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Given the matrix equation A X + X B + f (X) C = D in the unknown n × m matrix X , we analyze existence and uniqueness conditions, together with computational solution strategies for f : R n × m → R being a linear or nonlinear function. We characterize different properties of the matrix equation and of its solution, depending on the considered classes of functions f. Our analysis mainly concerns small dimensional problems, though several considerations also apply to large scale matrix equations. [ABSTRACT FROM AUTHOR]

The Haar wavelet collocation with iteration technique is applied for solving a class of time-fractional physical equations. The approximate solutions obtained by two dimensional Haar wavelet with iteration technique are compared with those obtained by analytical methods such as Adomian decomposition method (ADM) and variational iteration method (VIM). The results show that the present scheme is effective and appropriate for obtaining the numerical solution of the timefractional Modified Camassa-Holm equation and Time fractional Modified Degasperis-Procesi equation. [ABSTRACT FROM AUTHOR]

The split-step Padé approach is an extremely efficient tool for the integration of pseudodifferential parabolic equations, which are widely used for the modeling of acoustic wave propagation. In this study, a generalization of this method to the case of parabolic equations with unknown vector functions is presented. Such generalization requires an algorithm for efficient numerical evaluation of a function of a matrix with differential operators as its elements. After a finite-difference discretization this algorithm reduces to the solution of several Sylvester-like problems. The generalized split-step Padé method presented here is an attractive tool for solving 3D problems of sound propagation in the ocean within the framework of coupled mode parabolic equations theory. • A method is proposed for the numerical solution of the one-way counterpart of a system of coupled Helmholtz equations. • The method is based on a generalization of the split-step Padé algorithm to the case of unknown vector functions. • The method is an attractive tool for handling many practical problems where the propagation of coupled guided modes takes place. [ABSTRACT FROM AUTHOR]

In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation L M are bold, please chekc if bold neceaasry, if not, please remove all bold of equation − M K = r s T together with an evolution equation set of r and s , master function S (i , j) = s T K j C (I + M C) − 1 L i r is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ -function by τ = | I + M C |. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed. [ABSTRACT FROM AUTHOR]

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]

A linear quaternionic equation in one quaternionic variable q is of the form a 1 q b 1 + a 2 q b 2 + ⋯ + a m q b m = c , where the a i , b j , c are given quaternionic coefficients. If introducing basis elements i , j , k of pure quaternions, then the quaternionic equation becomes four linear equations in four unknowns over the reals, and solving such equations is trivial. On the other hand, finding a quaternionic rational function expression of the solution that involves only the input quaternionic coefficients and their conjugates, called a basis-free solution, is non-trivial. In 1884, Sylvester initiated the study of basis-free solution to linear quaternionic equation. He considered the three-termed equation a q + q b = c , and found its solution q = (a 2 + b b ¯ + a (b + b ¯)) − 1 (a c + c b ¯) by successive left and right multiplications. In 2013, Schwartz extended the technique to the four-termed equation, and obtained the basis-free solution in explicit form. This paper solves the general problem for arbitrary number of terms in the non-degenerate case. [ABSTRACT FROM AUTHOR]

In this paper we aim to derive solutions for the SU(N) self-dual Yang–Mills (SDYM) equation with arbitrary N. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev–Petviashvili hierarchy and the (matrix) Ablowitz–Kaup–Newell–Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU (N) SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one. • Solutions of the SU(N) SDYM equation with arbitrary N are constructed. • The SU(N) SDYM is formulated by the Cauchy matrix scheme of the KP hierarchy. • The SU(2N) SDYM is formulated by the Cauchy matrix scheme of the AKNS hierarchy. • Some solutions of the SU(2N) SDYM obtained in the two schemes are analyzed. [ABSTRACT FROM AUTHOR]

We consider the problem of efficiently solving Sylvester and Lyapunov equations of medium and large scale, in case of rank-structured data, i.e., when the coefficient matrices and the right-hand side have low-rank off-diagonal blocks. This comprises problems with banded data, recently studied in [A. Haber and M. Verhaegen, Automática J. IFAC, 73 (2016), pp. 256--268; D. Palitta and V. Simoncini, Numerical Methods for Large-Scale Lyapunov Equations with Symmetric Banded Data, preprint, arxiv, 1711.04187, 2017], which often arise in the discretization of elliptic PDEs. We show that, under suitable assumptions, the quasiseparable structure is guaranteed to be numerically present in the solution, and explicit novel estimates of the numerical rank of the off-diagonal blocks are provided. Efficient solution schemes that rely on the technology of hierarchical matrices are described, and several numerical experiments confirm the applicability and efficiency of the approaches. We develop a MATLAB toolbox that allows easy replication of the experiments and a ready-to-use interface for the solvers. The performances of the different approaches are compared, and we show that the new methods described are efficient on several classes of relevant problems. [ABSTRACT FROM AUTHOR]

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermitian positive definite or negative definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation A X + X B = C , in particular, it can be faster and more accurate than the built-in implementation of the Bartels–Stewart algorithm, when A and B are well conditioned and have very different size. [ABSTRACT FROM AUTHOR]

A novel state-space realization for parameterized macromodeling is proposed in this paper. A judicious choice of the state-space realization is required in order to account for the assumed smoothness of the state-space matrices with respect to the design parameters. This technique is used in combination with suitable interpolation schemes to interpolate a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parameterized macromodels. The key points of the novel state-space realizations are the choice of a proper pivot matrix and a well-conditioned solution of a Sylvester equation. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed Sylvester realization for parameterized macromodeling. [ABSTRACT FROM AUTHOR]

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: In this paper, we propose a new projection method based on global Arnoldi algorithm for solving large Sylvester matrix equations and the large generalized Sylvester matrix equations of the form . We show how to extract low-rank approximate solutions to Sylvester matrix equations and generalized Sylvester matrix equations. Some theoretical results are given. Numerical tests report the effectiveness of these methods. [Copyright &y& Elsevier]

Abstract: For when the Sylvester matrix equation has a unique solution, this work provides a closed form solution, which is expressed as a polynomial of known matrices. In the case of non-uniqueness, the solution set of the Sylvester matrix equation is a subset of that of a deduced equation, which is a system of linear algebraic equations. [Copyright &y& Elsevier]

Abstract: In the present paper, we propose an Arnoldi-based method for solving large and sparse Sylvester matrix equations with low rank right hand sides. We will show how to extract low-rank approximations via a matrix Krylov subspace method. We give some theoretical results such an expression of the exact solution and upper bounds for the norm of the error and for the residual. Numerical experiments will also be given to show the effectiveness of the proposed method. [Copyright &y& Elsevier]

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]

Abstract: The global GMRES and global FOM algorithm are recently proposed by Jbilou et al. for solving the linear equations with multiple right-hand sides [K. Jbilou, A. Messaoudi, H. Sados, Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math. 31 (1999) 49–63]. Like GMRES for the linear equations, they generally uses restarting, which slows the convergence. However, some information can be retained at the time of the restart and used in the next cycle. We present algorithms that use implicit restarting in order to retain this information as Morgan have proposed recently [R.B. Morgan, Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations, SIAM J. Matrix anal. Appl. 21 (2000) 1112–1135]. At the same time, we prove that global GMRES and global FOM methods for matrix equations are equivalent with the corresponding methods for linear equations and propose implicitly restarted global FOM and GMRES. Numerical examples show that our methods are efficient. [Copyright &y& Elsevier]

This paper proposes a new method for estimating the error in the solution of matrix equations. The estimate is based on the adjoint method in combination with small sample statistical theory. It can be implemented simply and is inexpensive to compute. Numerical examples are presented which illustrate the power and effectiveness of the new method. [Copyright &y& Elsevier]

In this paper we construct an approximation to the solution x of a linear system of equations Ax = b of tensor product structure as it typically arises for finite element and finite difference discretisations of partial differential operators on tensor grids. For a right-hand side b of tensor product structure we can prove that the solution x can be approximated by a sum of O (log(ε)2) tensor product vectors where ε is the relative approximation error. Numerical examples for systems of size 1024256 indicate that this method is suitable for high-dimensional problems. [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]

We present sufficient conditions for the unique solvability of the matrix equation A X B + C | X | D = E where A , B , C , D and E are given square or rectangular real matrices of appropriate size and X is unknown. We also give sufficient conditions for the non-solvability of the equation in special cases. Our sufficient conditions can be checked with a cubic complexity in terms of the size of the input matrices. [ABSTRACT FROM AUTHOR]

Various integrable systems, both continuous and discrete, having soliton solutions have been studied through the generalized Cauchy matrix method. Here we employ this approach to investigate a lattice CKP equation which is expressed by a τ function. We start from introducing a Sylvester equation and then show how to solve it. The variables in the Sylvester equation help us construct the τ function of the lattice CKP equation, which yields different types of exact solutions to the lattice CKP equation as well. [ABSTRACT FROM AUTHOR]

The key purpose of this study is to present two schemes based on Lagrange polynomials to deal with the numerical solution of second order two-dimensional telegraph equation (TDTE) with the Dirichlet boundary conditions. First, we convert the main equation into partial integro-differential equations (PIDEs) with the help of initial and boundary conditions. The operational matrices of differentiation and integration are then used to transform the PIDEs into algebraic generalized Sylvester equation. We compared the results obtained by the proposed schemes with Bernoulli matrix method and B-spline differential quadrature method which shows that the proposed schemes are accurate for small number of basis functions. [ABSTRACT FROM AUTHOR]

This article presents a reliable numerical approach for the fractional advection–dispersion equation by making use of Legendre scaling functions as a basis. The fractional advection–dispersion equation describes the anomalous transport in surface and subsurface water. Using two dimensional basis formed by Legendre scaling functions, we get operational matrix for fractional integrations and differentiations. Substituting these operational matrices in the equation leads linear algebraic equations whose solutions can be derived with the aid of Sylvester's approach; this in turn yields approximate solutions for advection–dispersion equation. Convergence analysis of the proposed scheme is presented. The potency and accuracy of the proposed numerical algorithm are shown by plotting error figures. • We consider a fractional advection–dispersion equation. • A reliable numerical approach is applied to examine the fractional problem. • Convergence analysis of the proposed scheme is shown. • Numerical examples are given to show the effectiveness of the numerical method. [ABSTRACT FROM AUTHOR]