Weighted least squares solutions to general coupled Sylvester matrix equations

Abstract

MSC: 15A09 15A12 15A24Keywords: Weighted least squares solutions Weighted generalized inverses Coupled Sylvester matrix equations Gradient based iterative algorithms Maximal convergence rate a b s t r a c t This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.

The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation AYB + CYTD = E

“…Zhou and Duan [27--30] studied the solution of the several generalized Sylvester matrix equations. Zhou et al [31] proposed an iterative method for finding weighted least squares solutions to coupled Sylvester matrix equations. Ding and Chen introduced the hierarchical least squares-iterative (HLSI) algorithms for several coupled matrix equations [32,33] and the hierarchical gradient-iterative (HGI) algorithms for general matrix equations [34,35].…”

Solving coupled matrix equations over generalized bisymmetric matrices

“…Kyrchei [29] considered systems of linear quaternionic equations and obtained Cramer's rules for right and left quaternionic systems of linear equations. Zhou et al [30] proposed an iterative method for finding weighted least squares solutions to coupled Sylvester matrix equations. In [31], Zhou et al analyzed the computational complexity of the Smith iteration and its variations for solving the Stein matrix equation.…”

Linear modeling and regression for exponential engineering functions by a generalized ordinary least squares method

“…Quan et al [23] proposed a weighted least square support vector machine (WLS-SVM) for the prediction of nonlinear time series. Zhou et al [24] proposed gradient based iterative algorithms (based on weighted least squares) to solve the general coupled Sylvester matrix equations. The objective of this paper is to illustrate the executive steps of linearization and optimization by the ordinary least squares method applying on the exponential engineering functions in order to determine the optimum parameters in a linear regression model that govern the investigated system (physical, chemical, or biochemical).…”