{"paper":{"title":"Optimal Solvers for Linear Systems with Fractional Powers of Sparse SPD Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Pencho Marinov, Raytcho Lazarov, Stanislav Harizanov, Svetozar Margenov, Yavor Vutov","submitted_at":"2016-12-14T21:25:51Z","abstract_excerpt":"In this paper we consider efficient algorithms for solving the algebraic equation ${\\mathcal A}^\\alpha {\\bf u}={\\bf f}$, $0< \\alpha <1$, where ${\\mathcal A}$ is a symmetric and positive definite matrix obtained form finite difference or finite element approximations of second order elliptic problems in ${\\mathbb R}^d$, $d=1,2,3$. The method is based on the best uniform rational approximation of the function $t^{\\beta-\\alpha}$ for $0 < t \\le 1$ and natural $\\beta$, and the assumption that one has at hand an efficient method (e.g. multigrid, multilevel, or other fast algorithm) for solving equat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04846","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"}