Implicitly Restarted Lanczos Method
  R. Lehoucq and D. Sorensen

The Lanczos process for a Hermitian matrix $A=A^{\ast}$ has been derived previously in §4.4. Here, we discuss how to apply implicit restart. Our starting point is a $k$-step Lanczos factorization (4.10):

$$A V_k = V_k T_k + r_k e_k^{\ast},$$

where $V_k \in {\cal C}^{n \times k}$ has orthonormal columns, $V_k^{\ast} r_k = 0$, and $T_k \in {\cal R}^{k \times k}$ is real, symmetric, and tridiagonal with nonnegative subdiagonal elements. The columns of $V_k$ are referred to as the Lanczos vectors. For implicit restart it is important that the columns of $V_k$ be made orthogonal to full accuracy.