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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 this paper, a modified conjugate gradient iterative method for solving Sylvester equation is presented. By using this iterative method, the least squares anti-bisymmetric solution and the optimal approximation solution can be obtained. Here we present the derivation and theoretical analysis of our iterative method. Numerical results illustrate the feasibility and effectiveness of the proposed iterative method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a wavelet operational method based on Haar wavelet is proposed to solve the fractional partial differential equations in the Caputo derivative sense. We give the Haar wavelet operational matrix of fractional order integration. A truncated Haar wavelet series together with the wavelet operational matrix are utilized to reduce the fractional partial differential equations to Sylvester equations. In addition, some examples are presented to show the efficiency and the accuracy of the approach. [Copyright &y& Elsevier]

Abstract: Fast algorithms for enclosing solutions of Sylvester equations are proposed. The results obtained by these algorithms are “verified” in the sense that all the possible rounding errors have been taken into account. For developing these algorithms, theories which directly supply error bounds for numerical solutions are established. The proposed algorithms require only operations if A and B are diagonalizable. Techniques for accelerating the enclosure and obtaining smaller error bounds are introduced. Numerical results show the properties of the proposed algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new algorithm to solve the Sylvester matrix equation . The technique consists of orthogonal reduction of the matrix to a block upper Hessenberg form and then solving the reduced equation, for through recurrence relation, where , and . We then recover the solution of the original problem via the relation . The numerical results show the accuracy and the efficiency of the proposed algorithm. In addition, how the technique described can be applied to other matrix equations was shown. [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]