大规模Sylvester矩阵方程在系统控制、信号处理、模型降阶、图像恢复、滤波器设计等实际问题中有着非常广泛的应用。基于Krylov子空间的投影类方法为大规模矩阵方程的求解提供了有效途径。然而，在运用块形式的扩展Krylov子空间方法时，随着矩阵规模及迭代步数的增大，算法将受到存储空间等的限制。本项目在前期研究的基础上，拟针对如下几个问题进行研究：(1)拟设计求解大规模Sylvester矩阵方程的重启动扩展块Krylov子空间算法，减少已有算法的存储量，并对其收敛性及相关理论进行分析；(2)针对具有特殊结构的Toeplitz矩阵，通过结合Toeplitz矩阵的结构性质，设计求解大规模Toeplitz矩阵方程的快速算法，并将其应用于图像去噪的问题中。本项目旨在研究一系列求解大规模矩阵方程的快速算法并进行理论分析，研究结果将为系统控制、图像恢复及信号处理等领域提供高效的计算方法和理论依据。

Large scale Sylvester matrix equations are ubiquitous in system and control theory, signal processing, model reduction, image restoration and filter design. Projection methods like Krylov subspace methods provide an efficient way for solving large scale matrix equations. However, as the size of matrices and step of iterations increase, those methods based on the block extended Krylov subspace will be limited by the memory of computers. Based on the previous research, we mainly focus on the following problems: (1) we will design a restarted-block extended Krylov subspace method for solving large scale matrix equations, it can reduce the storage, and the convergence theories will be established. (2) For the Toeplitz matrices with special structure, we will design some fast algorithms for solving large scale Toeplitz matrix equations combining the special structures and apply the algorithms to image restoration problems. This project aims at studying some efficient algorithms with theoretical analysis for solving large scale matrix equations. The research results will provide efficient algorithms and theoretical basis for a range of fields such as system control, image restoration and signal processing.