From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms
The problem of reducing an algebraic Riccati equation XCX − AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind P X 2 + QX + R = 0 is analyzed. New reductions are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm of B. a suitable UQME. A new algorithm obtained by complementing our reductions with the shrink-and-shift technique of Ramaswami is presented. Finally, faster algorithms which require some non-singularity conditions, are designed. The non-singularity restriction is relaxed by introducing a suitable similarity transformation of the Hamiltonian.