{"paper":{"title":"An algorithmic approach to direct spline products: procedures and computational aspects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Direct B-spline products are computed robustly by recasting the formula via the Oslo algorithm and factoring terms to cut costs.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Alessandra Sestini, Francesco Patrizi","submitted_at":"2026-01-24T12:07:27Z","abstract_excerpt":"We introduce an efficient algorithmic procedure for implementing the direct formula that represents the product of splines in the B-spline basis. We first demonstrate the relevance of this direct approach through numerical evidence showing that implicit methods, such as collocation, may fail in some instances due to severe ill-conditioning of the associated system matrices, whereas the direct formula remains robust. We then recast the direct formula into an algorithmic framework based on the Oslo Algorithm and subsequently enhance it, through a factorization of the terms to be computed, to dra"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We then recast the direct formula into an algorithmic framework based on the Oslo Algorithm and subsequently enhance it, through a factorization of the terms to be computed, to dramatically improve computational efficiency.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the factorization step preserves numerical stability for all knot vectors and degrees without introducing new cancellation errors or requiring extra safeguards beyond those discussed.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A factorization-enhanced Oslo-algorithm procedure computes B-spline products directly, avoiding ill-conditioned systems and cutting computational cost compared with collocation.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Direct B-spline products are computed robustly by recasting the formula via the Oslo algorithm and factoring terms to cut costs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"04e5ae67bb7fd18215027f94947058865f16ff897b5dc8ddb4a65177d1f7d13a"},"source":{"id":"2601.17432","kind":"arxiv","version":2},"verdict":{"id":"433a91de-c76e-4144-99e9-812b692b3469","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T11:28:51.262716Z","strongest_claim":"We then recast the direct formula into an algorithmic framework based on the Oslo Algorithm and subsequently enhance it, through a factorization of the terms to be computed, to dramatically improve computational efficiency.","one_line_summary":"A factorization-enhanced Oslo-algorithm procedure computes B-spline products directly, avoiding ill-conditioned systems and cutting computational cost compared with collocation.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the factorization step preserves numerical stability for all knot vectors and degrees without introducing new cancellation errors or requiring extra safeguards beyond those discussed.","pith_extraction_headline":"Direct B-spline products are computed robustly by recasting the formula via the Oslo algorithm and factoring terms to cut costs."},"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"}