Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces

We consider the numerical solution of the projected nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newton's method, arising in the refinement of estimates of invariant (or deflating subspaces) for a large and sparse real matrix A (or pencil A−λB). The engine of the method is the inversion of the matrix P₂P₂^⊤A−γI_n or P_l2P_l2^⊤(A−γB), for some orthonormal P₂ or P_l2 from R^n×(n−m), making use of the structures in A or A−λB and the Sherman–Morrison–Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples.