A New Dynamically Weighted Halley-like Iteration Method For Solving Algebraic Riccati Equations

Algebraic Riccati equations arise in many applications form different areas,including control theory, stochastic fluid model, queueing model, differential games.Solving the numerical solution of algebraic Riccati equations has important theory significance and application values. We transform the problem of solving algebraic Riccati equation into the problem of solving matrix sign function, and we know that Halley-like iteration method can solve matrix sign function. In this paper, we obtain a dynamically weighted Halley-like iteration method by means of introducing three parameters. In order to improve the convergence rate of dynamically weighted Halley-like iteration method, we obtain optimal parameters by means of solving a maximin problem.When the Hamiltonian matrix of algebraic Riccati equation have eigenvalues with large maginary number or zero eigenvalues. In two cases, the convergence rate of algorithms will be slow down. Then, we propose a shift technique to accelerate the convergence rate of algorithms. Numerical experiment shows that dynamically weighted Halley-like iteration is more efficient than the Halley-like iteration, and solving the shifted Riccati equation is more efficient than the original Riccati equation.