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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 ondoi:10.1093/imanum/drt015 fatcat:5n3rrax5inerpa7jtpbj3kdxhm