Guaranteed Eigenfunction Computation

This chapter is devoted to the guaranteed bounds for the approximation error encountered in eigenfunction calculations. The chapter explores three distinct algorithms depending on problem settings: the Rayleigh quotient-based algorithm, the residual-based algorithm, and the projection-based algorithm. Particular emphasis is placed on the residual-based error estimation, including an in-depth discussion on the Davis-Kahan theorem extended to weakly formulated eigenvalue problems. These algorithms have a common feature: they consider the angle between the exact eigenspace and the approximate one, thereby enabling them to handle cases where eigenvalues are tightly clustered.