{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:UQNUJGPZAIMCXJZWZQEJKUZRR3","short_pith_number":"pith:UQNUJGPZ","canonical_record":{"source":{"id":"2212.00302","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2022-12-01T06:19:47Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"468a653a5a4c3e5d9f1a25d69f9cc5832e2e7e5d504a3bba89f723156f7c1bd9","abstract_canon_sha256":"0a4cd9523f9c72a2e031f6d9aaa677f3e3f5a61dab583470384b330871c1f09c"},"schema_version":"1.0"},"canonical_sha256":"a41b4499f902182ba736cc089553318eeb5780724f053fa902503a5065ea5ab5","source":{"kind":"arxiv","id":"2212.00302","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2212.00302","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"arxiv_version","alias_value":"2212.00302v3","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2212.00302","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"pith_short_12","alias_value":"UQNUJGPZAIMC","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"UQNUJGPZAIMCXJZW","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"UQNUJGPZ","created_at":"2026-05-18T12:33:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:UQNUJGPZAIMCXJZWZQEJKUZRR3","target":"record","payload":{"canonical_record":{"source":{"id":"2212.00302","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2022-12-01T06:19:47Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"468a653a5a4c3e5d9f1a25d69f9cc5832e2e7e5d504a3bba89f723156f7c1bd9","abstract_canon_sha256":"0a4cd9523f9c72a2e031f6d9aaa677f3e3f5a61dab583470384b330871c1f09c"},"schema_version":"1.0"},"canonical_sha256":"a41b4499f902182ba736cc089553318eeb5780724f053fa902503a5065ea5ab5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:47.456563Z","signature_b64":"w2rNlTVCcFFvKYFovxeDE5i5i5RAXbUWNF6CLl1pRre6vfuJnLVi3bV0DwejYVv60L9o0WjzWe6SG4J6/QzEBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a41b4499f902182ba736cc089553318eeb5780724f053fa902503a5065ea5ab5","last_reissued_at":"2026-05-18T03:09:47.455649Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:47.455649Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2212.00302","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:09:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z5rYKpajzWJBFo7SVOiXh3TV/q3PNINUVVZOvT2K5KzSsd8H2HQyOGR3VvepdNeykZeia6S0iE1PNUvG67LdCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T14:21:22.676162Z"},"content_sha256":"510bd6f2ec12ccf0ecc6b4d6dbd3422b45684d4a2e3dee50171f355999237384","schema_version":"1.0","event_id":"sha256:510bd6f2ec12ccf0ecc6b4d6dbd3422b45684d4a2e3dee50171f355999237384"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:UQNUJGPZAIMCXJZWZQEJKUZRR3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An analysis of the Rayleigh-Ritz and refined Rayleigh-Ritz methods for regular nonlinear eigenvalue problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Qingqing Zheng, Zhongxiao Jia","submitted_at":"2022-12-01T06:19:47Z","abstract_excerpt":"We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(\\lambda_{*},x_{*})$ of a given analytic regular nonlinear eigenvalue problem (NEP). In terms of the deviation $\\varepsilon$ of $x_{*}$ from a given subspace $\\mathcal{W}$, we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector. The results show that, as $\\varepsilon\\rightarrow 0$, there exists a Ritz value that unconditionally converges to $\\lambda_*$ and the corresponding refined Ritz vector does so"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.00302","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:09:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vApKz/xzPCsLBvBCX5v22i4iPE+Q5Ck61RhVJK/f1Js2Z8n2zlAfiSiNRXFZtlQAR94QWkwcsx8CIzbhei1uCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T14:21:22.676649Z"},"content_sha256":"77625e339f1c6ca892c2da9b90c96d808fdb4defb8ded9aba6f93f300115fa18","schema_version":"1.0","event_id":"sha256:77625e339f1c6ca892c2da9b90c96d808fdb4defb8ded9aba6f93f300115fa18"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UQNUJGPZAIMCXJZWZQEJKUZRR3/bundle.json","state_url":"https://pith.science/pith/UQNUJGPZAIMCXJZWZQEJKUZRR3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UQNUJGPZAIMCXJZWZQEJKUZRR3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T14:21:22Z","links":{"resolver":"https://pith.science/pith/UQNUJGPZAIMCXJZWZQEJKUZRR3","bundle":"https://pith.science/pith/UQNUJGPZAIMCXJZWZQEJKUZRR3/bundle.json","state":"https://pith.science/pith/UQNUJGPZAIMCXJZWZQEJKUZRR3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UQNUJGPZAIMCXJZWZQEJKUZRR3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:UQNUJGPZAIMCXJZWZQEJKUZRR3","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":"0a4cd9523f9c72a2e031f6d9aaa677f3e3f5a61dab583470384b330871c1f09c","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2022-12-01T06:19:47Z","title_canon_sha256":"468a653a5a4c3e5d9f1a25d69f9cc5832e2e7e5d504a3bba89f723156f7c1bd9"},"schema_version":"1.0","source":{"id":"2212.00302","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2212.00302","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"arxiv_version","alias_value":"2212.00302v3","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2212.00302","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"pith_short_12","alias_value":"UQNUJGPZAIMC","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"UQNUJGPZAIMCXJZW","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"UQNUJGPZ","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:77625e339f1c6ca892c2da9b90c96d808fdb4defb8ded9aba6f93f300115fa18","target":"graph","created_at":"2026-05-18T03:09:47Z","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":"We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(\\lambda_{*},x_{*})$ of a given analytic regular nonlinear eigenvalue problem (NEP). In terms of the deviation $\\varepsilon$ of $x_{*}$ from a given subspace $\\mathcal{W}$, we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector. The results show that, as $\\varepsilon\\rightarrow 0$, there exists a Ritz value that unconditionally converges to $\\lambda_*$ and the corresponding refined Ritz vector does so","authors_text":"Qingqing Zheng, Zhongxiao Jia","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2022-12-01T06:19:47Z","title":"An analysis of the Rayleigh-Ritz and refined Rayleigh-Ritz methods for regular nonlinear eigenvalue problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.00302","kind":"arxiv","version":3},"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":"sha256:510bd6f2ec12ccf0ecc6b4d6dbd3422b45684d4a2e3dee50171f355999237384","target":"record","created_at":"2026-05-18T03:09:47Z","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":"0a4cd9523f9c72a2e031f6d9aaa677f3e3f5a61dab583470384b330871c1f09c","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2022-12-01T06:19:47Z","title_canon_sha256":"468a653a5a4c3e5d9f1a25d69f9cc5832e2e7e5d504a3bba89f723156f7c1bd9"},"schema_version":"1.0","source":{"id":"2212.00302","kind":"arxiv","version":3}},"canonical_sha256":"a41b4499f902182ba736cc089553318eeb5780724f053fa902503a5065ea5ab5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a41b4499f902182ba736cc089553318eeb5780724f053fa902503a5065ea5ab5","first_computed_at":"2026-05-18T03:09:47.455649Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:47.455649Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"w2rNlTVCcFFvKYFovxeDE5i5i5RAXbUWNF6CLl1pRre6vfuJnLVi3bV0DwejYVv60L9o0WjzWe6SG4J6/QzEBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:47.456563Z","signed_message":"canonical_sha256_bytes"},"source_id":"2212.00302","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:510bd6f2ec12ccf0ecc6b4d6dbd3422b45684d4a2e3dee50171f355999237384","sha256:77625e339f1c6ca892c2da9b90c96d808fdb4defb8ded9aba6f93f300115fa18"],"state_sha256":"fb8744b325a3c9ae9dacb6acbcd873db3aae9310d1eacfd9be3f862563200d44"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2qQ6IwfOXGIXvV1lIt8dM1oIxOPfBLyoWvlAIdqPYqEbSNllWS5o+eaB7VeI+U1YTd7giToDqv4USrnQxM/iDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T14:21:22.678788Z","bundle_sha256":"e58f3f4d05e93872f53a7eacb2001ad75b0612cb15db4c049777acae85ac4716"}}