From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms

2022/01/26 by Bini, Dario A., Meini, Beatrice, Poloni, Federico
#Algebraic Riccati Equation #Cyclic Reduction #Matrix Equation #Structured doubling algorithm

paper · doi:10.4230/dagsemproc.07461.7

Abstract

The problem of reducing an algebraic Riccati equation XCX-AX-XD+B=0 to a unilateral quadratic matrix equation (UQME) of the kind PX²+QX+R 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.D.O. Anderson [Internat. J. Control, 1978] is nothing else but the cyclic reduction algorithm applied to a suitable UQME. A new algorithm obtained by complementing our reductions with the shrink-and-shift tech- nique of Ramaswami is presented. Finally, faster algorithms which require some non-singularity conditions, are designed. The non-singularity re- striction is relaxed by introducing a suitable similarity transformation of the Hamiltonian.

Related

• Order reduction approaches for the algebraic Riccati equation and the LQR problem
• The Shifting Technique for Solving a Nonsymmetric Algebraic Riccati Equation
• Low-rank updates and divide-and-conquer methods for quadratic matrix equations
• A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory
• Iterative and doubling algorithms for Riccati-type matrix equations: a\n comparative introduction