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In this paper, a class of variable-order fractional convection diffusion equations have been solved with assistance of the second kind Chebyshev wavelets operational matrix. The operational matrix of variable-order fractional derivative is derived for the second kind Chebyshev wavelets. By implementing the second kind Chebyshev wavelets functions and also the associated operational matrix, the considered equations will be reduced to the corresponding Sylvester equation, which can be solved by some appropriate iterative solvers. Also, the convergence analysis of the proposed numerical method to the exact solutions and error estimation are given. A variety of numerical examples are considered to show the efficiency and accuracy of the presented technique.

The linear matrix equations A X B − C X D = E , A X − X ⁎ D = E , and A X B − X ⁎ = E are studied. In the case of uniqueness the solutions are expressed in terms of contour integrals.

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.

Given a Laurent polynomial with matrix coefficients that is positive semi-definite over the unit circle in the complex plane, the Fejer–Riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint. This paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation X + A ⁎ X − 1 A = Q . In particular, the nonlinear equation can be recast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation is readily obtainable from hermitian eigenvalue computation. The unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate-free Newton iteration for high precision calculation. This approach offers a complete parametrization of all solutions and, in contrast to most existent algorithms, makes it possible to find all solutions if so desired.

We consider a nonlinear eigenvalue problem (NEP) arising from absorbing boundary conditions in the study of a partial differential equation (PDE) describing a waveguide. We propose a new computational approach for this large-scale NEP based on residual inverse iteration (Resinv) with preconditioned iterative solves. Similar to many preconditioned iterative methods for discretized PDEs, this approach requires the construction of an accurate and efficient preconditioner. For the waveguide eigenvalue problem, the associated linear system can be formulated as a generalized Sylvester equation A X + X B + A 1 X B 1 + A 2 X B 2 + K ∘ X = C , where ∘ denotes the Hadamard product. The equation is approximated by a low-rank correction of a Sylvester equation, which we use as a preconditioner. The action of the preconditioner is efficiently computed by using the matrix equation version of the Sherman–Morrison–Woodbury (SMW) formula. We show how the preconditioner can be integrated into Resinv. The results are illustrated by applying the method to large-scale problems.

We provide necessary and sufficient conditions for the generalized $\star$-Sylvester matrix equation, $AXB + CX^\star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $\star$-Sylvester equations., This new version corrects some inaccuracies in corollaries 7 and 9

We provide an amended version of Corollaries 7 and 9 in [De Teran, Iannazzo, Poloni, Robol, "Solvability and uniqueness criteria for generalized Sylvester-type equations"]. These results characterize the unique solvability of the matrix equation AXB + CX*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. (C) 2017 Elsevier Inc. All rights reserved.

We develop a new matrix-based approach to the numerical solution of partial differential equations (PDE) and apply it to the numerical solution of partial fractional differential equations (PFDE). The proposed method is to discretize a given PFDE as a Sylvester Equation, and parameterize the integral surface using matrix algebra. The combination of these two notions results in an algorithm which can solve a general class of PFDE efficiently and accurately by means of an O ( n 3 ) algorithm for solving the Sylvester Matrix Equation (over an m i? n grid with m ~ n ). The proposed parametrization of the integral surface allows for the solution with the more general Robin boundary conditions, and allows for high-order approximations to derivative boundary conditions. To achieve our ends, we also develop a new matrix-based approximation to fractional order derivatives. The proposed method is demonstrated by the numerical solution of the fractional diffusion equation with fractional derivatives in both the temporal and spatial directions.

The ADI iteration is closely related to the rational Krylov projection methods for constructing low rank approximations to the solution of Sylvester equations. In this paper we show that the ADI and rational Krylov approximations are in fact equivalent when a special choice of shifts are employed in both methods. We will call these shifts pseudo H"2-optimal shifts. These shifts are also optimal in the sense that for the Lyapunov equation, they yield a residual which is orthogonal to the rational Krylov projection subspace. Via several examples, we show that the pseudo H"2-optimal shifts consistently yield nearly optimal low rank approximations to the solutions of the Lyapunov equations.

Recent results on the convergence of a Galerkin projection method for the Sylvester equation are extended to more general linear systems with tensor product structure. In the Hermitian positive definite case, explicit convergence bounds are derived for Galerkin projection based on tensor products of rational Krylov subspaces. The results can be used to optimize the choice of shifts for these methods. Numerical experiments demonstrate that the convergence rates predicted by our bounds appear to be sharp.

Given matrices Ai, Bi and Ci (i∈I) of corresponding dimensions over a field F, we prove that: (i) if AiCiOBi are simultaneously similar to AiOOBi, then there exists a simultaneous solution X to the matrix Sylvester equations AiX-XBi=Ci; and (ii) if AiCiOBi are simultaneously equivalent to AiOOBi, then there exist simultaneous solutions X,Y to the matrix equations AiX-YBi=Ci.We also show that analogous results hold for mixed pairs of matrix Sylvester equations A1X1-YB1=C1, A2X2-YB2=C2 and for generalized Stein equations X-AYB=C.

We develop methods for computing verified solutions of Sylvester matrix equations AX + XB = C . To this purpose we propose a variant of the Krawczyk interval operator with a factorized preconditioner so that the complexity is reduced to cubic when A and B are dense and diagonalizable. Block diagonalizations can be used in cases where A or B are not diagonalizable. The Lyapunov equation, as a special case, is also considered.

We investigate simultaneous solutions of the matrix Sylvester equations A i X - XB i = C i , i = 1 , 2 , … , k , where { A 1 , … , A k } and { B 1 , … , B k } are k -tuples of commuting matrices of order m × m and p × p , respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k -tuple of m × p matrices { C 1 , … , C k } if and only if the joint spectra σ ( A 1 , … , A k ) and σ ( B 1 , … , B k ) are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k -tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations.

Recently, Ding and Chen [F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006) 2269–2284] developed a gradient-based iterative method for solving a class of coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified, so that the iterative solutions are obtained by applying hierarchical identification principle. In this note, by considering the coupled Sylvester matrix equation as a linear operator equation we give a natural way to derive this algorithm. We also propose some faster algorithms and present some numerical results.

In this paper we suggest a new formula for the residual of Galerkin projection onto rational Krylov spaces applied to a Sylvester equation, and establish a relation to three different underlying extremal problems for rational functions. These extremal problems enable us to compare the size of the residual for the above method with that obtained by ADI. In addition, we deduce several new a priori error estimates for Galerkin projection onto rational Krylov spaces, both for the Sylvester and for the Lyapunov equation.

In this paper, we present two iterative methods for the solution of the low-rank Sylvester equation AX+XB+EF^T=0. These methods are projection methods that use the extended block Arnoldi (EBA) process and the extended global Arnoldi (EGA) process to generate orthonormal bases and F-orthonormal bases of extended Krylov subspaces. For each algorithm, we show how to stop the iterations by computing the residual norm or an upper bound without computing the approximate solution and without using expensive products with the matrices A and B. We also describe how to get the low rank solution of the Sylvester equation in a factored form. Finally, some numerical experiments are presented in order to show the efficiency and robustness of the proposed methods.

This paper is concerned with the numerical solution of large scale Sylvester equations AX−XB=C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) [22] and Li and White (2002) [20] demonstrated that the so-called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. An easily implementable extension of Penz’s shift strategy for the Lyapunov equation is presented for the current case. It is demonstrated that Galerkin projection via ADI subspaces often produces much more accurate solutions than ADI solutions.

A Haar wavelet based numerical method for solving singularly perturbed linear time invariant system is presented in this paper. The reduced pure slow and pure fast subsystems are obtained by decoupling the singularly perturbed system and differential matrix equations are converted into algebraic Sylvester matrix equations via Haar wavelet technique. The operational matrix of integration and its inverse matrix are utilized to reduce the computational time to the solution of algebraic matrix equations. Finally a numerical example is given to demonstrate the validity and applicability of the proposed method.

In this paper, by extending the well-known Jacobi and Gauss–Seidel iterations for Ax = b, we study iterative solutions of matrix equations AXB = F and generalized Sylvester matrix equations AXB + CXD = F (including the Sylvester equation AX + XB = F as a special case), and present a gradient based and a least-squares based iterative algorithms for the solution. It is proved that the iterative solution always converges to the exact solution for any initial values. The basic idea is to regard the unknown matrix X to be solved as the parameters of a system to be identified, and to obtain the iterative solutions by applying the hierarchical identification principle. Finally, we test the algorithms and show their effectiveness using a numerical example.

In this paper, we propose a new projection method based on global Arnoldi algorithm for solving large Sylvester matrix equations AX+XB+CD^T=0 and the large generalized Sylvester matrix equations of the form AXB+X+CD^T=0. 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.

This paper presents a new version of the successive approximations method for solving Sylvester equations AX − XB = C , where A and B are symmetric negative and positive definite matrices, respectively. This method is based on the block GMRES-Sylvester method. We also discuss the convergence of the new method. Some numerical experiments for obtaining the numerical solution of Sylvester equations are given. Numerical experiments show that the solution of Sylvester equations can be obtained with high accuracy and the new algorithm is a robust technique for solving Sylvester equations.

This paper presents preconditioned Galerkin and minimal residual algorithms for the solution of Sylvester equations AX-XB=C. Given two good preconditioner matrices M and N for matrices A and B, respectively, we solve the Sylvester equations MAXN-MXBN=MCN. The algorithms use the Arnoldi process to generate orthonormal bases of certain Krylov subspaces and simultaneously reduce the order of Sylvester equations. Numerical experiments show that the solution of Sylvester equations can be obtained with high accuracy by using the preconditioned versions of Galerkin and minimal residual algorithms and this versions are more robust and more efficient than those without preconditioning.

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.

In the paper we propose Galerkin and minimal residual methods for iteratively solving generalized Sylvester equations of the form AXB − X = C. The algorithms use Krylov subspace for which orthogonal basis are generated by the Arnoldi process and reduce the storage space required by using the structure of the matrix. We give some convergence results and present numerical experiments for large problems to show that our methods are efficient.

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.

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.

We consider numerical computation of Taylor expansions of invariant manifolds around equilibria of maps and flows. These expansions are obtained by writing the corresponding functional equation in a number of points, setting up a nonlinear system of equations and solving this system using a simplified Newton’s method. This approach will avoid symbolic or explicit numerical differentiation. The linear algebra issues of solving the resulting Sylvester equations are studied in detail.

We discuss convergence properties of the GMRES and FOM methods for solving large Sylvester equations of the form AX-XB=C. In particular we show the importance of the separation between the fields of values of A and B on the convergence behavior of GMRES. We also discuss the stagnation phenomenon in GMRES and its consequence on FOM. We generalize the issue of breakdown in the block-Arnoldi algorithm and explain its consequence on FOM and GMRES methods. Several numerical tests illustrate the theoretical results.

In the present paper, we propose block Krylov subspace methods for solving the Sylvester matrix equation AX−XB=C. We first consider the case when A is large and B is of small size. We use block Krylov subspace methods such as the block Arnoldi and the block Lanczos algorithms to compute approximations to the solution of the Sylvester matrix equation. When both matrices are large and the right-hand side matrix is of small rank, we will show how to extract low-rank approximations. We give some theoretical results such as perturbation results and bounds of the norm of the error. Numerical experiments will also be given to show the effectiveness of these block methods.

A numerical procedure is proposed to solve a matrix polynomial equation frequently encountered in discrete-time control and signal processing. The algorithm is based on a simple rewriting of the original equation in terms of a reduced Sylvester matrix. In contrast to previously published methods, it does not make use of elementary polynomial operations. Moreover, and most notably, it is numerically reliable. Basic examples borrowed from control and signal processing literature are aimed at illustrating the simplicity and efficiency of this new numerical method.

A recently published result describes a numerical procedure for solving a matrix Sylvester equation that is subject to certain constraints. It is quite possible that this Sylvester equation, or another intermediate one in the solution process, is nearly singular. As a result, certain computed parameters can have unexpectedly large norms and be very inaccurate. This paper incorporates an implicit deflation method for nearly singular matrix Sylvester equations to implement a reliable version of the published algorithm.

In this paper a necessary and sufficient condition for the existence of a solution for the interval Sylvester equation is given. A modified Oettli's inequality is derived to characterize the solution. Many direct methods for solving the equation are suggested and compared to each other. These methods are based on different techniques such as simulation, linear programming, correspondence between an interval Sylvester equation and an interval linear system as well as sensitivity analysis. An iterative technique for solving the interval Sylvester equation is provided with special conditions to guarantee the convergence. The square root of an interval matrix is calculated as an application to solving interval Sylvester equations.

Topological properties of a matrix equation of Sylvester type are considered. This equation is related to the block similarity between rectangular matrices, and the passage matrices in this equivalence relation are solutions. A local criterion for that similarity is exposed. The points of continuity of the map that associates with the coefficient matrices the solution space of that equation are determined.

The sensivtiity of the solution of the matrix Sylvester equation AX-XB=C is considered in the context of the classical perturbation theory. Our purpose is to find the most influent parameters in the sensitivity of the solution under perturbations in the data, and to compare the theoretical error bounds with numerical evidence.

The ADI iterative method for the solution of Sylvester's equationAX?XB=C proceeds by strictly alternating between the solution of the two equations $$\begin{gathered} \left( {A - \delta _{k + 1} I} \right)X_{2k + 1} = X_{2k} \left( {B - \delta _{k + 1} I} \right) + C, \hfill \\ X_{2k + 2} \left( {B - \tau _{k + 1} I} \right) = \left( {A - \tau _{k + 1} I} \right)X_{2k + 1} - C, \hfill \\ \end{gathered} $$ fork=0, 1, 2,... HereX o is a given initial approximate solution, and the ? k and? k are real or complex parameters chosen so that the computed approximate solutionsX k converge rapidly to the solutionX of the Sylvester equation ask increases. This paper discusses the possibility of solving one of the equations in the ADI iterative method more often than the other one, i.e., relaxing the strict alternation requirement, in order to achieve a higher rate of convergence. Our analysis based on potential theory shows that this generalization of the ADI iterative method can give faster convergence than when strict alternation is required.

The aim of the paper is to determine the solution of the Sylvester matric algebraic equation via the Kronecker algebra. The main result points out that, if the unknown matrix is ( p × q ), the solution can be determined through the inversion of a ( min ( p , q ) × min ( p , q )) matrix.

We present a numerical method of solving the time-varying Lyapunov and Sylvester matrix differential equations. The method is easy to program, fast, and accurate. It is shown that the error of computation is O(Δt 3) where Δt is the interval of computation. Two examples are given.

A complete, general and explicit solution to the generalized Sylvester matrix equation AX−XF=BY, with the matrix F in a companion form, is proposed. The solution is in an extremely neat form represented by a symmetric operator matrix, a Hankel matrix and the controllability matrix of the matrix pair (A,B). Furthermore, several equivalent forms of this solution are also presented. Based on these presented results, explicit solutions to the normal Sylvester equation and the well-known Lyapunov matrix equation are also established. The results provide great convenience to the analysis of the solution to the equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, a simple and effective approach for parametric pole assignment is proposed.

We study the well-known Sylvester equation XA − BX = R in the case when A and B are given and R is known up to its first n - 1 rows. We prove new results on the existence and uniqueness of X . Our results essentially state that, in case A is a nonderogatory matrix, there always exists a solution to this equation; a solution is uniquely determined by its first row x 1 ; and there is an interesting relationship between x 1 and the rows of R . We also give a complete characterization of the nonsingularity of X in this case. As applications of our results we develop direct methods for constructing symmetrizers and commuting matrices, computing the characteristic polynomial of a matrix, and finding the numbers of common eigenvalues between A and B . Some well-known important results on symmetrizers, Bezoutians, and inertia are recovered as special cases.

In this paper we consider the special Sylvester equation XM - NX = 0 for fixed n × n matrices M and N , where a positive definite solution X is sought. We show that the solution sets varying over ( M , N ) provide a new family of geodesic submanifolds in the symmetric Riemannian manifold P n of positive definite matrices which is stable under congruence transformations; it consists of geodesically complete convex cones of P n invariant under Cartan symmetries. It is further shown that the solution set is stable under the iterative means obtained by the weighted arithmetic, harmonic and geometric means.

The notion of skew primeness introduced by Wolovich in the context of polynomial matrices is extended to the context of inner functions. Skew primeness is related to a geometric condition as well as to the solvability, over H ∞ , of the Sylvester equation.