This paper addresses two problems: an image denoising problem assuming dense observations and an image reconstruction problem from sparse data. It shows that both problems can be solved by the Sylvester/Lyapunov algebraic equation. The Sylvester/Lyapunov equation has been extensively studied in Control Theory and it can be efficiently solved by well known numeric algorithms. This paper proposes...

An efficient computational method is presented for state space analysis of singular systems via Haar wavelets. Singular systems are those in which dynamics are governed by a combination of algebraic and differential equations. The corresponding differential-algebraic matrix equation is converted to a generalized Sylvester matrix equation by using Haar wavelet basis. First, an explicit expressio...

‎The global generalized minimum residual (Gl-GMRES)‎ ‎method is examined for solving the generalized Sylvester matrix equation‎ ‎[sumlimits_{i = 1}^q {A_i } XB_i = C.]‎ ‎Some new theoretical results are elaborated for‎ ‎the proposed method by employing the Schur complement‎. ‎These results can be exploited to establish new convergence properties‎ ‎of the Gl-GMRES method for solving genera...

Let A be a unital complex semisimple Banach algebra, and MA denote its maximal ideal space. For matrix M∈An×n, Mˆ denotes the obtained by taking entry-wise Gelfand transforms. M∈Cn×n, σ(M)⊂C set of eigenvalues M. It is shown that if A∈An×n B∈Am×m are such for all φ∈MA, σ(Aˆ(φ))∩σ(Bˆ(φ))=∅, then C∈An×m, Sylvester equation AX−XB=C has unique solution X∈An×m. As an application, Roth's removal rule...

The nonlinear matrix equation X+A∗X−1A = Q can be cast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation can be obtained by means of hermitian eigenvalue computation. The unitary constraint can be satisfied by means of either a straightforward alternating projection method or by a coordinate-free Newton iteration. The idea proposed in this paper originates fro...

Some complex quaternionic equations in the type AX - XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.

in this paper, an iterative method is proposed for solving large general sylvester matrix equation $axb+cxd = e$, where $a in r^{ntimes n}$ , $c in r^{ntimes n}$ , $b in r^{stimes s}$ and  $d in r^{stimes s}$ are given matrices and $x in r^{stimes s}$  is the unknown matrix. we present a global conjugate gradient (gl-cg) algo- rithm for solving linear system of equations with multiple right-han...

In this paper we study numerical methods for solving Sylvester matrix equations of the form AX +XB +CD = 0. A new projection method is proposed. The union of Krylov subspaces in A and its inverse and the union of Krylov subspaces in B and its inverse are used as the right and left projection subspaces, respectively. The Arnoldi-like process for constructing the orthonormal basis of the projecti...

In this talk 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 may deduce several new a pr...