## Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils (2002)

Venue: | SIAM J. Matrix Anal. Appl |

Citations: | 41 - 25 self |

### BibTeX

@ARTICLE{Benner02numericalcomputation,

author = {Peter Benner and Ralph Byers and Volker Mehrmann and Hongguo Xu},

title = {Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils},

journal = {SIAM J. Matrix Anal. Appl},

year = {2002},

volume = {24},

pages = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems, H1 optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deating subspaces of matrices and matrix pencils with Hamiltonian and/or skew-Hamiltonian structure. We extend the recently developed methods for Hamiltonian matrices to the general case of skew-Hamiltonian/Hamiltonian pencils. The algorithms circumvent problems with skew-Hamiltonian/Hamiltonian matrix pencils that lack structured Schur forms by embedding them into matrix pencils that always admit a structured Schur form. The rounding error analysis of the resulting algorithms is favorable. For the embedded matrix pencils, the algorithms use structure preserving unitary matrix computations and are strongly backwards stable, i.e., they compute the exact structured Schur form of a nearby matrix pencil with the same structure. Keywords. eigenvalue problem, deating subspace, Hamiltonian matrix, skew-Hamiltonian matrix, skew-Hamiltonian/Hamiltonian matrix pencil. AMS subject classication. 49N10, 65F15, 93B40, 93B36. 1.

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Citation Context ... fact that even when a Hamiltonian Schur form exists, there is no completely satisfactory structure preserving, numerical method to compute it. It has been argued in [2] that, except in special cases =-=[13, 14]-=-, QR-like algorithms are impractically expensive because of the lack of a Hamiltonian Hessenberg-like form. For this reason other methods like the multishift-method of [1], the structured implicit pro... |

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Citation Context ...rical simulation of elastic deformation [28, 34], and linear response theory [30]. Linearquadratic optimal control and H# optimization problems are related to skew-Hamiltonian /Hamiltonian pencils in =-=[4, 5]-=-. It is important to exploit and preserve algebraic structures (like symmetries in the matrix blocks or symmetries in the spectrum) as much as possible. Such algebraic structures typically arise from ... |

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Citation Context ... in the Hamiltonian Schur form described in [31]. (Similar di#culties arise in the skew-Hamiltonian/Hamiltonian pencil case for the Schur-like forms of skew-Hamiltonian /Hamiltonian matrix pencils in =-=[25, 26]-=- and for the other structures given in Definition 1.1 in [24].) A second problem comes from the fact that even when a Hamiltonian Schur form exists, there is no completely satisfactory structure-prese... |

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Citation Context ..., so L = L(U H (iJ T )U)L, where U = (1= p 2) h In iI n In iI n i is the unitary matrix of eigenvectors of iJ T . Hence, (2.2) holds with Z = ULW . The following immediate corollary also follows from =-=[15-=-]. Corollary 2.3. Every real skew-Hamiltonian matrix S is J -semidenite. Proof. If S is real, then JS is real and skew-symmetric. The eigenvalues of J S appear in complex conjugate pairs with zero rea... |

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Citation Context ...tes the eigenvalues of a complex skew-Hamiltonian /Hamiltonian matrix pencil #S - #H using an unusual embedding of C into SKEW-HAMILTONIAN/HAMILTONIAN PENCILS 173 R 2 , which was recently proposed in =-=[8]-=-. Let #S - #H be a complex skew-Hamiltonian /Hamiltonian matrix pencil with J -semidefinite skew-Hamiltonian part S = JZ H J T Z. Split the skew-Hamiltonian matrix N = iH # SH 2n as iH = N = N 1 + iN ... |

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Citation Context ...ugh the numerical computation of n-dimensional Lagrangian invariant subspaces of Hamiltonian matrices and the related problem of solving algebraic Riccati equations have been extensively studied (see =-=[12, 22, 27, 35]-=- and the references therein), completely satisfactory methods for general Hamiltonian matrices and matrix pencils are still an open problem. Completely satisfactory methods would be numerically backwa... |

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Citation Context ...of the form (2.2). A J -denite skew-Hamiltonian matrix is a skew-Hamiltonian matrix that is both J -semidenite and non-singular. The property of J -semideniteness arises frequently in applications [3,=-= 4, 5-=-]. We show below that all real skew-Hamiltonian matrices are J -semidenite. We also show that if a skew-Hamiltonian /Hamiltonian matrix pencil has a skew-Hamiltonian/Hamiltonian Schur form, then the s... |

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Citation Context ...ay neither the e#ects of perturbing each factor separately nor the e#ects of structured perturbations. Therefore, we make use of the perturbation analysis for formal products of matrices developed in =-=[9]-=-. If Algorithm 2 is applied to the skew-Hamiltonian/Hamiltonian matrix pencil #S - #H, then we compute the structured Schur form of the extended skew-Hamiltonian /Hamiltonian matrix pencil #B c S -#B ... |

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Citation Context ...ot in the Hamiltonian Schur form described in [31]. (Similar diculties arise in the skew-Hamiltonian/Hamiltonian pencil case for the Schur-like forms of skew-Hamiltonian/Hamiltonian matrix pencils in =-=[25, 26-=-] and for the other structures given in Denition 1.1 in [24].) A second problem comes from the fact that even when a Hamiltonian Schur form exists, there is no completely satisfactory structure preser... |

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9 |
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Citation Context ... optimization [18, 39]. Moreover, instances of skew-Hamiltonian/Hamiltonian pencils appear in several other areas of applied mathematics, computational physics and chemistry, e.g., gyroscopic systems =-=[20], nu-=-merical simulation of elastic deformation [28, 34], and linear response Working title was \Numerical Computation of De ating Subspaces of Embedded Hamiltonian and Symplectic Pencils". All author... |

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Citation Context ...ty arises when the Hamiltonian matrix or the skew-Hamiltonian/Hamiltonian matrix pencil has eigenvalues on the imaginary axis. In that case, the desired Lagrangian subspace is, in general, not unique =-=[29-=-]. Furthermore, ifsnite precision arithmetic or other errors perturb the matrix o the Lie algebra of Hamiltonian matrices, then it is typically the case that the perturbed matrix has no Lagrangian sub... |

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Citation Context ...ptimization [18, 39]. Moreover, instances of skew-Hamiltonian /Hamiltonian pencils appear in several other areas of applied mathematics, computational physics, and chemistry, e.g., gyroscopic systems =-=[20]-=-, numerical simulation of elastic deformation [28, 34], and linear response theory [30]. Linearquadratic optimal control and H# optimization problems are related to skew-Hamiltonian /Hamiltonian penci... |

5 |
Numerical computation of de subspaces of embedded Hamiltonian pencils
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(Show Context)
Citation Context ... was completed while this author was with the TU Chemnitz, Germany. 1 theory [30]. Linear-quadratic optimal control and H1 optimization problems are related to skewHamiltonian /Hamiltonian pencils in =-=[4, 5]-=-. It is important to exploit and preserve algebraic structures (like symmetries in the matrix blocks or symmetries in the spectrum) as much as possible. Such algebraic structures typically arise from ... |

2 |
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(Show Context)
Citation Context ...Hamiltonian /Hamiltonian pencils appear in several other areas of applied mathematics, computational physics, and chemistry, e.g., gyroscopic systems [20], numerical simulation of elastic deformation =-=[28, 34]-=-, and linear response theory [30]. Linearquadratic optimal control and H# optimization problems are related to skew-Hamiltonian /Hamiltonian pencils in [4, 5]. It is important to exploit and preserve ... |

1 |
Asymptotic expansions of elastic in domains with boundary and structural singularities, in Boundary element topics
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Citation Context ...w-Hamiltonian/Hamiltonian pencils appear in several other areas of applied mathematics, computational physics and chemistry, e.g., gyroscopic systems [20], numerical simulation of elastic deformation =-=[28, 34], an-=-d linear response Working title was \Numerical Computation of De ating Subspaces of Embedded Hamiltonian and Symplectic Pencils". All authors were partially supported by Deutsche Forschungsgemei... |