Doubling Algorithm for Nonsymmetric Algebraic Riccati Equations Based on a Generalized Transformation (Advances in Applied Mathematics and Mechanics, 2021)
We consider computing the minimal nonnegative solution of the nonsymmetricalgebraic Riccati equation with $M$-matrix. It is well known that such equationscan be efficiently solved via the structure-preserving doubling algorithm (SDA) withthe shift-and-shrink transformation or the generalized Cayley transformation. In thispaper, we propose a more generalized transformation of which the shift-and-shrinktransformation and the generalized Cayley transformation could be viewed as twospecial cases. Meanwhile, the doubling algorithm based on the proposed generalizedtransformation is presented and shown to be well-defined. Moreover, the convergenceresult and the comparison theorem on convergent rate are established. Preliminarynumerical experiments show that the doubling algorithm with the generalized transformationis efficient to derive the minimal nonnegative solution of nonsymmetric algebraicRiccati equation with $M$-matrix.