonian matrix, is used for solving two dual algebraic Riccati equations at the cost of essentially one. A fast-converging quadrupleshift bulge-chasing SR algorithm is also introduced for this purpose. Numerical examples confirm the quality of the reduced-order models over those from conventional schemes.

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Section: Solving Dual Caresmentioning

confidence: 99%

“…where Ψ = diag(Ii−1, P, I k−l−i+1 ) and P = I l −2ww T /w T w. Again, the choice of w ∈ l , 2 ≤ l ≤ k − i + 1, is standard [17,20]. Algorithm H is used to zero column vectors of length l on the upper half of the Hamiltonian matrix.…”

Section: Solving Dual Caresmentioning

confidence: 99%

Passivity-preserving model reduction via a computationally efficient project-and-balance scheme

Wong

Balakrishnan

Koh

2004

Proceedings of the 41st Annual Design Automation Conference

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“…In scientific computation, the eigenvalue problem of the Hamiltonian matrix is an important and well-studied topic [1,2,6]. One applications of the symplectic method is solving the Ricatti equation arising from control theory [3,5]. The symplectic method enables us to compute eigenvalues quickly by using the structure of a matrix, unlike the Householder qr or Lanczos methods.…”

Section: Introductionmentioning

confidence: 99%

Block symplectic Gram–Schmidt method

Matsuo

Nodera

2016

ANZIAMJ

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For large scale linear problems, it is common to use the symplectic Lanczos method which uses the symplectic Gram-Schmidt method to compute symplectic vectors. However, previous studies showed that the selection process of the parameter in the symplectic Gram-Schmidt method is flawed, as it results in a partially destroyed J-orthogonality of the J-orthogonal matrix. We explore a block type symplectic GramSchmidt method and a new condition for the reorthogonalization to maintain J-orthogonality and to more accurately compute symplectic factorization. Applying the block size scheme to this method, we develop a new procedure for computing symplectic vectors.

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“…For a detailed discussion on the choice of the spectral transformation function q, the choice of the shifts, and convergence properties, see [4], [11]. An algorithm for computing S and R explicitly is presented in [8]. As with explicit QR steps, the expense of explicit SR steps comes from the fact that q(B) has to be computed explicitly.…”

Section: Introductionmentioning

confidence: 99%

The parameterized 𝑆𝑅 algorithm for symplectic (butterfly) matrices

Faßbender

2000

Math. Comp.

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Abstract. The SR algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix B in butterfly form is uniquely determined by 4n − 1 parameters. Using these 4n − 1 parameters, we show how one step of the symplectic SR algorithm for B can be carried out in O(n) arithmetic operations compared to O(n 3 ) arithmetic operations when working on the actual symplectic matrix. Moreover, the symplectic structure, which will be destroyed in the numerical process due to roundoff errors when working with a symplectic (butterfly) matrix, will be forced by working just with the parameters.

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A symplectic QR like algorithm for the solution of the real algebraic Riccati equation

Cited by 108 publications

References 9 publications

Passivity-preserving model reduction via a computationally efficient project-and-balance scheme

Passivity-preserving model reduction via a computationally efficient project-and-balance scheme

Block symplectic Gram–Schmidt method

The parameterized 𝑆𝑅 algorithm for symplectic (butterfly) matrices

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A symplectic QR like algorithm for the solution of the real algebraic Riccati equation

Cited by 108 publications

References 9 publications

Passivity-preserving model reduction via a computationally efficient project-and-balance scheme

Passivity-preserving model reduction via a computationally efficient project-and-balance scheme

Block symplectic Gram&ndash;Schmidt method

The parameterized 𝑆𝑅 algorithm for symplectic (butterfly) matrices

Contact Info

Product

Resources

About

Block symplectic Gram–Schmidt method