{"paper":{"title":"N derivatives are necessary for order N+1 convergence in quadrature: a converse result","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Jeffrey Tsang","submitted_at":"2014-01-28T05:14:00Z","abstract_excerpt":"Results on the error bounds of quadrature methods are well known - most state that if the method has degree N, and the integrand has N derivatives, then the error is order N+1. We prove here a converse: that if the integrand fails to have N derivatives, even only at a finite number of points, no method, regardless of its degree, can guarantee convergence more than order N. Even if the integrand fails to have N derivatives at just 3 (for even N, 2) points, no method can produce order more than N+1 convergence. This is done by an adversarial proof: we explicitly construct the functions that exhi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7086","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"}