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@MISC{,
            author = {Liu, Xuefeng and Vejchodsk{\'{y}}, Tom{\'{a}}{\v s}},
             title = {Rigorous and fully computable a posteriori error bounds for eigenfunctions},
              year = {2019},
          abstract = {Guaranteed a posteriori estimates on the error of approximate
eigenfunctions in both energy and L2 norms are derived for the Laplace
eigenvalue problem. The problem of ill-conditioning of eigenfunctions
in case of tight clusters and multiple eigenvalues is solved by estimat-
ing the directed distance between the spaces of exact and approximate
eigenfunctions. The error estimates for approximate eigenfunctions
are based on rigorous lower and upper bounds on eigenvalues. Such
eigenvalue bounds can be computed for example by the nite element
method along with the recently developed explicit error estimation
[24] and the Lehmann{Goerisch method. The eciency of the derived
error bounds for eigenfunctions is illustrated by numerical examples.},
  Preprint project = {NCMM},
     Preprint year = {2019},
   Preprint number = {02},
       Preprint ID = {NCMM/2019/02}
}