Some numerical considerations and Newton´s method revisited for solving algebraic Riccati equations

Analyzes some of the numerical aspects of solving the algebraic Riccati equation (ARE). This analysis applies to both the symmetric and unsymmetric cases. The author reconsiders the numerically relevant problems of balancing the ARE and the conditioning properties of the ARE and shows how these can be exploited by a solution algorithm. He proposes an estimator for the condition number of the Sylvester equation AX+XB=C based on iterative refinement. Also, he interprets Newton´s method as a sequence of similarity transformations on the underlying system matrix. This closes the gap between so-called global and iterative methods for solving the ARE and also suggests an altogether revised implementation of Newton´s method. One of the advantages of this revised implementation is that, in the case where Newton´s method converges to a solution different from the desired solution, enough information emerges to allow a switch to the desired solution. The author examines the roundoff properties of the new algorithm and provides implementation considerations and numerical examples to highlight pros and cons