Convergence theory of iterative methods based on proper splittings and proper multisplittings for rectangular linear systems

Multisplitting methods are useful to solve differential-algebraic equations. In this connection, we discuss the theory of matrix splittings and multisplittings, which can be used for finding the iterative solution of a large class of rectangular (singular) linear system of equations of the form Ax = b. In this direction, many convergence results are proposed for different subclasses of proper splittings in the literature. But, in some practical cases, the convergence speed of the iterative scheme is very slow. To overcome this issue, several comparison results are obtained for different subclasses of proper splittings. This paper also presents a few such results. However, this idea fails to accelerate the speed of the iterative scheme in finding the iterative solution. In this regard, Climent and Perea [J. Comput. Appl. Math. 158 (2003), 43-48: MR2013603] introduced the notion of proper multisplittings to solve the system Ax = b on parallel and vector machines, and established convergence theory for a subclass of proper multisplittings. With the aim to extend the convergence theory of proper multisplittings, this paper further adds a few results. Some of the results obtained in this paper are even new for the iterative theory of nonsingular linear systems.

Optimal control of large-scale singular linear systems via hierarchical strategy

This paper states a hierarchical strategy of large-scale singular linear system in which the system is composed of J singular linear subsystems with interconnections. Among hierarchical strategies, the two-level optimization conditions based on Interaction Prediction Method (IPM) are derived such that whole large-scale singular system is optimized. Based on the two-level coordination method, the optimization analysis and controller design of large-scale singular linear system is discussed. For this purpose, two decentralized nonlinear state feedback controllers are designed for each subsystem, such that the whole closed-loop linear system is optimized. These conditions are used to extract two IPM algorithms. Then, by using both algorithms, two numerical examples are given to confirm the analytical results and illustrate the effectiveness of the proposed method.

Delayed over-relaxation in iterative schemes to solve rank deficient linear system of (matrix) equations

Recently in [Journal of Computational Physics, 321 (2016), 829-907], an approach has been developed for solving linear system of equations with nonsingular coefficient matrix. The method is derived by using a delayed over-relaxation step (DORS) in a generic (convergent) basic stationary iterative method. In this paper, we first prove semi-convergence of iterative methods with DORS to solve singular linear system of equations. In particular, we propose applying the DORS in the Modified HSS (MHSS) to solve singular complex symmetric systems and in the Richardson method to solve normal equations. Moreover, based on the obtained results, an algorithm is developed for solving coupled matrix equations. It is seen that the proposed method outperforms the relaxed gradient-based (RGB) method [Comput. Math. Appl. 74 (2017), no. 3, 597-604] for solving coupled matrix equations. Numerical results are examined to illustrate the validity of the established results and applicability of the presented algorithms.