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...

Presents a recurrent neural network for solving the Sylvester equation with time-varying coefficient matrices. The recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time-varying solution of a given Sylvester equation. Theoretical results of convergence and sensitivity analysis are presented to ...

We provide necessary and sufficient conditions for the uniqueness of solution of the generalized Sylvester and ⋆-Sylvester equations, AXB − CXD = E and AXB+CX ⋆ D = E, respectively, where the matrix coefficients are allowed to be rectangular. These conditions generalize existing results for the same equations with square coefficients. Our characterization reveals unexpected constraints regardin...

We propose a displacement structure based rank-revealing algorithm for Sylvester matrix, then apply it to compute approximate greatest common division of two univariate polynomials with floating-point coefficients. This structured rank-revealing method is based on a stabilized version of the generalized Schur algorithm [8], and is a fast rank-revealing method in the sense that, all computations...

We present a direct method for reordering eigenvalues in the generalized periodic real Schur form of a regular K-cyclic matrix pair sequence (Ak, Ek). Following and generalizing existing approaches, reordering consists of consecutively computing the solution to an associated Sylvester-like equation and constructingK pairs of orthogonal matrices. These pairs define an orthogonal K-cyclic equival...