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