Applications
Lanczos algorithms are very attractive because the multiplication by is the only large-scale linear operation. Since weighted-term text retrieval engines implement just this operation, the Lanczos algorithm can be applied efficiently to text documents (see Latent Semantic Indexing). Eigenvectors are also important for large-scale ranking methods such as the HITS algorithm developed by Jon Kleinberg, or the PageRank algorithm used by Google.
Lanczos algorithms are also used in Condensed Matter Physics as a method for solving Hamiltonians of strongly correlated electron systems.
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