{"paper":{"title":"Quadrature as a least-squares and minimax problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"M\\'ario M. Gra\\c{c}a","submitted_at":"2012-06-01T19:26:08Z","abstract_excerpt":"The vector of weights of an interpolatory quadrature rule with $n$ preassigned nodes is shown to be the least-squares solution $\\omega$ of an overdetermined linear system here called {\\em the fundamental system} of the rule. It is established the relation between $\\omega$ and the minimax solution $\\stackrel{\\ast}{z}$ of the fundamental system, and shown the constancy of the $\\infty$-norms of the respective residual vectors which are equal to the {\\em principal moment} of the rule. Associated to $\\omega$ and $\\stackrel{\\ast}{z}$ we define several parameters, such as the angle of a rule, in orde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0281","kind":"arxiv","version":1},"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"}