Condition Numbers On Symmetric Algebraic Riccati Equations And Its Estimation

Symmetric algebraic Riccati equations, including the continuous-time and discrete-time types, are non-linear matrix equations that arise in the context of infinite-horizonoptimal control problems in continuous time or discrete time. In numerical analysis, con-dition numbers describe the ill-posedness of the problems with respect to perturbations oninput data when we solve them by means of numerical algorithms. Normwise conditionnumbers are classical that do not take account of the relevances between perturbationsand their corresponding input data. In general normwise condition numbers can not tellthe true conditioning of the problems. From1980s, componentwise perturbation analysishas been introduced for linear systems, linear least squares and eigenvalue problems. Theresulted componentwise condition numbers can gauge the conditioning of the problem. Inthis thesis, we will adopt componentwise perturbation analysis to study symmetric alge-braic Riccati equations, especially because of the structure of the coefficient matrices, werestrict the perturbations have the same structure. The structured componentwise condi-tion numbers for symmetric algebraic Riccati equations are defined, and their expressionsare derived. Based on statistical condition estimation, we devise small sample conditionalgorithms to estimate the condition numbers for symmetric algebraic Riccati equations.Numerical examples show that our expressions and algorithms are reliable.