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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 − λ B ). The engine of the method is the inversion of the matrix P 2 P 2 ⊤ A − γ I n or P l 2 P l 2 ⊤ ( A − γ B ) , for some orthonormal P 2 or P l 2 from R n × ( n − m ) , making use of the structures in A or A − λ B and the Sherman–Morrison–Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples. [ABSTRACT FROM AUTHOR]