Fully computable a posteriori error bounds for eigenfunctions

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both apply well the case of tight clusters and multiple eigenvalues, under settings target eigenvalue problems. Algorithm I is based on Rayleigh quotient min-max principle that characterizes The formula provided by easy compute applies problems with limited information quotients. II, as an extension Davis–Kahan method, takes advantage dual formulation differential along Prager–Synge technique provides greatly improved accuracy estimate, especially finite element approximations eigenfunctions. Numerical examples matrices Laplace over convex non-convex domains illustrate efficiency algorithms.