Generalized conjugate direction algorithm for solving general coupled Sylvester matrix equations

In this paper, a generalized conjugate direction algorithm (GCD) is proposed for solving general coupled Sylvester matrix equations. The GCD algorithm is an improved gradient algorithm, which can realize gradient descent by introducing matrices P / (k) and T / (k) to construct parameters & alpha; (k) and & beta; (k) . The matrix P / (k) and T / (k) are iterated in a cross way to accelerate the convergence rate. In addition, it is further proved that the algorithm converges to the exact solution in finite iteration steps in the absence of round-off errors if the system is consistent. Also, the sufficient conditions for least squares solutions and the minimum F-norm solutions are obtained. Finally, numerical examples are given to demonstrate the effectiveness of the GCD algorithm. & COPY; 2023 The Franklin Institute. Published by Elsevier Inc. All rights reserved.