Flexible fixed-point iteration and its applications for nonsymmetric algebraic Riccati equations
Zhen-Chen Guo, Xin Liang
TL;DR
This paper studies the nonsymmetric algebraic Riccati equation (NARE) $\mathscr{R}(X)=XCX-XD-AX+B=0$ and shows that its unique stabilizing solution can be characterized by a Toeplitz-structured closed form under low-rank factorizations of $B$ and $C$. It develops a shift-involved fixed-point iteration based on a generalized Cayley transformation, yielding a Toeplitz-structured closed form and motivating a nonsymmetric RADI-type method with flexible shifts to handle large-scale sparse problems. The approach integrates low-rank residual factorizations, implicit updates of large matrices, and real-arithmetic implementations for complex shifts, and extends to generalized and strengthened NAREs. Numerical experiments on large-scale problems validate the method and highlight effective shift strategies such as generalized Leja points. Overall, the work contributes scalable tools for NAREs and suggests applications to related eigenvalue problems like Bethe–Salpeter equations.
Abstract
In this paper, we reveal the intrinsic Toeplitz structure in the unique stabilizing solution for nonsymmetric algebraic Riccati equations by employing a shift-involved fixed-point iteration, and propose an RADI-type method for computing this solution for large-scale equations of this type with sparse and low-rank structure by incorporating flexible shifts into the fixed-point iteration. We present a shift-selection strategy, termed Leja shifts, based on rational approximation theory, which is incorporated into the RADI-type method. We further discuss important implementation aspects for the method, such as low-rank factorization of residuals, implicit update of large-scale sparse matrices, real arithmetics with complex shifts, and related equations of other type. Numerical experiments demonstrate the efficiency of both the proposed method and the introduced shift-selection strategy.