Local Reorthogonalization and Detecting Spurious Ritz Values.

If we choose, or are forced to, use only local reorthogonalization, some of the eigenvalues of the tridiagonal matrix $T$ will be new copies of already converged eigenvalues and we will also get spurious eigenvalues of $T$. Such an eigenvalue occurs suddenly at a certain step $j$ only to disappear at the next step.

Cullum [90] has devised a way to weed out such extra copies and spurious values. She takes the tridiagonal matrix $T_j$ and another $\hat{T}_2$, which is obtained from $T_j$ by deleting the first row and column. All eigenvalues of $T_j$ that are very close to eigenvalues of $\hat{T}_2$ need special consideration. If such an eigenvalue is a multiple eigenvalue of $T_j$, 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 $T_j$ is also an eigenvalue of $\hat{T}_2$, it is spurious and should be discarded.