A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation

Authors:
Xiao-Xia Guo;Wen-Wei Lin;Shu-Fang Xu
Affiliations:
State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy o ...;Department of Mathematics, National Tsing Hua University, P.O. Box 2719, 300, Hsinchu, Taiwan;LMAM, School of Mathematical Sciences, Peking University, P.O. Box 2719, 100871, Beijing, China
Venue:
Numerische Mathematik
Year:
2006

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Abstract

In this paper, we propose a structure-preserving doubling algorithm (SDA) for the computation of the minimal nonnegative solution to the nonsymmetric algebraic Riccati equation (NARE), based on the techniques developed for the symmetric cases. This method allows the simultaneous approximation to the minimal nonnegative solutions of the NARE and its dual equation, requiring only the solutions to two linear systems and several matrix multiplications per iteration. Similar to Newton's method and the fixed-point iteration methods for solving NAREs, we also establish global convergence for SDA under suitable conditions, using only elementary matrix theory. We show that sequences of matrices generated by SDA are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the NARE and its dual equation. Numerical experiments show that the SDA algorithm is feasible and effective, and outperforms Newton's iteration and the fixed-point iteration methods.