SOME NUMERICAL CONSIDERATIONS AND NEWTON METHOD REVISITED FOR SOLVING ALGEBRAIC RICCATI-EQUATIONS

In this note, we analyze some of the numerical aspects involved in solving the algebraic Riccati equation (ARE). Our analysis applies to both the symmetric and unsymmetric cases. We reconsider the numerically relevant problems of balancing the ARE and the conditioning properties of the ARE, and see how these can be exploited by a solution algorithm. We also propose an estimator for the condition number of the Sylvester equation AX + XB = C based on interative refinement. Also, we interpret Newton's method as a sequence of similarity transformations on the underlying system matrix. On the one hand, this closes the gap between so-called global and iterative methods for solving the ARE, and on the other hand it also suggests a revised implementation of Newton's method altogether. One of the advantages of this revised implementation is that, in case Newton's method converges to a solution different than the desired solution, enough information emerges so to allow a switch to the desired solution. We examine the roundoff properties of the new algorithm and provide implementation considerations and numerical examples to highlight pros and cons.