{"doi":"10.1002/nla.2427","title":"Toward efficient polynomial preconditioning for GMRES","abstract":"<jats:title>Abstract</jats:title><jats:p>We present a polynomial preconditioner for solving large systems of linear equations. The polynomial is derived from the minimum residual polynomial (the GMRES polynomial) and is more straightforward to compute and implement than many previous polynomial preconditioners. Our current implementation of this polynomial using its roots is naturally more stable than previous methods of computing the same polynomial. We implement further stability control using added roots, and this allows for high degree polynomials. We discuss the effectiveness and challenges of root‐adding and give an additional check for stability. In this article, we study the polynomial preconditioner applied to GMRES; however it could be used with any Krylov solver. This polynomial preconditioning algorithm can dramatically improve convergence for some problems, especially for difficult problems, and can reduce dot products by an even greater margin.</jats:p>","journal":"Numerical Linear Algebra with Applications","year":2022,"id":23928,"datarank":0.5472587537708035,"base_score":2.639057329615259,"endowment":2.639057329615259,"self_citation_contribution":0.3958585994422889,"citation_network_contribution":0.15140015432851456,"self_endowment_contribution":0.3958585994422889,"citer_contribution":0.15140015432851456,"corpus_percentile":null,"corpus_rank":null,"citation_count":13,"citer_count":12,"citers_with_citation_signal":8,"citers_with_endowment":8,"datacite_reuse_total":1,"is_dataset":false,"is_dataset_confidence":null,"is_data_producer":false,"deposit_databanks":null,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":null,"fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":145151,"name":"Ronald B. Morgan","orcid":"0000-0002-0623-3740","position":1,"is_corresponding":false},{"id":145150,"name":"Jennifer A. Loe","orcid":"0000-0002-3018-7190","position":0,"is_corresponding":false}],"reference_count":0,"raw_metadata":{"has_enrichment":true,"base_score":2.639057329615259,"endowment":2.639057329615259,"datacite_reuse_total":1,"file_count":0,"downloads":0,"views":0,"has_version_chain":false,"is_dataset":false,"is_oa":false,"pmid":"18998881","pmcid":null,"openalex_id":"https://openalex.org/W4205471980","authors":[],"funders":[{"funder_name":"National Science Foundation of Sri Lanka","grant_id":"DMS‐1418677","title":null},{"funder_name":"National Science Foundation","grant_id":"1418677","title":"Krylov Multigrid Methods for Eigenvalues and Linear Equations"}],"total_grants":2,"fwci":1.3697,"citation_percentile":0.84439703,"influential_citations":3,"citation_trend":[{"year":2023,"count":3},{"year":2024,"count":6},{"year":2025,"count":3},{"year":2026,"count":1}],"oa_status":"green","license":"Wiley Online Library User Agreement","oa_locations":[{"url":"https://arxiv.org/pdf/1911.07065","host_type":"repository"},{"url":"https://arxiv.org/pdf/1911.07065","host_type":"GREEN"},{"url":"https://arxiv.org/pdf/1911.07065","host_type":"repository"},{"url":"https://onlinelibrary.wiley.com/doi/pdf/10.1002/nla.2427","host_type":"publisher"},{"url":"https://onlinelibrary.wiley.com/doi/full-xml/10.1002/nla.2427","host_type":"publisher"},{"url":"http://arxiv.org/abs/1911.07065","host_type":"repository"},{"url":"https://doi.org/10.1002/nla.2427","host_type":"journal"},{"url":"https://www.osti.gov/biblio/1838187","host_type":"repository"},{"url":"http://arxiv.org/pdf/1911.07065","host_type":""},{"url":"https://dx.doi.org/10.48550/arxiv.1911.07065","host_type":""},{"url":"https://zbmath.org/7584135","host_type":""}],"fields_of_study":["Matrix Theory and Algorithms","Electromagnetic Scattering and Analysis","Advanced Numerical Methods in Computational Mathematics","Computer Science","Mathematics","0101 mathematics","01 natural sciences"],"mesh_terms":[],"keywords":["Generalized minimal residual method","Preconditioner","Polynomial","Mathematics","Matrix polynomial","Solver","Convergence (economics)","Minimal polynomial (linear algebra)","Reciprocal polynomial","Stable polynomial","Degree of a polynomial","Applied mathematics","Square-free polynomial","Residual","Iterative method","Mathematical optimization","Algorithm","Mathematical analysis","Iterative numerical methods for linear systems","Computational methods for sparse matrices","polynomial preconditioning","FOS: Mathematics","Preconditioners for iterative methods","harmonic Ritz values","Mathematics - Numerical Analysis","Numerical Analysis (math.NA)","GMRES","linear equations","Orthogonalization in numerical linear algebra"],"sdg_mappings":[],"linked_datasets":[{"doi":"10.48550/arxiv.1911.07065","title":"Toward Efficient Polynomial Preconditioning for GMRES","publisher":"arXiv","resource_type":"Text"}],"clinical_trials":[],"software_tools":[],"database_accessions":[],"source":"live","citation_network_status":"fetched"},"created_at":"2026-06-07T20:55:53.794129Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}