  We consider the numerical solution of the continuous algebraic Riccati
equation $A^*X+XA-XFX+G=0$, with $F=F^*, G=G^*$ of low rank and $A$ large and
sparse. We develop an algorithm for the low rank approximation of $X$ by means
of an invariant subspace iteration on a function of the associated Hamiltonian
matrix. We show that the sought after approximation can be obtained by a low
rank update, in the style of the well known ADI iteration for the linear
equation, from which the new method inherits many algebraic properties.
Moreover, we establish new insightful matrix relations with emerging
projection-type methods, which will help increase our understanding of this
latter class of solution strategies.
