{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:I3BMRA4HHZEBA2ESSO7FG262NC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b01bc2dbf3e1d1fb8b91ca8c2878f68f4be38a625847a8e2147a076f76f0583a","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-01-07T23:35:48Z","title_canon_sha256":"877526d7292e8b916a4b9beea68bace0c8aa3e7c9af478f222cc6cac706a184d"},"schema_version":"1.0","source":{"id":"2401.03608","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.03608","created_at":"2026-07-05T07:31:11Z"},{"alias_kind":"arxiv_version","alias_value":"2401.03608v1","created_at":"2026-07-05T07:31:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.03608","created_at":"2026-07-05T07:31:11Z"},{"alias_kind":"pith_short_12","alias_value":"I3BMRA4HHZEB","created_at":"2026-07-05T07:31:11Z"},{"alias_kind":"pith_short_16","alias_value":"I3BMRA4HHZEBA2ES","created_at":"2026-07-05T07:31:11Z"},{"alias_kind":"pith_short_8","alias_value":"I3BMRA4H","created_at":"2026-07-05T07:31:11Z"}],"graph_snapshots":[{"event_id":"sha256:85bff6aaa540b31792ad6e95f6b8f8cef88eb5945c2d35c23cc0e95023997fca","target":"graph","created_at":"2026-07-05T07:31:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2401.03608/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop the ultraspherical rectangular collocation (URC) method, a collocation implementation of the sparse ultraspherical method of Olver \\& Townsend for two-point boundary-value problems. The URC method is provably convergent, the implementation is simple and efficient, the convergence proof motivates a preconditioner for iterative methods, and the modification of collocation nodes is straightforward. The convergence theorem applies to all boundary-value problems when the coefficient functions are sufficiently smooth and when the roots of certain ultraspherical polynomials are used as col","authors_text":"Thomas Trogdon","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-01-07T23:35:48Z","title":"The ultraspherical rectangular collocation method and its convergence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.03608","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5d37de30c33382ea69726762dc7b58b9f9c77586ae81b3395409f450948514f3","target":"record","created_at":"2026-07-05T07:31:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b01bc2dbf3e1d1fb8b91ca8c2878f68f4be38a625847a8e2147a076f76f0583a","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-01-07T23:35:48Z","title_canon_sha256":"877526d7292e8b916a4b9beea68bace0c8aa3e7c9af478f222cc6cac706a184d"},"schema_version":"1.0","source":{"id":"2401.03608","kind":"arxiv","version":1}},"canonical_sha256":"46c2c883873e4810689293be536bda68b5f7ba721f2549861d5e02476cafb2ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"46c2c883873e4810689293be536bda68b5f7ba721f2549861d5e02476cafb2ad","first_computed_at":"2026-07-05T07:31:11.479493Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:31:11.479493Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/6/hKTdPq7eKcflzKQaaLEpez+NQ/ZoR7N/wa9cfHLHhvb4x1rQDCf4//OfJHQkOTww+qCD9CH14w8zLFfEUAA==","signature_status":"signed_v1","signed_at":"2026-07-05T07:31:11.479923Z","signed_message":"canonical_sha256_bytes"},"source_id":"2401.03608","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d37de30c33382ea69726762dc7b58b9f9c77586ae81b3395409f450948514f3","sha256:85bff6aaa540b31792ad6e95f6b8f8cef88eb5945c2d35c23cc0e95023997fca"],"state_sha256":"a588bf083d2145454e7c8523e4d98e85174a817bbcaebe7c5879eba1b48b1ba2"}