A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation

@article{Bini2008AFN,
  title={A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation},
  author={Dario Bini and Bruno Iannazzo and Federico Poloni},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2008},
  volume={30},
  pages={276-290}
}
A special instance of the algebraic Riccati equation $XCX-XE-AX+B=0$ where the $n\times n$ matrix coefficients $A,B,C,E$ are rank structured matrices is considered. Relying on the structural properties of Cauchy-like matrices, an algorithm is designed for performing the customary Newton iteration in $O(n^2)$ arithmetic operations (ops). The same technique is used to reduce the cost of the algorithm proposed by L.-Z. Lu in [Numer. Linear Algebra Appl., 12 (2005), pp. 191-200] from $O(n^3)$ to $O… 

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