{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:JB3XVCVW3O3S57XYHTOLKD7XZH","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":"ccbe56399132400b1739f5064a145bfba74bda3162da18a2c7345d990ec6be97","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NA","submitted_at":"2018-08-08T17:48:19Z","title_canon_sha256":"a1b4097d30e4657cb0e8bd0be2b31d28209edf97768bf44cae3e8abe66b28fd0"},"schema_version":"1.0","source":{"id":"1808.03229","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.03229","created_at":"2026-05-18T00:08:29Z"},{"alias_kind":"arxiv_version","alias_value":"1808.03229v1","created_at":"2026-05-18T00:08:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.03229","created_at":"2026-05-18T00:08:29Z"},{"alias_kind":"pith_short_12","alias_value":"JB3XVCVW3O3S","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"JB3XVCVW3O3S57XY","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"JB3XVCVW","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:f3ee5a3327bccd69d9ba262e8deda1213a15f4bc4a118e1948d5c1950880f20e","target":"graph","created_at":"2026-05-18T00:08:29Z","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"},"paper":{"abstract_excerpt":"In the paper \"A Chaotic Search for $i$\"~(\\cite{strang1991chaotic}), Strang completely explained the behaviour of Newton's method when using real initial guesses on $f(x) = x^{2}+1$, which has only a pair of complex roots $\\pm i$. He explored an exact symbolic formula for the iteration, namely $x_{n}=\\cot{ \\left( 2^{n} \\theta_{0} \\right) }$, which is valid in exact arithmetic. In this paper, we extend this to to $k^{th}$ order Householder methods, which include Halley's method, and to the secant method. Two formulae, $x_{n}=\\cot{ \\left( \\theta_{n-1}+\\theta_{n-2} \\right) }$ with $\\theta_{n-1}=\\m","authors_text":"Ao Li, Robert M. Corless","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NA","submitted_at":"2018-08-08T17:48:19Z","title":"Revisiting Gilbert Strang's \"A Chaotic Search for $i$\""},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.03229","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:aa1f5ce4b2192541af2e7ad3f6031c9b306691cd5687b84fcfe6cbba3dda83f4","target":"record","created_at":"2026-05-18T00:08:29Z","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":"ccbe56399132400b1739f5064a145bfba74bda3162da18a2c7345d990ec6be97","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NA","submitted_at":"2018-08-08T17:48:19Z","title_canon_sha256":"a1b4097d30e4657cb0e8bd0be2b31d28209edf97768bf44cae3e8abe66b28fd0"},"schema_version":"1.0","source":{"id":"1808.03229","kind":"arxiv","version":1}},"canonical_sha256":"48777a8ab6dbb72efef83cdcb50ff7c9c72e6d9c3e0c79d6040e689c7606f649","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48777a8ab6dbb72efef83cdcb50ff7c9c72e6d9c3e0c79d6040e689c7606f649","first_computed_at":"2026-05-18T00:08:29.852732Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:29.852732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0qlvZwN4dCLYBZGWKw85Qrb/uECjwy+8K7zdeP1jgUPtBDVnxhb6O/PGT4Ki2/w5kq42NcC73lCQF2AzPXIFBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:29.853157Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.03229","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa1f5ce4b2192541af2e7ad3f6031c9b306691cd5687b84fcfe6cbba3dda83f4","sha256:f3ee5a3327bccd69d9ba262e8deda1213a15f4bc4a118e1948d5c1950880f20e"],"state_sha256":"6f22286923e90fa11f80f6acf4cc7599361bc384fd6f2485c9b6036adde93383"}