{"paper":{"title":"Randomized algorithms for Generalized Hermitian Eigenvalue Problems with application to computing Karhunen-Lo\\`{e}ve expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Arvind K. Saibaba, Jonghyun Lee, Peter K. Kitanidis","submitted_at":"2013-07-25T22:13:42Z","abstract_excerpt":"We describe randomized algorithms for computing the dominant eigenmodes of the Generalized Hermitian Eigenvalue Problem (GHEP) $Ax=\\lambda Bx$, with $A$ Hermitian and $B$ Hermitian and positive definite. The algorithms we describe only require forming operations $Ax$, $Bx$ and $B^{-1}x$ and avoid forming square-roots of $B$ (or operations of the form, $B^{1/2}x$ or $B^{-1/2}x$). We provide a convergence analysis and a posteriori error bounds that build upon the work of~\\cite{halko2011finding,liberty2007randomized,martinsson2011randomized} (which have been derived for the case $B=I$). Additiona"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6885","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}