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Let $\\textit{P}^{(1)}_{k}$ be the regular monic polynomial of degree $k$ belonging to the family of formal orthogonal polynomials (FOP) with respect to $c^{(1)}$ defined as $c^{(1)}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1403.0323","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.NA","submitted_at":"2014-03-03T06:29:12Z","cross_cats_sorted":[],"title_canon_sha256":"fdf1deac7c5c1781090ea706c81a0e00b3e7e033518ebcd952e27b777406f126","abstract_canon_sha256":"760717aef0be4f1bcd87d68fcd122ef82a35ad9f8267065a99630a2910738c5c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:25.612335Z","signature_b64":"Mh4j/c8BPFw5Pvu8q7QYVWxj+Vuf38KioS2x+RwJtc4Q4DJju/va4+r2AtTFaN9SZSjLKVbsEziR+fkbYFe9BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b119871e49e15fd4016bac123c13135ae0b18419234a5ea224b98f3d8cfdb1ab","last_reissued_at":"2026-05-18T02:52:25.611617Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:25.611617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New Recurrence Relationships between Orthogonal Polynomials which Lead to New Lanczos-type Algorithms","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Abdellah Salhi, Muhammad Farooq","submitted_at":"2014-03-03T06:29:12Z","abstract_excerpt":"Lanczos methods for solving $\\textit{A}\\textbf{x}=\\textbf{b}$ consist in constructing a sequence of vectors $(\\textbf{x}_k), k=1,...$ such that $\\textbf{r}_{k}=\\textbf{b}-\\textit{A}\\textbf{x}_{k}=\\textit{P}_{k}(\\textit{A})\\textbf{r}_{0}$,, where $\\textit{P}_{k}$ is the orthogonal polynomial of degree at most $k$ with respect to the linear functional $c$ defined as $c(\\xi^i)=(\\textbf{y},\\textit{A}^i\\textbf{r}_{0})$. 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