The block Lanczos method, to be described in this section, is a variation of the Lanczos procedure as presented in §7.8. It uses block versions of the three-term recursions (7.34) and (7.33). As a general thrust, block algorithms substitute matrix block multiplies and block solvers for matrix-vector products and simple solvers in unblocked algorithms. In other words, higher level BLAS are used in the inner loop of the block algorithms. This decreases the I/O costs essentially by a factor of the block size. In addition, the block algorithms are generally more robust and efficient for matrices with multiple or closely clustered eigenvalues. The adaptive block method, called ABLE, presented in this section is designed to overcome possible breakdowns and to eliminate the causes of slow convergence by either the loss of biorthogonality of the computed Lanczos vectors or a block size that is smaller than the size of the cluster of desired eigenvalues. The ABLE method was first presented in [29] by Bai, Day, and Ye.
In §7.10, a band Lanczos method is presented. It is an alternative formulation of a block Lanczos method. In this variant, in contrast to the methods described in this section, the sizes of the right and left blocks of Lanczos vectors need not necessarily be the same. In particular, this method can be applied when the right and left starting blocks have different sizes, a situation that often arises in reduced-order modeling of linear dynamical systems, a topic closely related to matrix eigenvalue problems.