{"id":"https://openalex.org/W1970334228","doi":"https://doi.org/10.1137/s1064827594271512","title":"Multigrid Methods for Symmetric Positive Definite Block Toeplitz Matrices with Nonnegative Generating Functions","display_name":"Multigrid Methods for Symmetric Positive Definite Block Toeplitz Matrices with Nonnegative Generating Functions","publication_year":1996,"publication_date":"1996-09-01","ids":{"openalex":"https://openalex.org/W1970334228","doi":"https://doi.org/10.1137/s1064827594271512","mag":"1970334228"},"language":"en","primary_location":{"id":"doi:10.1137/s1064827594271512","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827594271512","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101575002","display_name":"Giuseppe Fiorentino","orcid":"https://orcid.org/0000-0002-5016-5458"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Giuseppe Fiorentino","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5053268193","display_name":"Stefano Serra\u2010Capizzano","orcid":"https://orcid.org/0000-0001-9477-109X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Stefano Serra","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5101575002"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.0703,"has_fulltext":false,"cited_by_count":114,"citation_normalized_percentile":{"value":0.96411279,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"17","issue":"5","first_page":"1068","last_page":"1081"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9943000078201294,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/toeplitz-matrix","display_name":"Toeplitz matrix","score":0.8804498910903931},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8570314645767212},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.6521823406219482},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6017199754714966},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.5315890908241272},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5266860723495483},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.5006539821624756},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.4834943115711212},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4534558951854706},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.4144250154495239},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.4131006896495819},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2619650959968567},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.2525298297405243},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22723332047462463},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.21114811301231384},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.13421469926834106}],"concepts":[{"id":"https://openalex.org/C147710293","wikidata":"https://www.wikidata.org/wiki/Q849428","display_name":"Toeplitz matrix","level":2,"score":0.8804498910903931},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8570314645767212},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.6521823406219482},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6017199754714966},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.5315890908241272},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5266860723495483},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.5006539821624756},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.4834943115711212},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4534558951854706},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.4144250154495239},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.4131006896495819},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2619650959968567},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.2525298297405243},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22723332047462463},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.21114811301231384},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.13421469926834106},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s1064827594271512","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827594271512","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W126699582","https://openalex.org/W208025404","https://openalex.org/W1976865075","https://openalex.org/W1999886256","https://openalex.org/W2003977554","https://openalex.org/W2025799417","https://openalex.org/W2034367460","https://openalex.org/W2041394080","https://openalex.org/W2045463928","https://openalex.org/W2057439610","https://openalex.org/W2060594652","https://openalex.org/W2073393685","https://openalex.org/W2095018839","https://openalex.org/W2501171808","https://openalex.org/W4230055019","https://openalex.org/W4388534025"],"related_works":["https://openalex.org/W2361901126","https://openalex.org/W2155025153","https://openalex.org/W4253880741","https://openalex.org/W2361205607","https://openalex.org/W2070080181","https://openalex.org/W4235544854","https://openalex.org/W2674186696","https://openalex.org/W2015339619","https://openalex.org/W1826911152","https://openalex.org/W4237208466"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3],"introduce":[4],"a":[5,28,39],"generalized":[6],"multigrid":[7],"method":[8],"for":[9,96,101,167,174],"solving":[10],"linear":[11],"systems":[12],"${\\bf":[13],"T}_{N,M}":[14,79],"{\\bf":[15,18],"x}":[16],"=":[17],"b}$where":[19],"$T_{N,M}":[20],"\\in":[21],"\\Re":[22],"^{NM":[23],"\\times":[24],"NM}":[25],"$":[26],"is":[27,93,120],"symmetric":[29,34],"block":[30,176],"Toeplitz":[31,35,177,181],"matrix":[32],"with":[33,55,67,104,123,179],"blocks.":[36,182],"We":[37],"use":[38],"special":[40],"choice":[41,62],"of":[42,59,71,82,85],"the":[43,51,56,72,83,86,108,124],"projection":[44],"operator":[45],"whose":[46],"coefficients":[47],"simply":[48],"depend":[49],"on":[50],"generating":[52],"functions":[53],"associated":[54],"proposed":[57,129],"class":[58],"matrices.":[60,88],"This":[61,118],"leads":[63],"to":[64],"iterative":[65],"methods":[66,128],"convergence":[68],"rates":[69],"independent":[70],"Euclidean":[73],"condition":[74],"number":[75],"$\\kappa":[76],"_2":[77],"({\\bf":[78],")$":[80],"and":[81,99,132,143,160,170],"dimension":[84],"involved":[87],"The":[89],"total":[90],"arithmetic":[91],"cost":[92],"$O(NM\\log":[94],"(NM))$":[95,113],"dense":[97,168],"matrices":[98,103,169,178],"$O(NM)$":[100],"band":[102,105,175,180],"blocks;":[106],"in":[107],"PRAM":[109],"model":[110],"only":[111],"$O(\\log":[112],"parallel":[114],"steps":[115],"are":[116],"required.":[117],"algorithm":[119],"therefore":[121],"competitive":[122],"preconditioned":[125],"conjugate":[126],"gradient":[127],"by":[130],"Chan":[131],"Jin":[133],"[SIAM":[134,145,155],"J":[135],"Sci.":[136,147,157],"Statist.":[137,148],"Comput.,13":[138,149],"(1992),":[139,150],"pp.":[140,151,165],"1218\u20131235],":[141],"Ku":[142],"Kuo":[144],"J.":[146,156],"948\u2013966],":[152],"Di":[153],"Benedetto":[154],"Comput.,17":[158],"(1996)],":[159],"Serra":[161],"[BIT,":[162],"34":[163],"(1994),":[164],"579\u2013594]":[166],"improves":[171],"those":[172],"results":[173]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}