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In the numerical treatment of large-scale Sylvester and Lyapunov equations, projection methods require solving a reduced problem to check convergence. As the approximation space expands, this solution takes an increasing portion of the overall computational effort. When data are symmetric, we show that the Frobenius norm of the residual matrix can be computed at significantly lower cost than with available methods, without explicitly solving the reduced problem. For certain classes of problems, the new residual norm expression combined with a memory-reducing device make classical Krylov strategies competitive with respect to more recent projection methods. Numerical experiments illustrate the effectiveness of the new implementation for standard and extended Krylov subspace methods. [ABSTRACT FROM AUTHOR]

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]