Summary: We present an a posteriori finite element procedure that provides rigorous, constant-free, asymptotic lower and upper bounds for smooth nonlinear-functional outputs of general elliptic partial differential equations. This new abstract framework includes not only our earlier bound procedures for coercive linear (symmetric or nonsymmetric), noncoercive linear (e.g., Helmholtz), and nonlinear (e.g., Burgers) equations, but also the (symmetric) generalized eigenvalue problem. The latter -- which provides a posteriori bounds both for the eigenvalues and for functionals of the eigenvectors -- is described in detail, and sustained by illustrative numerical results.