## A multishift algorithm for the numerical solution of algebraic Riccati equations

Gregory Ammar, Peter Benner, and Volker Mehrmann

### Abstract

We study an algorithm for the numerical solution of algebraic matrix Riccati equations that arise in linear optimal control problems. The algorithm can be considered to be a multishift technique, which uses only orthogonal symplectic similarity transformations to compute a Lagrangian invariant subspace of the associated Hamiltonian matrix. We describe the details of this method and compare it with other numerical methods for the solution of the algebraic Riccati equation.

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### Key words

algebraic matrix Riccati equation, Hamiltonian matrix, Lagrangian invariant subspace.

### AMS subject classifications

65F15, 15A24, 93B40.

 Vol. 8 (1999), pp. 115-126 Peter Benner, Volker Mehrmann, and Hongguo Xu: A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems Vol. 26 (2007), pp. 121-145 H. Faßbender: The parametrized $SR$ algorithm for Hamiltonian matrices