{"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 $c(\\xi^i)=(\\textbf{y},\\textit{A}^i\\textbf{r}_{0})$. 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)}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0323","kind":"arxiv","version":1},"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"}