On exact line search method for a polynomial matrix equation

Abstract

In this work, we investigate the elementwise minimal non-negative (EMN) solution of the matrix polynomial equation using an exact line search (ELS) technique to enhance the convergence of the Newton method. Nonnegative solutions to matrix equations are essential in engineering, optimization, signal processing, and data mining, driving advancements and improving efficiency in these fields. While recent advancements in solving matrix equations with nonnegative constraints have emphasized iterative methods, optimization strategies, and theoretical developments, efficiently finding the EMN solution remains a significant challenge. The proposed method integrates the Newton method with an exact line search (ELS) strategy to accelerate convergence and improve solution accuracy. Numerical experiments demonstrate that this approach requires fewer iterations to reach the EMN solution compared to the standard Newton method. Moreover, the method shows improved stability, particularly when dealing with ill-conditioned input matrices and very small tolerance errors.