Title: Accurately Computing Eigenvalues of Extremely Ill-conditioned Matrices, with an Application to the Biharmonic Operator.



Abstract: We are primarily concerned with computing smaller eigenvalues of large, extremely ill-conditioned matrices. After discussing where the standard algorithms fail to compute such eigenvalues with any accuracy, we offer a solution to the problem for diagonally dominant matrices. We will then apply this solution to accurately compute an eigenvalue of the biharmonic operator on the unit circle.