{"id":"https://openalex.org/W2018574635","doi":"https://doi.org/10.1090/s0025-5718-1965-0184442-5","title":"On Optimal Alternating Direction Parameters for Singular Matrices","display_name":"On Optimal Alternating Direction Parameters for Singular Matrices","publication_year":1965,"publication_date":"1965-01-01","ids":{"openalex":"https://openalex.org/W2018574635","doi":"https://doi.org/10.1090/s0025-5718-1965-0184442-5","mag":"2018574635"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-1965-0184442-5","doi":"https://doi.org/10.1090/s0025-5718-1965-0184442-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-1965-0184442-5","pdf_url":"https://www.ams.org/mcom/1965-19-091/S0025-5718-1965-0184442-5/S0025-5718-1965-0184442-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/1965-19-091/S0025-5718-1965-0184442-5/S0025-5718-1965-0184442-5.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015318062","display_name":"R. Bruce Kellogg","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"R. B. Kellogg","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5063465661","display_name":"Jerome Spanier","orcid":"https://orcid.org/0000-0002-9875-039X","observed_orcids":["https://orcid.org/0000-0002-9875-039X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"J. Spanier","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.8278,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.72265967,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"19","issue":"91","first_page":"448","last_page":"448"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9984999895095825,"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":0.9984999895095825,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.00039999998989515007,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.00019999999494757503,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/generalized-inverses","display_name":"generalized inverses","score":0.6790000200271606},{"id":"https://openalex.org/keywords/singular-matrices","display_name":"singular matrices","score":0.7979999780654907},{"id":"https://openalex.org/keywords/alternating-direction-method","display_name":"alternating direction method","score":0.6620000004768372},{"id":"https://openalex.org/keywords/matrix-inversion","display_name":"matrix inversion","score":0.9539999961853027},{"id":"https://openalex.org/keywords/convergence-of-iterative-sequences","display_name":"convergence of iterative sequences","score":0.6330000162124634}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8367642164230347},{"id":"https://openalex.org/C200106649","wikidata":"https://www.wikidata.org/wiki/Q223683","display_name":"Transpose","level":3,"score":0.679731011390686},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.678804337978363},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.6497533321380615},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5950341820716858},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5788162350654602},{"id":"https://openalex.org/C76134657","wikidata":"https://www.wikidata.org/wiki/Q370634","display_name":"Generalized inverse","level":3,"score":0.5521602630615234},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5164432525634766},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4604654014110565},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.41269102692604065},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40592125058174133},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.253152459859848},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.20796096324920654},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.17387863993644714},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.09635412693023682},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08375093340873718},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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},{"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1090/s0025-5718-1965-0184442-5","doi":"https://doi.org/10.1090/s0025-5718-1965-0184442-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-1965-0184442-5","pdf_url":"https://www.ams.org/mcom/1965-19-091/S0025-5718-1965-0184442-5/S0025-5718-1965-0184442-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"doi:10.2307/2003675","doi":"https://doi.org/10.2307/2003675","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2003675","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-1965-0184442-5","doi":"https://doi.org/10.1090/s0025-5718-1965-0184442-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-1965-0184442-5","pdf_url":"https://www.ams.org/mcom/1965-19-091/S0025-5718-1965-0184442-5/S0025-5718-1965-0184442-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2018574635.pdf","grobid_xml":"https://content.openalex.org/works/W2018574635.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W1491365776","https://openalex.org/W1994869272","https://openalex.org/W2007017590","https://openalex.org/W2029411308","https://openalex.org/W2043857934","https://openalex.org/W2046380039","https://openalex.org/W2053262745","https://openalex.org/W2095036901","https://openalex.org/W2569133694","https://openalex.org/W3035235874"],"related_works":["https://openalex.org/W1984755089","https://openalex.org/W2017726114","https://openalex.org/W1991717161","https://openalex.org/W2077400379","https://openalex.org/W1993091830","https://openalex.org/W2371790906","https://openalex.org/W1989221315","https://openalex.org/W1984437100","https://openalex.org/W2149121185","https://openalex.org/W2058609598"],"abstract_inverted_index":{"The":[0,127],"iterative":[1,18],"process,":[2],"X+":[3],"=":[4,43,47,138],"Xn(21":[5],"AX.),":[6],"for":[7,20],"computing":[8],"A-1,":[9],"is":[10,28,67],"generalized":[11,15,68,79,122],"to":[12,25,51,69,76,158],"obtain":[13],"the":[14,31,34,52,64,78,98,104,109,115,121],"inverse.":[16],"An":[17],"method":[19],"inverting":[21],"a":[22,71],"matrix,":[23,92],"due":[24],"Schulz":[26],"[1],":[27],"based":[29],"on":[30,103,114,148],"convergence":[32],"of":[33,36,55,73,81,101,106,112,117,124,129],"sequence":[35,72,128],"matrices,":[37],"defined":[38,131],"recursively":[39],"by":[40,132],"(1)":[41,66],"X,+1":[42],"X,(21":[44],"AXn)":[45],"(n":[46],"0,":[48],"1,":[49],"**)":[50],"inverse":[53,80,123],"A`":[54],"A,":[56,107],"whenever":[57],"X0":[58],"approximates":[59],"A-'.":[60],"In":[61],"this":[62],"note":[63],"process":[65],"yield":[70],"matrices":[74,130],"converging":[75],"A+,":[77],"A":[82,85],"[2].":[83],"Let":[84],"denote":[86],"an":[87],"n":[88,90],"X":[89],"complex":[91],"A*":[93],"its":[94],"conjugate":[95],"transpose,":[96],"PR(A)":[97],"perpendicular":[99,110],"projection":[100,111],"Em":[102],"range":[105,116],"PR(A.)":[108],"En":[113],"A*,":[118],"and":[119],"A+":[120],"A.":[125],"THEOREM.":[126],"(2)":[133],"XYn+1":[134],"X.(2PR(4)":[135],"AX,)":[136],"(t":[137],"01,":[139],"Received":[140],"November":[141],"9,1964.":[142],"This":[143],"content":[144],"downloaded":[145],"from":[146],"157.55.39.220":[147],"Fri,":[149],"02":[150],"Sep":[151],"2016":[152],"04:12:02":[153],"UTC":[154],"All":[155],"use":[156],"subject":[157],"http://about.jstor.org/terms":[159]},"counts_by_year":[{"year":2018,"cited_by_count":1}],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2016-06-24T00:00:00"}