Spectral division methods for block generalized Schur decompositions

Sun, Xiaobai; Quintana–Ort́ı, Enrique S.

doi:10.1090/s0025-5718-04-01667-9

Cited by 33 publications

(30 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This can be done by the QZ algorithm with subsequent eigenvalue reordering [19], the GUPTRI algorithm [16,17], or the disk function method [4,47]. By setting B :…”

Section: Choice Of the System Normmentioning

confidence: 99%

Spectral characterization and enforcement of negative imaginariness for descriptor systems

Benner

Voigt

2013

Linear Algebra and its Applications

View full text Add to dashboard Cite

Submitted by Peter SemrlAMS classification: 93B99 65L80 15A22 15A23 Keywords: Descriptor system Even matrix pencil Kronecker canonical form Negative imaginariness Skew-Hamiltonian/Hamiltonian matrix pencil Structure enforcementSystems with counterclockwise input-output dynamics (or negative imaginary transfer functions) arise in various applications such as the modeling of flexible mechanical structures or electrical circuits when certain kinds of measurements are taken. In this paper we introduce descriptor systems with such an additional structure. We state various of their properties and prove algebraic characterizations of negative imaginariness in terms of spectral conditions of certain structured matrix pencils. For this purpose we also analyze particular boundary cases which are characterized by properties of a structured Kronecker canonical form. Finally, we describe a method which can be used to restore the negative imaginary property in case that it is lost. This happens, e.g., when a system with theoretically negative imaginary transfer function is obtained by, e.g., model order reduction methods, linearization, or other approximations. The method is illustrated by numerical examples.

show abstract

“…This can be done by the QZ algorithm with subsequent eigenvalue reordering [19], the GUPTRI algorithm [16,17], or the disk function method [4,47]. By setting B :…”

Section: Choice Of the System Normmentioning

confidence: 99%

Spectral characterization and enforcement of negative imaginariness for descriptor systems

Benner

Voigt

2013

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…The (1, 2) block of (46) now follows from (47). The (2, 2) block of (46) follows from (50). Equation (49) now simplifies to W 2 = I .…”

Section: Theorem 36 If E J and A J Obeymentioning

confidence: 99%

“…(Example 3 from [50] is slightly different.) Let Q be a random orthogonal matrix generated as described in [45].…”

Section: Example 43 This Is Example 42 Frommentioning

confidence: 99%

An Arithmetic for Matrix Pencils: Theory and New Algorithms

Benner

Byers

2006

Numer. Math.

View full text Add to dashboard Cite

This paper introduces arithmetic-like operations on matrix pencils. The pencil-arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case. These operations give an unusual perspective on a variety of pencil related computations. We derive generalizations of monodromy matrices and the matrix exponential. A new algorithm for computing a pencil-arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces.

show abstract

“…Owing to the ultimate quadratic convergence of the iteration this ensures that, in most cases, the best attainable accuracy is obtained while avoiding convergence stagnation problems. The pair at convergence (A ∞ , E ∞ ) := lim j →∞ (A j , E j ) can be used for spectral division as described in [16]. There, a subspace extraction technique is proposed that provides both Q and Z in Equation (3) from a single inverse-free iteration, thus saving half of the computational cost.…”

Section: Spectral Division Using the Matrix Disk Functionmentioning

confidence: 99%

Stabilizing large‐scale generalized systems on parallel computers using multithreading and message‐passing

Benner¹,

Dolz²,

Mayo³

et al. 2006

Concurrency and Computation

View full text Add to dashboard Cite

SUMMARYWe discuss the parallelization of an efficient algorithm for the partial stabilization of large-scale linear control systems in generalized state-space form. The algorithm is composed of highly parallel iterative schemes that appear in the computation of certain matrix functions. Here we evaluate different approaches to exploit parallelism at two levels, based on threads and processes. Our experimental results on a cluster of symmetric multiprocessors and a CC-NUMA platform show that the efficiency of the matrix operations underlying the iterative schemes carry over to the parallel implementation of the stabilization algorithm.

show abstract

Spectral division methods for block generalized Schur decompositions

Cited by 33 publications

References 26 publications

Spectral characterization and enforcement of negative imaginariness for descriptor systems

Spectral characterization and enforcement of negative imaginariness for descriptor systems

An Arithmetic for Matrix Pencils: Theory and New Algorithms

Stabilizing large‐scale generalized systems on parallel computers using multithreading and message‐passing

Contact Info

Product

Resources

About