Backward error,sensitivity, and refinement of computed solutions of algebraic Riccati equations

In this paper, a new backward error criterion, together with a sensitivity measure, is presented for assessing solution accuracy of nonsymmetric and symmetric algebraic Riccati equations (AREs). The usual approach to assessing reliability of computed solutions is to employ standard perturbation and sensitivity results for linear systems and to extend them further to AREs. However, such methods are not altogether appropriate since they do not take account of the underlying structure of these matrix equations. The approach considered here is to first compute the backward error of a computed solution X? that measures the amount by which data must be perturbed so that X? is the exact solution of the perturbed original system. Conventional perturbation theory is used to define structured condition numbers that fully respect the special structure of these matrix equations. The new condition number, together with the backward error of computed solutions, provides accurate estimates for the sensitivity of solutions. Optimal perturbations are then used in an iterative refinement procedure to give further more accurate approximations of actual solutions. The results are derived in their most general setting for nonsymmetric and symmetric AREs. This in turn offers a unifying framework through which it is possible to establish similar results for Sylvester equations, Lyapunov equations, linear systems, and matrix inversions.