A New Subspace Iteration method for the Algebraic Riccati Equation

We consider the numerical solution of the continuous algebraic Riccati equation $A^{\ast }X+XA-XFX+G=0$, with $F=F^{\ast },G=G^{\ast }$ 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.