Notes and References

A block version of the Lanczos method in the symmetric case appeared in the works of Cullum and Donath [89] and Golub and Underwood [197]. The so-called band Lanczos algorithm proposed by Ruhe is an implementation variation of the block Lanczos method [375]. The implementation of the block-symmetric Lanczos method by Grimes, Lewis, and Simon represents the state of the art [206]; this software is proprietary.

A block version of the nonsymmetric Lanczos method is straightforward, but has many pitfalls awaiting the unwary implementor. Bai studied a simple implementation of the block Lanczos method [27]. Aliaga et al. generalized Ruhe's band Lanczos implementation to the nonsymmetric case [5]. This is presented in §7.10.

The ABLE method for nonsymmetric eigenvalue problems proposed in this section was presented by Bai, Day, and Ye [29]. ABLE also implements a block version of the algorithm presented in [104,105], to monitor the biorthogonality loss and maintain semi-biorthogonality among the computed Lanczos vectors. A so-called $s$-step biorthogonal Lanczos method was proposed by Kim and Chronopoulus [262] for better data locality on parallel vector computers.