The shift techniques for a nonsymmetric algebraic Riccati equation

In this paper, we want to analyze a special instance of a nonsymmetric algebraic matrix Riccati equation arising from transport theory. Traditional approaches for finding its minimal nonnegative solution are based on fixed point iterations and the speed of the convergence is linear. Recently, iterative methods such as Newton method and the structure‐preserving doubling algorithm with quadratic convergence are designed for improving the speed of convergence. But, in some case, the speed of convergence will significantly decrease so that linear convergence becomes sublinear convergence and quadratic convergence becomes linear convergence. Our contribution in this work is to provide a thorough analysis to show that after the shift techniques, the speed of linear or quadratic convergence is preserved. Finally, we apply the shift procedures to the discussion of the simple iteration algorithm, improve its speed of convergence, and reduce its total elapsed CPU time.