Results 1  10
of
35
Structured polynomial eigenvalue problems: Good vibrations from good linearizations
 SIAM J. Matrix Anal. Appl
"... Abstract. Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomial (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of lineariz ..."
Abstract

Cited by 38 (14 self)
 Add to MetaCart
Abstract. Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomial (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of linearizations that reflect the structure of these polynomials, and therefore preserve symmetries in their spectra, is introduced and investigated. We analyze the existence and uniqueness of such linearizations, and show how they may be systematically constructed.
Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods
, 2004
"... We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems. We present the currently available solution methods such as the JacobiDavidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new li ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems. We present the currently available solution methods such as the JacobiDavidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new linearization technique and demonstrate how it can be used to improve structure preservation and with this the accuracy and efficiency of linearization based methods. We present several recent applications where structured and unstructured nonlinear eigenvalue problems arise and some numerical results.
Palindromic polynomial eigenvalue problems: Good vibrations from good linearizations
 DFG Research Center Matheon, Mathematics for
, 2005
"... Abstract. Palindromic polynomial eigenvalue problems and related classes of structured eigenvalue problems are considered. These structures generalize the concepts of symplectic and Hamiltonian matrices to matrix polynomials. We discuss several applications where these matrix polynomials arise, and ..."
Abstract

Cited by 22 (8 self)
 Add to MetaCart
Abstract. Palindromic polynomial eigenvalue problems and related classes of structured eigenvalue problems are considered. These structures generalize the concepts of symplectic and Hamiltonian matrices to matrix polynomials. We discuss several applications where these matrix polynomials arise, and show how linearizations can be derived that reflect the structure of all these structured matrix polynomials and therefore preserve symmetries in the spectrum.
Product eigenvalue problems
 SIAM Review
, 2005
"... Abstract. Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors: A = AkAk−1 ···A1. Usually more accurate results are obtained by working with ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
Abstract. Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors: A = AkAk−1 ···A1. Usually more accurate results are obtained by working with the factors rather than forming A explicitly. For example, if we want eigenvalues/vectors of B T B, it is better to work directly with B and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic GR algorithm applied to a related cyclic matrix.
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 ..."
Abstract

Cited by 13 (6 self)
 Add to MetaCart
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.
The sensitivity of computational control problems
 IEEE Control Syst. Mag
, 2004
"... What factors contribute to the accurate and efficient numerical solution of problems in control systems analysis and design? Although numerical methods have been used for many centuries to solve problems in science and engineering, the importance of computation grew tremendously with the advent of d ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
What factors contribute to the accurate and efficient numerical solution of problems in control systems analysis and design? Although numerical methods have been used for many centuries to solve problems in science and engineering, the importance of computation grew tremendously with the advent of digital computers. It became immediately clear that many of the classical analytical and numerical methods and algorithms could not be implemented directly as computer codes, although they were well suited for hand computations. What was the reason? When doing computations by hand a person can choose the accuracy of each elementary calculation and then estimate, based on intuition and experience, its influence on the final result. In contrast, when computations are done automatically, intuitive error control is usually not possible and the effect of errors on the intermediate calculations must be estimated in a more systematic way. Due to this observation, starting
Numerical solution of a quadratic eigenvalue problem
 Linear Algebra Appl
"... Dedicated to Peter Lancaster on the occasion of his 75th birthday We consider the quadratic eigenvalue problem (QEP) (λ 2 M + λG + K)x = 0, where M = M T is positive definite, K = K T is negative definite, and G = −G T. The eigenvalues of the QEP occur in quadruplets (λ, λ, −λ, −λ) or in real or pur ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
Dedicated to Peter Lancaster on the occasion of his 75th birthday We consider the quadratic eigenvalue problem (QEP) (λ 2 M + λG + K)x = 0, where M = M T is positive definite, K = K T is negative definite, and G = −G T. The eigenvalues of the QEP occur in quadruplets (λ, λ, −λ, −λ) or in real or purely imaginary pairs (λ, −λ). We show that all eigenvalues of the QEP can be found efficiently and with the correct symmetry, by finding a proper solvent X of the matrix equation MX 2 + GX + K = 0, as long as the QEP has no eigenvalues on the imaginary axis. This solvent approach works well also for some cases where the QEP has eigenvalues on the imaginary axis. AMS classification: 15A18; 15A24; 65F30
Robust numerical methods for robust control
 INSTITUT FÜR MATHEMATIK, TU BERLIN, STR. DES 17. JUNI 136, D10623
, 2004
"... We present numerical methods for the solution of the optimal H∞ control problem. In particular, we investigate the iterative part often called the γiteration. We derive a method with better robustness in the presence of rounding errors than other existing methods. It remains robust in the presence ..."
Abstract

Cited by 9 (7 self)
 Add to MetaCart
We present numerical methods for the solution of the optimal H∞ control problem. In particular, we investigate the iterative part often called the γiteration. We derive a method with better robustness in the presence of rounding errors than other existing methods. It remains robust in the presence of rounding errors even as γ approaches its optimal value. For the computation of a suboptimal controller, we avoid solving algebraic Riccati equations with their problematic matrix inverses and matrix products by adapting recently suggested methods for the computation of deflating subspaces of skewHamiltonian/Hamiltonian pencils. These methods are applicable even if the pencil has eigenvalues on the imaginary axis. We compare the new method with older methods and present several examples.
Numerical Solution of Large Scale Structured Polynomial or Rational Eigenvalue Problems
 In Foundations of Computational Mathematics
, 2003
"... This paper deals with the numerical solution of large scale polynomial or rational eigenvalue problems with Hamiltonian or symplectic symmetry in the spectrum. Applications where such problems arise are introduced briey. It is shown how these problems may be formulated as linear generalized eige ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
This paper deals with the numerical solution of large scale polynomial or rational eigenvalue problems with Hamiltonian or symplectic symmetry in the spectrum. Applications where such problems arise are introduced briey. It is shown how these problems may be formulated as linear generalized eigenvalue problems that have either symmetric/skew symmetric, skew Hamiltonian/Hamiltonian or symplectic pencils. The presented numerical methods are designed to preserve these structures.
Block Algorithms for Orthogonal Symplectic Factorizations
 BIT
, 2002
"... On the basis of a new WYlike representation block algorithms for orthogonal symplectic matrix factorizations are presented. Special emphasis is placed on symplectic QR and URV factorizations. The block variants mainly use level 3 (matrixmatrix) operations that permit data reuse in the higher level ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
On the basis of a new WYlike representation block algorithms for orthogonal symplectic matrix factorizations are presented. Special emphasis is placed on symplectic QR and URV factorizations. The block variants mainly use level 3 (matrixmatrix) operations that permit data reuse in the higher levels of a memory hierarchy. Timing results show that our new algorithms outperform standard algorithms by a factor 34 for sufficiently large problems.