Invariant Subspace Methods for the Numerical Solution of Riccati Equations

Laub, Alan J.

doi:10.1007/978-3-642-58223-3_7

Cited by 96 publications

(75 citation statements)

References 159 publications

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“…A special case of this decomposition is an appropriately ordered Jordan decomposition of M as was used by Vaughan (1970) in developing an invariant subspace algorithm for computing P y . Laub (1991) traces this solution strategy back to the 19th century and credits MacFarlane (1963) and Potter (1966) with introducing it to the control literature. As emphasized by Laub (1991), it is preferable to build algorithms based on other upper triangular decompositions that are more stable numerically.…”

Section: Nonsingular a Yymentioning

confidence: 99%

“…Laub (1991) traces this solution strategy back to the 19th century and credits MacFarlane (1963) and Potter (1966) with introducing it to the control literature. As emphasized by Laub (1991), it is preferable to build algorithms based on other upper triangular decompositions that are more stable numerically. The Jordan decomposition is particularly problematic when the symplectic matrix M has eigenvalues with multiplicities greater than one (see also Golub and Wilkinson 1976).…”

Section: Nonsingular a Yymentioning

confidence: 99%

“…2 Our survey of these methods draws heavily on Anderson (1978), Gardiner and Laub (1986), Golub, Nash and Van Loan (1979), Laub (1979Laub ( ,1991, and Pappas, Laub and Sandell (1980). recent algorithms for solving these sub-problems.…”

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confidence: 99%

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Modeling the search for tolerant-to-uncertainty monetary and fiscal policies under their interaction

Serkov¹

2017

EATP

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Section: Nonsingular a Yymentioning

confidence: 99%

Section: Nonsingular a Yymentioning

confidence: 99%

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Modeling the search for tolerant-to-uncertainty monetary and fiscal policies under their interaction

Serkov¹

2017

EATP

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“…Concerning the numerical stability of Algorithm 5 we notice that various suggestions have been made in the literature to calculate solutions of Riccati equations in a numerically reliable way (see e.g. Laub (1979Laub ( , 1991, Paige and van Loan (1981), van Dooren (1981), Mehrmann (1991) and Abou-Kandil and Bertrand (1986) for a more general survey on various types of Riccati equations). These methods can also be used to improve the numerical stability of Algorithm 5.…”

Section: Algorithmmentioning

confidence: 99%

A Numerical Toolbox to Solve N-Player Affine LQ Open-Loop Differential Games

2011

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We present an algorithm and a corresponding MATLAB numerical toolbox to solve any form of infinite-planning horizon affine linear quadratic open-loop differential games. By rewriting a specific application into the standard framework one can use the toolbox to calculate and verify the existence of both the open-loop noncooperative Nash equilibrium (equilibria) and cooperative Pareto equilibrium (equilibria). In case there is more than one equilibrium for the non-cooperative case, the toolbox determines all solutions that can be implemented as a feedback strategy. Alternatively, the toolbox can apply a number of choice methods in order to discriminate between multiple equilibria. The user can predefine a set of coalition structures for which they would like to calculate the non-cooperative Nash solution(s). It is also possible to specify the relative importance of each player in any coalition structure. Furthermore, the toolbox offers plotting facilities as well as other options to analyse the outcome of the game. For instance, it is possible to disaggregate each player's total loss into its contributing elements. The toolbox is available as a freeware from the authors of this paper.

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“…The computation of eigenvalues and eigenvectors of such matrices is an important task in applications like the discrete linearquadratic regulator problem, discrete Kalman filtering, the solution of discrete-time algebraic Riccati equations and certain large, sparse quadratic eigenvalue problems. See, e.g., [82,84,94,96] for applications and further references. Symplectic matrices also occur when solving linear Hamiltonian difference systems [14].…”

Section: Symplectic Matricesmentioning

confidence: 99%

Structured Eigenvalue Problems

Faßbender

Kreßner

2006

GAMM-Mitteilungen

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Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew‐symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Invariant Subspace Methods for the Numerical Solution of Riccati Equations

Cited by 96 publications

References 159 publications

Modeling the search for tolerant-to-uncertainty monetary and fiscal policies under their interaction

Modeling the search for tolerant-to-uncertainty monetary and fiscal policies under their interaction

A Numerical Toolbox to Solve N-Player Affine LQ Open-Loop Differential Games

Structured Eigenvalue Problems

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