engleski

A Low-Rank Quadratic ADI Algorithm for Algebraic Riccati Equations

In recent years, several new approaches for solving the large-scale continuous-time algebraic Riccati equation have appeared in the literature. Amodei and Buchot suggest computing a low-dimensional invariant subspace of the associated Hamiltonian matrix. Simoncini and Lin also target the Hamiltonian matrix, but in a different way: they iterate on the Cayley-transformed matrix with various shifts. Wong and Balakrishnan directly generalize the Lyapunov ADI-method to the Riccati equation. In this talk we introduce another method, inspired by the Cholesky-factored variant of the Lyapunov ADI-method. The advantage of the new algorithm is in its immediate and efficient low-rank formulation, and a simpler implementation compared to the other three algorithms mentioned above. We show that the residuals can be easily computed, and discuss on other theoretical properties of the new method. Finally, we show that all of the seemingly different methods listed above in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.

matrix equations ; algebraic Riccati equations ; ADI iteration ; low rank approximation

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