Notes and References

Lanczos's original 1950 paper studied the eigenvalue problem $Ax = \lambda x$ for a general matrix $A$ [285].Cullum and Willoughby [93] demonstrated the success of the nonsymmetric Lanczos method with no reorthogonalization for a number of nonsymmetric eigenproblems from applications. Special devices were proposed for solving the eigenvalue problem of the reduced tridiagonal matrices and to identify good and spurious Ritz values.

Taylor [434] and Parlett, Taylor, and Liu [364] were the first to attempt to fix the breakdown problem. The terminology ``look-ahead'' was coined in their papers. A new study of the look-ahead scheme in order to cure breakdown started with the work of Freund, Gutknecht, and Nachtigal [178]. Ye also proposed a scheme to cure the breakdown by choosing a new starting vector to generate another Krylov subspace that includes the old one in an appropriate way [465]. The basic idea is incorporated in the ABLE method proposed in 1995 by Bai, Day, and Ye [29] (see §7.9). Theoretical studies of the breakdown phenomena and connections with the underlying polynomial theory can be found in papers by Gutknecht [213,214] and Brezinski, Redivo Zaglia, and Sadok [65].

Since the Lanczos method is an oblique projection method, the general convergence analysis of projection methods by Saad [384] and Jia [243] is applicable, with significant limitations in practical usage.

An analogue of Paige's theory on the relationship between the loss of orthogonality among the computed Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos method is extended to the nonsymmetric Lanczos method by Bai [26]. Day has published an error analysis of the nonsymmetric Lanczos process in finite precision arithmetic [104,105]. A number of well-known results on the symmetric Lanczos method are extended to the nonsymmetric case. For example, a nonsymmetric counterpart of Simon's partial reorthogonalization scheme for symmetric Lanczos [403] is presented. A partial reorthogonalization scheme was also published recently by van der Veen and Vuik [443].