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 Givens transformations and with a 2-norm version of the LINPACK condition estimator. An EISPACK-compatible code has been developed, and its performance is discussed. © 1987.
Van Loan, C. (1987). On estimating the condition of eigenvalues and eigenvectors. Linear Algebra and Its Applications, 88–89(C), 715–732. https://doi.org/10.1016/0024-3795(87)90131-5