Cullum [90] has devised a way to weed out such extra copies and spurious values. She takes the tridiagonal matrix and another , which is obtained from by deleting the first row and column. All eigenvalues of that are very close to eigenvalues of need special consideration. If such an eigenvalue is a multiple eigenvalue of , keep one of them and discard the rest as copies, remembering that an unreduced tridiagonal matrix by definition has only simple eigenvalues. If a simple eigenvalue of is also an eigenvalue of , it is spurious and should be discarded.