A note on Sylvester-type equations
For researchers in numerical linear algebra, this provides a novel theoretical framework and algorithm for solving Sylvester-type equations, though the practical impact is not demonstrated with concrete benchmarks.
This work establishes a connection between generalized eigenvalue problems and Sylvester-type equations, showing that the solution of the $\star$-Sylvester matrix equation is uniquely determined under regularity and can be obtained via deflating subspaces. An iterative method with quadratic convergence is proposed using the palindromic doubling algorithm.
This work is to provide a comprehensive treatment of the relationship between the theory of the generalized (palindromic) eigenvalue problem and the theory of the Sylvester-type equations. Under a regularity assumption for a specific matrix pencil, we show that the solution of the $\star$-Sylvester matrix equation is uniquely determined and can be obtained by considering its corresponding deflating subspace. We also propose an iterative method with quadratic convergence to compute the stabilizing solution of the $\star$-Sylvester matrix equation via the well-developed palindromic doubling algorithm. We believe that our discussion is the first which implements the tactic of the deflating subspace for solving Sylvester equations and could give rise to the possibility of developing an advanced and effective solver for different types of matrix equations.