A method is developed for estimating the accuracy of computed eigenvalues and eigenvectors that are obtained via certain EISPACK subroutines. It does this at a cost of $O(n2)$ flops per eigenpair assuming that the eigenpair is known and assuming that the original matrix has been reduced to Hessenberg form. The heart of the technique involves estimating the smallest singular value of a certain nearly triangular submatrix. This is accomplished by some standard "zero-chasing" with Hivens transformations and with a 2-norm version of the LINPACK condition estimator. An EISPACK compatible code has been developed and its performance is discussed. Suggestions for extending the current work to general invariant subspaces and to the generalized eigenvalue problem are given.