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Backward error, sensitivity, and refinement of computed solutions of algebraic Riccati equations. (English) Zbl 0829.65042

A new backward error criterion, together with a sensitivity measure, is presented for assessing solution accuracy of algebraic Riccati equations of the form $AX+XB-XFX+G=0$. The usual approach is to employ standard perturbation and sensitivity results for linear systems and to extend them for Riccati equations. The approach considered here is to take account of the underlying structure of these matrix equations. The backward error of an approximation to the solution is computed, and conventional perturbation theory is used to define structured condition numbers for these matrix equations. Backward error and condition numbers provide accurate estimates for the sensitivity of solutions. Optimal perturbations are used in an intuitive iterative procedure to refine approximations of actual solutions.

The results are derived for symmetric and non-symmetric algebraic Riccati equations and make it possible to establish similar results for Sylvester equations, Lyapunov equations and linear systems.

##### MSC:
 65F30 Other matrix algorithms 15A24 Matrix equations and identities 65F35 Matrix norms, conditioning, scaling (numerical linear algebra)