Fast Iterative Schemes for Nonsymmetric Algebraic Riccati Equations Arising from Transport Theory

Abstract

For a class of nonsymmetric algebraic Riccati equations arising from transport theory, Lu [SIAM J. Matrix Anal. Appl., 26 (2005), pp. 679-685] reformulated them into a couple of fixed-point equations in a vector form and constructed an iterative method for finding their minimal positive solutions. In this paper, based on this fixed-point equation, we propose several fast and effective iterative methods for solving the above-mentioned special type of nonsymmetric algebraic Riccati equations. These iterative methods can be categorized into the class of nonlinear block relaxation iterations and can also be considered as accelerated variants of the known fixed-point iteration according to the principle of using the currently available information as promptly as possible. The monotone convergence theorems about these new iterative methods are established, and the sharper bounds about the solutions of the nonsymmetric algebraic Riccati equations are derived. Numerical implementations show that the new iterative methods are feasible and effective solvers for the nonsymmetric algebraic Riccati equations arising from transport theory.