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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.


Susan Blackford 2000-11-20