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We discuss the use of a matrix‐oriented approach for numerically solving the dense matrix equation AX + XAT + M1XN1 + ... + MℓXNℓ = F, with ℓ ≥ 1, and Mi, Ni, i = 1, ... , ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix‐oriented method is reported. The application of the procedure as the core step in the solution of the large‐scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of large scale singular (continuous‐time) Lyapunov equations of the form AX + XA⊤ + BB⊤ = 0, where A is semistable, that is, its spectrum is contained in the left half plane, with the exception of a few semisimple eigenvalues at zero. We also consider the case of a few semisimple eigenvalues on the imaginary axis. We assume that we know these few eigenvalues (zero or imaginary), and that we have or can compute the corresponding invariant subspaces. We use this information to build an appropriate newly proposed subspace on which to project the Lyapunov equations, and then compute a low‐rank approximation to the least squares solution. Selected illustrative numerical examples are provided. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a c-stable linear equation in the tensor product space [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]

SUMMARY In this paper, we propose a block Arnoldi method for solving the continuous low-rank Sylvester matrix equation AX + XB = EF T. We consider the case where both A and B are large and sparse real matrices, and E and F are real matrices with small rank. We first apply an alternating directional implicit preconditioner to our equation, turning it into a Stein matrix equation. We then apply a block Krylov method to the Stein equation to extract low-rank approximate solutions. We give some theoretical results and report numerical experiments to show the efficiency of this method. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a method for solving large-scale Sylvester equations is presented. The method is based on the sign function iteration and is particularly effective for Sylvester equations with factorized right-hand side. In this case, the solution will be computed in factored form as it is for instance required in model reduction. The hierarchical matrix format and the corresponding formatted arithmetic are integrated in the iteration scheme to make the method feasible for large-scale computations. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This issue contains papers dealing with a variety of linear and nonlinear matrix equations arising in systems and control theory, model reduction, and in various other areas of applied mathematics, economics, engineering, and the sciences. Special emphasis is given to the numerical treatment of large-scale problems. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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]

The design of a Luenberger observer for large control systems is an important problem in Control Theory. Recently, several computational methods have been proposed by Datta and collaborators. The present paper discusses numerical aspects of one of these methods, described by Datta and Saad (1991). Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]