In this paper we propose a structure-preserving doubling algorithm (SDA) for computing the minimal nonnegative solutions to the nonsymmetric algebraic Riccati equation (NARE) based on the techniques developed in the symmetric cases. This method allows the simultaneous approximation of the minimal nonnegative solutions of the NARE and its dual equation, only requires the solutions of two linear systems, and does not need to choose any initial matrix, thus it overcomes all the defaults of the Newton iteration method and the fixed-point iteration methods. Under suitable conditions, we establish the convergence theory by using only the knowledge from elementary matrix theory. The theory shows that the SDA iteration matrix sequences are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the NARE and its dual equation, respectively. Numerical experiments show that the SDA algorithm is feasible and effective, and can outperform the Newton iteration method and the fixed-point iteration methods.

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In this paper, a structure-preserving transformation of a symplectic pencil is introduced, referred to as the doubling transformation, and its some basic properties are presented. Based on the nice properties of this kind of transformations, a unified convergence theory for the structure-preserving doubling algorithms for solving a class of Riccati-type matrix equations is established by using only the knowledge from elementary matrix theory.

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