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.

翻译：本文研究了区间广义Sylvester矩阵方程${\bf{A}}X{\bf{B}}+{\bf{C}}X{\bf{D}}={\bf{F}}$，并开发了若干技术以获得该区间系统所谓联合解集的外部估计。首先，我们提出了一种Krawczyk算子的改进变体，相较于Kronecker积形式，该算子将计算复杂度降低至三次。随后，我们提出了一种用于封闭解集的迭代技术。这些方法基于${\bf{A}}$、${\bf{B}}$、${\bf{C}}$和${\bf{D}}$中点的谱分解，并且在两种方法中，我们假设${\bf{A}}$和${\bf{C}}$的中点可同时对角化，同时${\bf{B}}$和${\bf{D}}$的中点也可同时对角化。数值实验展示了所提出方法的性能。