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Description
Consider the algebraic Riccati equation
$$A^{*}X + XA + C^*C - XBB^{*}X = 0$$
with large and sparse $A\in \mathbb{C}^{n\times n}$ and $B\in \mathbb{C}^{n\times m}$, $C\in \mathbb{C}^{p\times n}$. The goal is to find an approximate solution $X$ such that the rank of the Riccati residual remains equal to $p$. A method to determine an approximate solution to Riccati equations with this rank property is given by the RADI method [1], a generalization of the ADI method for linear matrix equations.
We present a new approach in which solves with $(A^*-\mu I_n)$ and the explicitly known residual factors are sufficient to obtain the unique low-rank residual solution.
[1] Benner, P., Bujanović, Z., Kürschner, P. et al. RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations. Numer. Math. (2018) 138: 301. https://doi.org/10.1007/s00211-017-0907-5
