A cyclic iterative approach and its modified version to solve coupled Sylvester-transpose matrix equations

Abstract

Recently, Tang et al. [Numer Algorithms. 2014;66(2):379–397] have offered a cyclic iterative method for determining the unique solution of the coupled matrix equationsAnalogues to the gradient-based algorithm, the proposed algorithm relies on a fixed parameter whereas it has wider convergence region. Nevertheless, the application of the algorithm to find the centro-symmetric solution of the mentioned problem has been left as a project to be investigated and the optimal value for the fixed parameter has not been derived. In this paper, we focus on a more general class of the coupled linear matrix equations that incorporate the mentioned ones in the earlier refereed work. More precisely, we first develop the authors’ propounded algorithm to resolve our considered coupled linear matrix equations over centro-symmetric matrices. Afterwards, we disregard the restriction of the existence of the unique (centro-symmetric) solution and also modify the authors’ algorithm by applying an oblique projection technique which allows to produce a sequence of approximate solutions which gratify an optimality property. Numerical results are reported to confirm the validity of the established results and to demonstrate the superior performance of the modified version of the cyclic iterative algorithm.