{"paper":{"title":"Piecewise linear secant approximation via Algorithmic Piecewise Differentiation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Andreas Griewank, Lutz Lehmann, Manuel Radons, Richard Hasenfelder, Tom Streubel","submitted_at":"2017-01-16T17:43:54Z","abstract_excerpt":"It is shown how piecewise differentiable functions $F: \\mathbb R^n \\mapsto \\mathbb R^m $ that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points $\\check x$ and $\\hat x$. We show that the discrepancy between function and model at any point $x$ is of the bilinear order $O(\\|x-\\check x\\| \\|x-\\hat x\\|)$. This is a little surprising since $x \\in \\mathbb R^n$ may vary over the whole Euclidean space, and we utilize only two function samples $\\check F=F(\\check x)$ and $\\hat F=F(\\hat x)$, as well as the intermediates computed duri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04368","kind":"arxiv","version":2},"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"}