Efficient Approaches for Enclosing the United Solution Set of the Interval Generalized Sylvester Matrix Equation

In this work, we investigate the interval generalized Sylvester matrix equation ${\bf{A}}X{\bf{B}}+{\bf{C}}X{\bf{D}}={\bf{F}}$ and develop some techniques for obtaining outer estimations for the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator which causes reducing computational complexity to cubic, compared to Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of ${\bf{A}}$, ${\bf{B}}$, ${\bf{C}}$ and ${\bf{D}}$ and in both of them we suppose that the midpoints of ${\bf{A}}$ and ${\bf{C}}$ are simultaneously diagonalizable as well as for the midpoints of the matrices ${\bf{B}}$ and ${\bf{D}}$. Some numerical experiments are given to illustrate the performance of the proposed methods

A generalized preconditioned MHSS method for a class of complex symmetric linear systems

Based on the MHSS (Modified Hermitian and skew-Hermitian splitting) and preconditioned MHSS methods, we will present a generalized preconditioned MHSS method for solving a class of complex symmetric linear systems. The new method (GPMHSS) is essentially a two-parameter iteration method where the iterative sequence is unconditionally convergent to the unique solution of the linear system. A parameter region of the convergence for our method is provided. An efficient preconditioner is presented for the actual implementation of the new method. Some numerical results are given to show its effectiveness