{"id":"https://openalex.org/W2165802699","doi":"https://doi.org/10.1155/2013/707381","title":"On Optimal Backward Perturbation Analysis for the Linear System with Skew Circulant Coefficient Matrix","display_name":"On Optimal Backward Perturbation Analysis for the Linear System with Skew Circulant Coefficient Matrix","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2165802699","doi":"https://doi.org/10.1155/2013/707381","mag":"2165802699","pmid":"https://pubmed.ncbi.nlm.nih.gov/24369488"},"language":"en","primary_location":{"id":"doi:10.1155/2013/707381","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/707381","pdf_url":"https://downloads.hindawi.com/journals/cmmm/2013/707381.pdf","source":{"id":"https://openalex.org/S36980176","display_name":"Computational and Mathematical Methods in Medicine","issn_l":"1748-670X","issn":["1748-670X","1748-6718"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and Mathematical Methods in Medicine","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://downloads.hindawi.com/journals/cmmm/2013/707381.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100421061","display_name":"Juan Li","orcid":"https://orcid.org/0000-0003-1046-571X"},"institutions":[{"id":"https://openalex.org/I15823474","display_name":"Linyi University","ror":"https://ror.org/01knv0402","country_code":"CN","type":"education","lineage":["https://openalex.org/I15823474"]},{"id":"https://openalex.org/I28006308","display_name":"Shandong Normal University","ror":"https://ror.org/01wy3h363","country_code":"CN","type":"education","lineage":["https://openalex.org/I28006308"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Juan Li","raw_affiliation_strings":["Department of Mathematics, Linyi University, Linyi, Shandong 276000, China","Department of Mathematics, Shandong Normal University, Ji\u2019nan, Shandong 250014, China"],"raw_orcid":"https://orcid.org/0000-0003-1046-571X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Linyi University, Linyi, Shandong 276000, China","institution_ids":["https://openalex.org/I15823474"]},{"raw_affiliation_string":"Department of Mathematics, Shandong Normal University, Ji\u2019nan, Shandong 250014, China","institution_ids":["https://openalex.org/I28006308"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024310786","display_name":"Zhaolin Jiang","orcid":"https://orcid.org/0000-0002-9521-9835"},"institutions":[{"id":"https://openalex.org/I15823474","display_name":"Linyi University","ror":"https://ror.org/01knv0402","country_code":"CN","type":"education","lineage":["https://openalex.org/I15823474"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhaolin Jiang","raw_affiliation_strings":["Department of Mathematics, Linyi University, Linyi, Shandong 276000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Linyi University, Linyi, Shandong 276000, China","institution_ids":["https://openalex.org/I15823474"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014714019","display_name":"Nuo Shen","orcid":null},"institutions":[{"id":"https://openalex.org/I15823474","display_name":"Linyi University","ror":"https://ror.org/01knv0402","country_code":"CN","type":"education","lineage":["https://openalex.org/I15823474"]},{"id":"https://openalex.org/I28006308","display_name":"Shandong Normal University","ror":"https://ror.org/01wy3h363","country_code":"CN","type":"education","lineage":["https://openalex.org/I28006308"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Nuo Shen","raw_affiliation_strings":["Department of Mathematics, Linyi University, Linyi, Shandong 276000, China","Department of Mathematics, Shandong Normal University, Ji\u2019nan, Shandong 250014, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Linyi University, Linyi, Shandong 276000, China","institution_ids":["https://openalex.org/I15823474"]},{"raw_affiliation_string":"Department of Mathematics, Shandong Normal University, Ji\u2019nan, Shandong 250014, China","institution_ids":["https://openalex.org/I28006308"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046354456","display_name":"Jianwei Zhou","orcid":"https://orcid.org/0000-0002-1288-5905"},"institutions":[{"id":"https://openalex.org/I15823474","display_name":"Linyi University","ror":"https://ror.org/01knv0402","country_code":"CN","type":"education","lineage":["https://openalex.org/I15823474"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jianwei Zhou","raw_affiliation_strings":["Department of Mathematics, Linyi University, Linyi, Shandong 276000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Linyi University, Linyi, Shandong 276000, China","institution_ids":["https://openalex.org/I15823474"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5024310786"],"corresponding_institution_ids":["https://openalex.org/I15823474"],"apc_list":{"value":2100,"currency":"USD","value_usd":2100},"apc_paid":{"value":2100,"currency":"USD","value_usd":2100},"fwci":3.3515,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.92590598,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"2013","issue":null,"first_page":"1","last_page":"7"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9975000023841858,"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":0.9975000023841858,"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":0.9887999892234802,"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/T13487","display_name":"Statistical and numerical algorithms","score":0.9861999750137329,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/skew","display_name":"Skew","score":0.750805139541626},{"id":"https://openalex.org/keywords/coefficient-matrix","display_name":"Coefficient matrix","score":0.6846954822540283},{"id":"https://openalex.org/keywords/circulant-matrix","display_name":"Circulant matrix","score":0.6808736324310303},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.602769672870636},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5354917645454407},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.472226619720459},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.4168170392513275},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37842655181884766},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3375888168811798},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3333154320716858},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.23814073204994202},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.23708775639533997},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.09928548336029053},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.07774969935417175}],"concepts":[{"id":"https://openalex.org/C43711488","wikidata":"https://www.wikidata.org/wiki/Q7534783","display_name":"Skew","level":2,"score":0.750805139541626},{"id":"https://openalex.org/C60866291","wikidata":"https://www.wikidata.org/wiki/Q5140577","display_name":"Coefficient matrix","level":3,"score":0.6846954822540283},{"id":"https://openalex.org/C115973184","wikidata":"https://www.wikidata.org/wiki/Q245457","display_name":"Circulant matrix","level":2,"score":0.6808736324310303},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.602769672870636},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5354917645454407},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.472226619720459},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.4168170392513275},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37842655181884766},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3375888168811798},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3333154320716858},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.23814073204994202},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.23708775639533997},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.09928548336029053},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.07774969935417175},{"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/C43617362","wikidata":"https://www.wikidata.org/wiki/Q170050","display_name":"Chromatography","level":1,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0}],"mesh":[{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000000981","qualifier_name":"diagnostic imaging","is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000000981","qualifier_name":"diagnostic imaging","is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000000981","qualifier_name":"diagnostic imaging","is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000473","qualifier_name":"pathology","is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000473","qualifier_name":"pathology","is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000473","qualifier_name":"pathology","is_major_topic":false},{"descriptor_ui":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D010477","descriptor_name":"Perfusion","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D010477","descriptor_name":"Perfusion","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D010477","descriptor_name":"Perfusion","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D011857","descriptor_name":"Radiographic Image Interpretation, Computer-Assisted","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D011857","descriptor_name":"Radiographic Image Interpretation, Computer-Assisted","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D011857","descriptor_name":"Radiographic Image Interpretation, Computer-Assisted","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D012984","descriptor_name":"Software","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D012984","descriptor_name":"Software","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D012984","descriptor_name":"Software","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D014057","descriptor_name":"Tomography, X-Ray Computed","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D014057","descriptor_name":"Tomography, X-Ray Computed","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D014057","descriptor_name":"Tomography, X-Ray Computed","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D016014","descriptor_name":"Linear Models","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D016014","descriptor_name":"Linear Models","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D016014","descriptor_name":"Linear Models","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false}],"locations_count":4,"locations":[{"id":"doi:10.1155/2013/707381","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/707381","pdf_url":"https://downloads.hindawi.com/journals/cmmm/2013/707381.pdf","source":{"id":"https://openalex.org/S36980176","display_name":"Computational and Mathematical Methods in Medicine","issn_l":"1748-670X","issn":["1748-670X","1748-6718"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and Mathematical Methods in Medicine","raw_type":"journal-article"},{"id":"pmid:24369488","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/24369488","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and mathematical methods in medicine","raw_type":null},{"id":"pmh:oai:doaj.org/article:7c9216e30ebf4ed9b6bda502d6fa1833","is_oa":false,"landing_page_url":"https://doaj.org/article/7c9216e30ebf4ed9b6bda502d6fa1833","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Computational and Mathematical Methods in Medicine, Vol 2013 (2013)","raw_type":"article"},{"id":"pmh:oai:europepmc.org:2854809","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/3863576","pdf_url":null,"source":{"id":"https://openalex.org/S4306400806","display_name":"Europe PMC (PubMed Central)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1303153112","host_organization_name":"European Bioinformatics Institute","host_organization_lineage":["https://openalex.org/I1303153112"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1155/2013/707381","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/707381","pdf_url":"https://downloads.hindawi.com/journals/cmmm/2013/707381.pdf","source":{"id":"https://openalex.org/S36980176","display_name":"Computational and Mathematical Methods in Medicine","issn_l":"1748-670X","issn":["1748-670X","1748-6718"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and Mathematical Methods in Medicine","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2119021428","display_name":null,"funder_award_id":"2012GGX10115","funder_id":"https://openalex.org/F4320323252","funder_display_name":"Linyi University"},{"id":"https://openalex.org/G2457281420","display_name":null,"funder_award_id":"2012GGX10115","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2967278857","display_name":null,"funder_award_id":"11201212","funder_id":"https://openalex.org/F4320323252","funder_display_name":"Linyi University"},{"id":"https://openalex.org/G3609563803","display_name":null,"funder_award_id":"61203047","funder_id":"https://openalex.org/F4320323252","funder_display_name":"Linyi University"},{"id":"https://openalex.org/G6896570429","display_name":null,"funder_award_id":"61203047","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8215812334","display_name":null,"funder_award_id":"11201212","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320323252","display_name":"Linyi University","ror":"https://ror.org/01knv0402"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2165802699.pdf","grobid_xml":"https://content.openalex.org/works/W2165802699.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W859245233","https://openalex.org/W1502940993","https://openalex.org/W1971404264","https://openalex.org/W1971578649","https://openalex.org/W1974172531","https://openalex.org/W2001774794","https://openalex.org/W2015673307","https://openalex.org/W2025688688","https://openalex.org/W2026927759","https://openalex.org/W2028933557","https://openalex.org/W2051015700","https://openalex.org/W2064318771","https://openalex.org/W2092085096","https://openalex.org/W2093425374","https://openalex.org/W2120400713","https://openalex.org/W2128749318"],"related_works":["https://openalex.org/W2980165407","https://openalex.org/W1532878563","https://openalex.org/W2037738710","https://openalex.org/W1584148758","https://openalex.org/W2998530998","https://openalex.org/W4385417781","https://openalex.org/W2051403509","https://openalex.org/W2130795715","https://openalex.org/W2332399844","https://openalex.org/W1988602319"],"abstract_inverted_index":{"We":[0],"first":[1,36],"give":[2],"the":[3,17,29,33,42,51,64,70,79],"style":[4,18,83],"spectral":[5,84],"decomposition":[6,19],"of":[7,20,28,35,54,81],"a":[8],"special":[9,43],"skew":[10,22,39,44,55,74],"circulant":[11,23,40,45,56,75],"matrix":[12,24,57,77],"C":[13],"and":[14,41],"then":[15],"get":[16],"arbitrary":[21],"by":[25],"making":[26],"use":[27],"Kronecker":[30],"products":[31],"between":[32],"elements":[34],"row":[37],"in":[38],"C.":[46],"Besides":[47],"that,":[48],"we":[49,61],"obtain":[50],"singular":[52],"value":[53],"as":[58],"well.":[59],"Finally,":[60],"deal":[62],"with":[63,73],"optimal":[65],"backward":[66],"perturbation":[67],"analysis":[68],"for":[69],"linear":[71],"system":[72],"coefficient":[76],"on":[78],"base":[80],"its":[82],"decomposition.":[85]},"counts_by_year":[{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":4}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}