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14
Structured factorizations in scalar product spaces
 SIAM J. Matrix Anal. Appl
"... Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. Fo ..."
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Cited by 21 (7 self)
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Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors and singular values that persists across a wide range of scalar products. A key feature of our analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of different characterizations of these scalar product classes is given.
SkewHamiltonian and Hamiltonian eigenvalue problems: Theory, algorithms and applications
 Proceedings of ApplMath03, Brijuni (Croatia
"... SkewHamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computation ..."
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Cited by 12 (4 self)
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SkewHamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computational methods. We will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures. Applications of Hamiltonian and skewHamiltonian eigenproblems are briefly described.
A chart of backward errors for singly and doubly structured eigenvalue problems
 SIAM J. MATRIX ANAL. APPL
, 2003
"... We present a chart of structured backward errors for approximate eigenpairs of singly and doubly structured eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. We identify a number of problems for which the ..."
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Cited by 11 (0 self)
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We present a chart of structured backward errors for approximate eigenpairs of singly and doubly structured eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. We identify a number of problems for which the structured backward error is within a factor √ 2 of the unstructured backward error. This paper collects, unifies, and extends existing work on this subject.
Stability of Structured Hamiltonian Eigensolvers
 SIAM J. Matrix Anal. Appl
, 2001
"... Various applications give rise to eigenvalue problems for which the matrices are Hamiltonian or skewHamiltonian and also symmetric or skewsymmetric. We define structured backward errors that are useful for testing the stability of numerical methods for the solution of these four classes of structu ..."
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Cited by 11 (3 self)
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Various applications give rise to eigenvalue problems for which the matrices are Hamiltonian or skewHamiltonian and also symmetric or skewsymmetric. We define structured backward errors that are useful for testing the stability of numerical methods for the solution of these four classes of structured eigenproblems. We introduce the symplectic quasiQR factorization and show that for three of the classes it enables the structured backward error to be efficiently computed. We also give a detailed rounding error analysis of some recently developed Jacobilike algorithms of Fabender, Mackey and Mackey for these eigenproblems. Based on the direct solution of 4 \Theta 4, and in one case 8 \Theta 8, structured subproblems these algorithms produce a complete basis of symplectic orthogonal eigenvectors for the two symmetric cases and a symplectic orthogonal basis for all the real invariant subspaces for the two skewsymmetric cases. We prove that, when the rotations are implemented using suitable formulae, the algorithms are strongly backward stable and we show that the QR algorithm does not have this desirable property.
A Chart of Backward Errors and Condition Numbers for Singly and Doubly Structured Eigenvalue Problems
 SIAM J. Matrix Anal. Appl
, 2001
"... We present a chart of structured backward errors for approximate eigenpairs of singly and doubly structured eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. We identify a number of problems for which the ..."
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Cited by 6 (4 self)
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We present a chart of structured backward errors for approximate eigenpairs of singly and doubly structured eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. We identify a number of problems for which the structured backward error is within a factor 2 of the unstructured backward error. This paper collects, uni es and extends existing work on this subject.
ON ASYMPTOTIC CONVERGENCE OF NONSYMMETRIC JACOBI ALGORITHMS
"... Abstract. The asymptotic convergence behavior of cyclic versions of the nonsymmetric Jacobi algorithm for the computation of the Schur form of a general complex matrix is investigated. Similar to the symmetric case, the nonsymmetric Jacobi algorithm proceeds by applying a sequence of rotations that ..."
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Cited by 3 (1 self)
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Abstract. The asymptotic convergence behavior of cyclic versions of the nonsymmetric Jacobi algorithm for the computation of the Schur form of a general complex matrix is investigated. Similar to the symmetric case, the nonsymmetric Jacobi algorithm proceeds by applying a sequence of rotations that annihilate a pivot element in the strict lower triangular part of the matrix until convergence to the Schur form of the matrix is achieved. In this paper, it is shown that the cyclic nonsymmetric Jacobi method converges locally and asymptotically quadratically under mild hypotheses if special ordering schemes are chosen, namely ordering schemes that lead to socalled northeast directed sweeps. The theory is illustrated by the help of numerical experiments. In particular, it is shown that there are ordering schemes that lead to asymptotic quadratic convergence for the cyclic symmetric Jacobi method, but only to asymptotic linear convergence for the cyclic nonsymmetric Jacobi method. Finally, a generalization of the nonsymmetric Jacobi method to the computation of the Hamiltonian Schur form for Hamiltonian matrices is introduced and investigated.
On Systems of Linear Quaternion Functions
, 2008
"... A method of reducing general quaternion functions of first degree, i.e., linear quaternion functions, to quaternary canonical form is given. Linear quaternion functions, once reduced to canonical form, can be maintained in this form under functional composition. Furthermore, the composition operatio ..."
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A method of reducing general quaternion functions of first degree, i.e., linear quaternion functions, to quaternary canonical form is given. Linear quaternion functions, once reduced to canonical form, can be maintained in this form under functional composition. Furthermore, the composition operation is symbolically identical to quaternion multiplication, making manipulation and reduction of systems of linear quaternion functions straight forward. 1
Symplectic structure of Jacobi systems on time scales
 DEPARTMENT OF MATHEMATICS AND STATISTICS, FACULTY OF SCIENCE, MASARYK UNIVERSITY, KOTLÁˇRSKÁ 2, CZ61137 BRNO, CZECH REPUBLIC EMAIL ADDRESSES: HILSCHER@MATH.MUNI.CZ, ZEMANEKP@MATH.MUNI.CZ EJQTDE, PROC. 9TH COLL. QTDE
, 2010
"... In this paper we study the structure of the Jacobi system for optimal control problems on time scales. Under natural and minimal invertibility assumptions on the coefficients we prove that the Jacobi system is a time scale symplectic system and not necessarily a Hamiltonian system. These new inverti ..."
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In this paper we study the structure of the Jacobi system for optimal control problems on time scales. Under natural and minimal invertibility assumptions on the coefficients we prove that the Jacobi system is a time scale symplectic system and not necessarily a Hamiltonian system. These new invertibility conditions are weaker than those considered in the current literature. This shows that the theory of time scale symplectic systems, rather than the theory of linear Hamiltonian systems, is fundamental for optimal control problems. Our results in this paper are new even for the Jacobi equations arising in the time scale calculus of variation and, in particular, for the discrete time calculus of variations and optimal control problems. We also show that nonlinear time scale Hamiltonian systems possess symplectic structure, that is, the Jacobian of the evolution mapping satisfies a time scale symplectic system.
THE PARAMETRIZED ALGORITHM FOR HAMILTONIAN MATRICES
"... Abstract. The heart of the implicitly restarted symplectic Lanczos method for Hamiltonian matrices consists of the algorithm, a structurepreserving algorithm for computing the spectrum of Hamiltonian matrices. The symplectic Lanczos method projects the large, sparse Hamiltonian matrix onto a small, ..."
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Abstract. The heart of the implicitly restarted symplectic Lanczos method for Hamiltonian matrices consists of the algorithm, a structurepreserving algorithm for computing the spectrum of Hamiltonian matrices. The symplectic Lanczos method projects the large, sparse Hamiltonian matrix onto a small, dense HamiltonianHessenberg matrix,. This Hamiltonian matrix is uniquely determined by parameters. Using these parameters, one step of the algorithm can be carried out in arithmetic operations (compared to#$&%'!arithmetic operations when working on the actual Hamiltonian matrix). As in the context of the implicitly restarted symplectic Lanczos method the usual assumption, that the Hamiltonian eigenproblem to be solved is stable, does not hold, the case of purely imaginary eigenvalues in the algorithm is treated here. Key words. Hamiltonian matrix, eigenvalue problem, algorithm AMS subject classifications. 65F15 1. Introduction. Renewed interest [36, 49, 10] in the implicitly restarted symplectic Lanczos method for computing a few eigenvalues of a large, sparse Hamiltonian matrix [9] led us to reconsider that algorithm. It sparse(*),+ projects the large, matrix