Low memory and low complexity iterative schemes for a nonsymmetric algebraic Riccati equation arising from transport theory

We reconsider Newton’s method and two fixed-point methods for finding the minimal positive solution of a nonsymmetric algebraic Riccati equation arising from transport theory. We rewrite the subproblem of the Newton and fixed-point iterative schemes into an equivalent form with some special structure. By the use of the particular structure of the subproblem, we then present low memory and low complexity versions of these iterative methods with a factored alternating-direction-implicit iteration. Some properties of eigenvalues for iterative coefficient matrices in solving the subproblem are derived and the convergence of the proposed methods is established. Numerical experiments show that the new iterative schemes are highly efficient to obtain the minimal positive solution. The proposed low memory and low complexity Newton’s method is particularly efficient for solving large scale Riccati equation arising from transport theory.