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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]

In this paper, we present the Sherman–Morrison–Woodbury-type formula for the solution of the Sylvester equation of the formas well as for the solution of theT-Sylvester equation of the formwhereU1,U2,V1,V2are low-rank matrices. Although the matrix version of this formula for the Sylvester equation has been used in several different applications (but not for the case of aT-Sylvester equation), we present a novel approach using a proper operator representation. This novel approach allows us to derive a matrix version of the Sherman–Morrison–Woodbury-type formula for the Sylvester equation as well as for theT-Sylvester equation which seems to be new. We also present algorithms for the efficient calculation of the solution of structured Sylvester andT-Sylvester equations by using these formulas and illustrate their application in several examples. [ABSTRACT FROM PUBLISHER]

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]