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Let A be a unital complex semisimple Banach algebra, and M A denote its maximal ideal space. For a matrix M ∈ A n × n , M ˆ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M ∈ C n × n , σ (M) ⊂ C denotes the set of eigenvalues of M. It is shown that if A ∈ A n × n and B ∈ A m × m are such that for all φ ∈ M A , σ (A ˆ (φ)) ∩ σ (B ˆ (φ)) = ∅ , then for all C ∈ A n × m , the Sylvester equation A X − X B = C has a unique solution X ∈ A n × m. As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra. [ABSTRACT FROM AUTHOR]

We provide an amended version of Corollaries 7 and 9 in [De Terán, Iannazzo, Poloni, Robol, “Solvability and uniqueness criteria for generalized Sylvester-type equations”]. These results characterize the unique solvability of the matrix equation A X B + C X ⋆ D = E (where the coefficients need not be square) in terms of an equivalent condition on the spectrum of certain matrix pencils of the same size as one of its coefficients. [ABSTRACT FROM AUTHOR]

The matrix Sylvester equation for congruence, or T-Sylvester equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning TSylvester equations is rather well understood, and there are stable and efficient numerical algorithms which solve these equations for small- to medium-sized matrices. However, developing numerical algorithms for solving large-scale TSylvester equations still remains an open problem. In this paper, we present several projection algorithms based on different Krylov spaces for solving this problem when the right-hand side of the T-Sylvester equation is a low-rank matrix. The new algorithms have been extensively tested, and the reported numerical results show that they work very well in practice, offering clear guidance on which algorithm is the most convenient in each situation. [ABSTRACT FROM AUTHOR]

We consider the evaluation of the sensitivity or condition number of (generalized) eigenvalue problems for a large and sparse real matrix (or matrix pair) in R n × n , through some (coupled) Sylvester equation using Newton’s method. The technique of the statistical condition estimation has been adapted to the sensitivity of symmetric matrices as well as general matrices with special structures under some assumptions on various types of perturbations. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions for the existence of a unique solution of the generalized ⋆-Sylvester matrix equation A X B + C X ⋆ D = E , where A , B , C , D , E are square matrices of the same size with real or complex entries, and where ⋆ stands for either the transpose or the conjugate transpose. This generalizes several previous uniqueness results for specific equations like the ⋆-Sylvester or the ⋆-Stein equations. [ABSTRACT FROM AUTHOR]

In various applications, for instance, in the detection of a Hopf bifurcation or in solving separable boundary value problems using the two-parameter eigenvalue problem, one has to solve a generalized eigenvalue problem with 2 × 2 operator determinants of the form (B1 ⊗ A2 - A1 ⊗ B2)z = µ B1 ⊗ C2 - C1 ⊗ B2)z. We present efficient methods that can be used to compute a small subset of the eigenvalues. For full matrices of moderate size, we propose either the standard implicitly restarted Arnoldi or Krylov-Schur iteration with shift-and-invert transformation, performed efficiently by solving a Sylvester equation. For large problems, it is more efficient to use subspace iteration based on low-rank approximations of the solution of the Sylvester equation combined with a Krylov-Schur method for the projected problems. [ABSTRACT FROM AUTHOR]

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems. [ABSTRACT FROM AUTHOR]

This work concerns the distance in the 2-norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Initially, we consider perturbations of the constant coefficient matrix only. A singular value optimization characterization is derived for the associated distance. We also consider the distance in the general setting, when all of the coefficient matrices are perturbed. In this general setting, we obtain a lower bound in terms of another singular value optimization problem. The singular value optimization problems derived facilitate the numerical computation of the distances. [ABSTRACT FROM AUTHOR]

In stabilization studies of linear parabolic control systems, a successful approach is a scheme employing dynamic compensators in the feedback loop. An essential reason is the fact that both sensors and actuators cannot be designed freely, especially in the case of boundary observation/boundary feedback. Most fundamental in this scheme is a simple stabilization result under the static feedback control scheme. In this scheme, little attention has been paid to how to assign new eigenvalues of the feedback system. In this article, we show a new feature of pole assignment that shows some choices of new eigenvalues cause a deterioration of the stability property. An algebraic growth rate is added to the feedback system in such a choice. [ABSTRACT FROM AUTHOR]

A new substructure preservation Sylvester-based model order reduction technique with application to power systems is presented in this article. The new approach is intended for multiple-input–multiple-output linear time invariant systems, given in the form of state-space realization with the objective of obtaining a proper reduced-order model (complexity reduction), preserving the dominant eigenvalues of the full-order model as a subset in the reduced model, and maintaining a minimum steady-state error. The proposed reduction method is performed based on transforming the system state matrix into a special form, taking into account the dominant eigenvalues, while the rest of the model transformation is derived utilizing the Sylvester equation formula. Once the system is transformed, the reduced-order model is obtained by truncating the less dominant eigenvalues using the singular perturbation technique. To evaluate the potential of the new approach, results of the proposed technique are compared to some of the well-known methods for model order reduction and relatively recently published work. Results comparison shows the superiority of the new method especially in terms of time convergence. [ABSTRACT FROM PUBLISHER]

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in Boutry et al. (2005, SIAM J. Matrix Anal. Appl., 27, 582–601) regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of Broyden-Fletcher-Goldfarb-Shanno and Lipschitz-based global optimization algorithms. [ABSTRACT FROM AUTHOR]

Abstract: This work is focused on the computation of the invariant subspace associated with a separated group of eigenvalues near a specified shift of a large sparse matrix. First, we consider the inverse subspace iteration with the preconditioned GMRES method. It guarantees a convergence to the desired invariant subspace but the rate of convergence is at best linear. We propose to use it as a preprocessing for a Newton scheme which necessitates, at each iteration, the solution of a Sylvester type equation for which an iterative algorithm based on the preconditioned GMRES method is specially devised. This combination results in a fast and reliable method. We discuss the implementation aspects and propose a theory of convergence. Numerical tests are given to illustrate our approach. [Copyright &y& Elsevier]

Abstract: The solution to a general Sylvester equation with a low-rank right-hand side is analyzed quantitatively through the Low-rank Alternating-Directional-Implicit method (LR-ADI) with exact shifts. New bounds and perturbation bounds on are obtained. A distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of and and the right-hand side factors and . Numerical examples suggest that because of this inclusion of details, new perturbation bounds are much sharper than the existing ones. [Copyright &y& Elsevier]

We present and analyze a new iterative scheme for large-scale solution of the well-known Sylvester equation. The proposed scheme is based on fixed point iteration approach and can make good use of the recently developed methods for solving block linear systems. It is shown mathematically that the iterative process converges under some assumptions on the coefficient matrices. Results on our numerical experiments with large-scale matrices are quite encouraging. In particular, the method compares favorably with the other block methods and a recently proposed method for Sylvester equation based on low-rank approximation of the right hand side matrix C. [ABSTRACT FROM AUTHOR]

Abstract: We give new lower bounds on the Rayleigh–Ritz approximations of a part of the spectrum of an elliptic operator. Furthermore, we present bounds for the accompanying Ritz vectors. The bounds include a form of a relative gap between the Ritz values and the rest of the spectrum of the operator. A model example shows that the obtained bounds may be very sharp. [Copyright &y& Elsevier]

The distance to uncontrollability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. We present an algorithm based on methods of Gu and Burke-Lewis-Overton that estimates the distance to uncontrollability to any prescribed accuracy. The new method requires O(n4) operations on average, which is an improvement over previous methods which have complexity O(n6), where n is the order of the system. Numerical experiments indicate that the new method is reliable in practice. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new method for obtaining a matrix M which is an approximate inverse preconditioner for a given matrix A, where the eigenvalues of A all either have negative real parts or all have positive real parts. This method is based on the approximate solution of the special Sylvester equation AX + XA =2I. We use a Krylov subspace method for obtaining an approximate solution of this Sylvester matrix equation which is based on the Arnoldi algorithm and on an integral formula. The computation of the preconditioner can be carried out in parallel and its implementation requires only the solution of very simple and small Sylvester equations. The sparsity of the preconditioner is preserved by using a proper dropping strategy. Some numerical experiments on test matrices from Harwell–Boing collection for comparing the numerical performance of the new method with an available well-known algorithm are presented. [Copyright &y& Elsevier]

We consider the Sylvester equation AX-XB+C=0 where the matrix C∈ℂn×m is of low rank and the spectra of A∈ℂn×n and B∈ℂm×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ℇ∈(0,1) there exists a matrix &Xtilde; of rank k=O(log(1/ℇ)) such that ∥X-&Xtilde;∥2⩽ℇ∥X∥2. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62: 89–108). The blockwise rank of the approximation is again proportional to log(1/ℇ). Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]