{"id":"https://openalex.org/W1982346954","doi":"https://doi.org/10.1080/00207160902722357","title":"An iterative method for the bisymmetric solutions of the consistent matrix equations<i>A</i><sub>1</sub><i>XB</i><sub>1</sub>=<i>C</i><sub>1</sub>,<i>A</i><sub>2</sub><i>XB</i><sub>2</sub>=<i>C</i><sub>2</sub>","display_name":"An iterative method for the bisymmetric solutions of the consistent matrix equations<i>A</i><sub>1</sub><i>XB</i><sub>1</sub>=<i>C</i><sub>1</sub>,<i>A</i><sub>2</sub><i>XB</i><sub>2</sub>=<i>C</i><sub>2</sub>","publication_year":2010,"publication_date":"2010-06-04","ids":{"openalex":"https://openalex.org/W1982346954","doi":"https://doi.org/10.1080/00207160902722357","mag":"1982346954"},"language":"en","primary_location":{"id":"doi:10.1080/00207160902722357","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160902722357","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","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/A5029033092","display_name":"Jing Cai","orcid":null},"institutions":[{"id":"https://openalex.org/I66867065","display_name":"East China Normal University","ror":"https://ror.org/02n96ep67","country_code":"CN","type":"education","lineage":["https://openalex.org/I66867065"]},{"id":"https://openalex.org/I4210087590","display_name":"Huzhou Vocational and Technical College","ror":"https://ror.org/002hfez23","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210087590"]},{"id":"https://openalex.org/I3018263800","display_name":"Huzhou University","ror":"https://ror.org/04mvpxy20","country_code":"CN","type":"education","lineage":["https://openalex.org/I3018263800"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jing Cai","raw_affiliation_strings":["Department of Mathematics , East China Normal University , Shanghai, 200062,  China ;     School of Science , Huzhou Teachers College , Huzhou, 313000,  China","Department of Mathematics , East China Normal University , Shanghai, 200062,  China","School of Science , Huzhou Teachers College , Huzhou, 313000,  China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , East China Normal University , Shanghai, 200062,  China ;     School of Science , Huzhou Teachers College , Huzhou, 313000,  China","institution_ids":["https://openalex.org/I3018263800","https://openalex.org/I66867065","https://openalex.org/I4210087590"]},{"raw_affiliation_string":"Department of Mathematics , East China Normal University , Shanghai, 200062,  China","institution_ids":["https://openalex.org/I66867065"]},{"raw_affiliation_string":"School of Science , Huzhou Teachers College , Huzhou, 313000,  China","institution_ids":["https://openalex.org/I3018263800"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100668045","display_name":"Guoliang Chen","orcid":"https://orcid.org/0000-0003-0227-7444"},"institutions":[{"id":"https://openalex.org/I66867065","display_name":"East China Normal University","ror":"https://ror.org/02n96ep67","country_code":"CN","type":"education","lineage":["https://openalex.org/I66867065"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Guoliang Chen","raw_affiliation_strings":["Department of Mathematics , East China Normal University , Shanghai, 200062,  China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , East China Normal University , Shanghai, 200062,  China","institution_ids":["https://openalex.org/I66867065"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5082448553","display_name":"Qingbing Liu","orcid":"https://orcid.org/0000-0003-1105-1690"},"institutions":[{"id":"https://openalex.org/I1315279114","display_name":"Zhejiang Wanli University","ror":"https://ror.org/00rjdhd62","country_code":"CN","type":"education","lineage":["https://openalex.org/I1315279114"]},{"id":"https://openalex.org/I66867065","display_name":"East China Normal University","ror":"https://ror.org/02n96ep67","country_code":"CN","type":"education","lineage":["https://openalex.org/I66867065"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qingbing Liu","raw_affiliation_strings":["Department of Mathematics , Zhejiang Wanli University , Ningbo, 315100,  China ;     Department of Mathematics , East China Normal University , Shanghai, 200062,  China","Department of Mathematics , East China Normal University , Shanghai, 200062,  China","Department of Mathematics , Zhejiang Wanli University , Ningbo, 315100,  China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Zhejiang Wanli University , Ningbo, 315100,  China ;     Department of Mathematics , East China Normal University , Shanghai, 200062,  China","institution_ids":["https://openalex.org/I1315279114","https://openalex.org/I66867065"]},{"raw_affiliation_string":"Department of Mathematics , East China Normal University , Shanghai, 200062,  China","institution_ids":["https://openalex.org/I66867065"]},{"raw_affiliation_string":"Department of Mathematics , Zhejiang Wanli University , Ningbo, 315100,  China","institution_ids":["https://openalex.org/I1315279114"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5100668045"],"corresponding_institution_ids":["https://openalex.org/I66867065"],"apc_list":null,"apc_paid":null,"fwci":0.3383,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.59532661,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"87","issue":"12","first_page":"2706","last_page":"2715"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9998999834060669,"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":0.9998999834060669,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9886000156402588,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9825000166893005,"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/mathematics","display_name":"Mathematics","score":0.8631339073181152},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.7148779034614563},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.6201248168945312},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.6038296222686768},{"id":"https://openalex.org/keywords/matrix-splitting","display_name":"Matrix splitting","score":0.6012417078018188},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5249466896057129},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4726112186908722},{"id":"https://openalex.org/keywords/solution-set","display_name":"Solution set","score":0.47180095314979553},{"id":"https://openalex.org/keywords/matrix-free-methods","display_name":"Matrix-free methods","score":0.45196446776390076},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.41835296154022217},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.37022340297698975},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.33915892243385315},{"id":"https://openalex.org/keywords/state-transition-matrix","display_name":"State-transition matrix","score":0.2605457901954651},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.21450254321098328},{"id":"https://openalex.org/keywords/sparse-matrix","display_name":"Sparse matrix","score":0.2079009711742401},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.06648623943328857},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.06332516670227051}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8631339073181152},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.7148779034614563},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.6201248168945312},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.6038296222686768},{"id":"https://openalex.org/C133589720","wikidata":"https://www.wikidata.org/wiki/Q6787891","display_name":"Matrix splitting","level":5,"score":0.6012417078018188},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5249466896057129},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4726112186908722},{"id":"https://openalex.org/C60033838","wikidata":"https://www.wikidata.org/wiki/Q1751685","display_name":"Solution set","level":3,"score":0.47180095314979553},{"id":"https://openalex.org/C54219892","wikidata":"https://www.wikidata.org/wiki/Q4080929","display_name":"Matrix-free methods","level":4,"score":0.45196446776390076},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.41835296154022217},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.37022340297698975},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.33915892243385315},{"id":"https://openalex.org/C22629506","wikidata":"https://www.wikidata.org/wiki/Q7602993","display_name":"State-transition matrix","level":4,"score":0.2605457901954651},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.21450254321098328},{"id":"https://openalex.org/C56372850","wikidata":"https://www.wikidata.org/wiki/Q1050404","display_name":"Sparse matrix","level":3,"score":0.2079009711742401},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.06648623943328857},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.06332516670227051},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer 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/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160902722357","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160902722357","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320309612","display_name":"Natural Science Foundation of Shanghai","ror":null},{"id":"https://openalex.org/F4320321885","display_name":"Science and Technology Commission of Shanghai Municipality","ror":"https://ror.org/03kt66j61"},{"id":"https://openalex.org/F4320338464","display_name":"Natural Science Foundation of Zhejiang Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W1664553927","https://openalex.org/W1971734187","https://openalex.org/W1985940203","https://openalex.org/W2010552701","https://openalex.org/W2014548081","https://openalex.org/W2015155239","https://openalex.org/W2040534832","https://openalex.org/W2062691903","https://openalex.org/W2072863821","https://openalex.org/W2096450582","https://openalex.org/W2161216573","https://openalex.org/W2388871042"],"related_works":["https://openalex.org/W2161209494","https://openalex.org/W2366222664","https://openalex.org/W2374901252","https://openalex.org/W2005314356","https://openalex.org/W1664553927","https://openalex.org/W2050716149","https://openalex.org/W4235085014","https://openalex.org/W2112461225","https://openalex.org/W3046754636","https://openalex.org/W1982346954"],"abstract_inverted_index":{"In":[0,66],"this":[1,88],"paper,":[2],"an":[3],"iterative":[4,89],"method":[5],"is":[6,97],"presented":[7],"for":[8],"finding":[9],"the":[10,43,49,67,71,74,94],"bisymmetric":[11,33,77],"solutions":[12],"of":[13,16,45,63,70,93],"a":[14,32,60,80],"pair":[15],"consistent":[17],"matrix":[18,72,82],"equations":[19],"A":[20,25],"1":[21],"XB":[22,27],"1=C":[23],"1,":[24],"2":[26],"2=C":[28],"2,":[29],"by":[30,58,87,99],"which":[31],"solution":[34,50,68,78],"can":[35,55,83],"be":[36,56,85],"obtained":[37,57],"in":[38,42],"finite":[39],"iteration":[40],"steps":[41],"absence":[44],"round-off":[46],"errors.":[47],"Moreover,":[48],"with":[51],"least":[52],"Frobenius":[53],"norm":[54],"choosing":[59],"special":[61],"kind":[62],"initial":[64],"matrix.":[65],"set":[69],"equations,":[73],"optimal":[75],"approximation":[76],"to":[79],"given":[81],"also":[84],"derived":[86],"method.":[90],"The":[91],"efficiency":[92],"proposed":[95],"algorithm":[96],"shown":[98],"some":[100],"numerical":[101],"examples.":[102]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}