The inexact Newton-Kleinman method is an iterative scheme for numerically solving large scale algebraic Riccati equations. At each iteration, the approximate solution of a Lyapunov linear equation is required. Specifically designed projection of the Riccati equation onto an iteratively generated approximation space provides a possible alternative. Our numerical experiments with enriched approximation spaces seem to indicate that this latter approach is superior to Newton-type strategies on

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... stic problems, thus giving experimental ground for recent developments in this direction. As part of an explanation of why this is so, we derive several matrix relations between the iterates produced by the same projection approach applied to both the (quadratic) Riccati equation and its linear counterpart, the Lyapunov equation.