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A PrimalDual Potential Reduction Method for Problems Involving Matrix Inequalities
 in Protocol Testing and Its Complexity&quot;, Information Processing Letters Vol.40
, 1995
"... We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worstcase analysis shows that the number of iterat ..."
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Cited by 102 (20 self)
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We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worstcase analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interiorpoint methods the overall computational effort is therefore dominated by the leastsquares system that must be solved in each iteration. A type of conjugategradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugategradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration.
SLICOT  A Subroutine Library in Systems and Control Theory
 Applied and Computational Control, Signals, and Circuits
, 1997
"... This article describes the subroutine library SLICOT that provides Fortran 77 implementations of numerical algorithms for computations in systems and control theory. Around a nucleus of basic numerical linear algebra subroutines, this library builds methods for the design and analysis of linear cont ..."
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Cited by 84 (56 self)
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This article describes the subroutine library SLICOT that provides Fortran 77 implementations of numerical algorithms for computations in systems and control theory. Around a nucleus of basic numerical linear algebra subroutines, this library builds methods for the design and analysis of linear control systems. A brief history of the library is given together with a description of the current version of the library and the ongoing activities to complete and improve the library in several aspects. 1 Introduction Systems and control theory are disciplines widely used to describe, control, and optimize industrial and economical processes. There is now a huge amount of theoretical results available which has lead to a variety of methods and algorithms used throughout industry and academia. Although based on theoretical results, these methods often fail when applied to reallife problems, which often tend to be illposed or of high dimensions. This failing is frequently due to the lack of...
An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems
, 1997
"... We discuss an inversefree, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB. This algorithm is based on earlier ones of Bulgakov, Godunov ..."
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Cited by 70 (11 self)
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We discuss an inversefree, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB. This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm.
Guaranteed Passive Balancing Transformations for Model Order Reduction
, 2002
"... The major concerns in stateoftheart model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteedpassive models ..."
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Cited by 65 (8 self)
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The major concerns in stateoftheart model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteedpassive models, for systems with special internal structure, using numerically stable and efficient Krylovsubspace iterations. Truncated Balanced Realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper we show how to construct TBRlike methods that guarantee passive reduced models and in addition are applicable to statespace systems with arbitrary internal structure.
A new method for computing the stable invariant subspace of a real Hamiltonian matrix
, 1997
"... A new backward stable, structure preserving method of complexity O(n 3 ) is presented for computing the stable invariant subspace of a real Hamiltonian matrix and the stabilizing solution of the continuoustime algebraic Riccati equation. The new method is based on the relationship between the inv ..."
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Cited by 57 (34 self)
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A new backward stable, structure preserving method of complexity O(n 3 ) is presented for computing the stable invariant subspace of a real Hamiltonian matrix and the stabilizing solution of the continuoustime algebraic Riccati equation. The new method is based on the relationship between the invariant subspaces of the Hamiltonian matrix H and the extended matrix 0 H H 0 and makes use of the symplectic URVlike decomposition that was recently introduced by the authors. Keywords. Eigenvalue problem, Hamiltonian matrix, algebraic Riccati equation, sign function, invariant subspace. AMS subject classification. 65F15, 93B40, 93B36, 93C60. 1 Introduction It is a well accepted fact in numerical analysis that a numerical algorithm should reflect as many of the structural properties of the physical problem or the resulting mathematical model. For the solution of eigenvalue problems this means that use of the symmetry structures of the matrix or the spectrum is made. While for symme...
A Collection of Benchmark Examples for the Numerical Solution of Algebraic Riccati Equations II: DiscreteTime Case
 FAK. F. MATHEMATIK, TU CHEMNITZZWICKAU
, 1995
"... This is the second part of a collection of benchmark examples for the numerical solution of algebraic Riccati equations. After presenting examples for the continuoustime case in Part I, our concern in this paper is discretetime algebraic Riccati equations. This collection may serve for testing pur ..."
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Cited by 56 (30 self)
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This is the second part of a collection of benchmark examples for the numerical solution of algebraic Riccati equations. After presenting examples for the continuoustime case in Part I, our concern in this paper is discretetime algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.
Solving Algebraic Riccati Equations on Parallel Computers Using Newton's Method with Exact Line Search
, 1999
"... We investigate the numerical solution of continuoustime algebraic Riccati equations via Newton's method on serial and parallel computers with distributed memory. We apply and extend the available theory for Newton's method endowed with exact line search to accelerate convergence. We also ..."
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Cited by 53 (9 self)
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We investigate the numerical solution of continuoustime algebraic Riccati equations via Newton's method on serial and parallel computers with distributed memory. We apply and extend the available theory for Newton's method endowed with exact line search to accelerate convergence. We also discuss a new stopping criterion based on recent observations regarding condition and error estimates. In each iteration step of Newton's method a stable Lyapunov equation has too be solved. We propose to solve these Lyapunov equations using iterative schemes for computing the matrix sign function. This approach can be efficiently implemented on parallel computers using ScaLAPACK. Numerical experiments on an ibm sp2 multicomputer report the accuracy, scalability, and speedup of the implemented algorithms.
A Multishift Algorithm For The Numerical Solution Of Algebraic Riccati Equations
 Electr. Trans. Num. Anal
, 1993
"... We study an algorithm for the numerical solution of algebraic matrix Riccati equations that arise in linear optimal control problems. The algorithm can be considered to be a multishift technique, which uses only orthogonal symplectic similarity transformations to compute a Lagrangian invariant subsp ..."
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Cited by 42 (27 self)
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We study an algorithm for the numerical solution of algebraic matrix Riccati equations that arise in linear optimal control problems. The algorithm can be considered to be a multishift technique, which uses only orthogonal symplectic similarity transformations to compute a Lagrangian invariant subspace of the associated Hamiltonian matrix. We describe the details of this method and compare it with other numerical methods for the solution of the algebraic Riccati equation.
Kalman filtering and Riccati equations for descriptor systems
 IEEE Trans. Automat. Contr
, 1992
"... AbstractIn this paper, we consider a general formulation of a discretetime filtering problem for descriptor systems. It is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems. Using a “dual approach ” to estimation we derive a socalled “3 ..."
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Cited by 41 (14 self)
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AbstractIn this paper, we consider a general formulation of a discretetime filtering problem for descriptor systems. It is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems. Using a “dual approach ” to estimation we derive a socalled “3block ” form for the optimal filter and a corresponding 3block Riccati equation for a general class of timevarying descriptor models which need not represent a wellposed system in that the dynamics may be either over or under constrained. Specializing to the timeinvariant case we examine the asymptotic properties of the 3block filter, and in particular analyze in detail the resulting 3block algebraic Riccati equation, generalizing significantly the results in 1231, 1281, 1331. Finally, the noncausal nature of discretetime descriptor dynamics implies that future dynamics may provide some information about the present state. We present a modified
An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem
 Linear Algebra Appl
, 1997
"... An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. The inherent numerical difficulties of the symplectic Lanczos method are addressed by inexpensive implicit restarts. The method is us ..."
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Cited by 40 (13 self)
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An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. The inherent numerical difficulties of the symplectic Lanczos method are addressed by inexpensive implicit restarts. The method is used to compute eigenvalues, eigenvectors, and invariant subspaces of large and sparse Hamiltonian matrices and lowrank approximations to the solution of continuoustime algebraic Riccati equations with large and sparse coefficient matrices.