{"id":"https://openalex.org/W2077091631","doi":"https://doi.org/10.1137/050631707","title":"Shifted Kronecker Product Systems","display_name":"Shifted Kronecker Product Systems","publication_year":2006,"publication_date":"2006-12-26","ids":{"openalex":"https://openalex.org/W2077091631","doi":"https://doi.org/10.1137/050631707","mag":"2077091631"},"language":"en","primary_location":{"id":"doi:10.1137/050631707","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050631707","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Matrix Analysis and Applications","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/A5015229483","display_name":"Carla D. Martin","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Carla D. Moravitz Martin","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5006610823","display_name":"Charles F. Van Loan","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Charles F. Van Loan","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5015229483"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.6741,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.71262436,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"29","issue":"1","first_page":"184","last_page":"198"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9973000288009644,"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":0.9973000288009644,"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/T11476","display_name":"Graph theory and applications","score":0.9614999890327454,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11435","display_name":"Polynomial and algebraic computation","score":0.9549000263214111,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/kronecker-product","display_name":"Kronecker product","score":0.9080678820610046},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.769584059715271},{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.7343106269836426},{"id":"https://openalex.org/keywords/triangular-matrix","display_name":"Triangular matrix","score":0.6988086104393005},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6116010546684265},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.5946530699729919},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5404714941978455},{"id":"https://openalex.org/keywords/schur-decomposition","display_name":"Schur decomposition","score":0.5377353429794312},{"id":"https://openalex.org/keywords/decomposition","display_name":"Decomposition","score":0.4939272999763489},{"id":"https://openalex.org/keywords/schur-complement","display_name":"Schur complement","score":0.47907334566116333},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.46519672870635986},{"id":"https://openalex.org/keywords/flops","display_name":"FLOPS","score":0.4503815770149231},{"id":"https://openalex.org/keywords/substitution","display_name":"Substitution (logic)","score":0.41418805718421936},{"id":"https://openalex.org/keywords/matrix-multiplication","display_name":"Matrix multiplication","score":0.41377392411231995},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.35382580757141113},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.34341251850128174},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1840924620628357},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11681574583053589},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09286868572235107},{"id":"https://openalex.org/keywords/parallel-computing","display_name":"Parallel computing","score":0.09276509284973145}],"concepts":[{"id":"https://openalex.org/C46030957","wikidata":"https://www.wikidata.org/wiki/Q1238125","display_name":"Kronecker product","level":3,"score":0.9080678820610046},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.769584059715271},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.7343106269836426},{"id":"https://openalex.org/C104528550","wikidata":"https://www.wikidata.org/wiki/Q506265","display_name":"Triangular matrix","level":3,"score":0.6988086104393005},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6116010546684265},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.5946530699729919},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5404714941978455},{"id":"https://openalex.org/C99383703","wikidata":"https://www.wikidata.org/wiki/Q1064218","display_name":"Schur decomposition","level":4,"score":0.5377353429794312},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.4939272999763489},{"id":"https://openalex.org/C2731191","wikidata":"https://www.wikidata.org/wiki/Q1778169","display_name":"Schur complement","level":3,"score":0.47907334566116333},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.46519672870635986},{"id":"https://openalex.org/C3826847","wikidata":"https://www.wikidata.org/wiki/Q188768","display_name":"FLOPS","level":2,"score":0.4503815770149231},{"id":"https://openalex.org/C2778220771","wikidata":"https://www.wikidata.org/wiki/Q1522579","display_name":"Substitution (logic)","level":2,"score":0.41418805718421936},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.41377392411231995},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35382580757141113},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.34341251850128174},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1840924620628357},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11681574583053589},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09286868572235107},{"id":"https://openalex.org/C173608175","wikidata":"https://www.wikidata.org/wiki/Q232661","display_name":"Parallel computing","level":1,"score":0.09276509284973145},{"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/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","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/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","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},{"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/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.0},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/050631707","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050631707","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.125.3138","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.125.3138","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.cornell.edu/cv/researchpdf/kronprod.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":4,"referenced_works":["https://openalex.org/W1986380039","https://openalex.org/W2047017077","https://openalex.org/W2134572726","https://openalex.org/W2798909945"],"related_works":["https://openalex.org/W1986380039","https://openalex.org/W2949407439","https://openalex.org/W2360296850","https://openalex.org/W2315991876","https://openalex.org/W567465994","https://openalex.org/W2009982207","https://openalex.org/W2952081580","https://openalex.org/W1904938324","https://openalex.org/W2363085855","https://openalex.org/W2077091631"],"abstract_inverted_index":{"A":[0,91],"fast":[1],"method":[2,93],"for":[3,138],"solving":[4],"a":[5,66],"linear":[6],"system":[7],"of":[8,59,113],"the":[9,37,55,60,73,97,118],"form":[10,41],"$(A^{(p)}":[11],"\\otimes":[12,14,43,45,126,128],"\\cdots":[13,44,81,89,127],"A^{(1)}":[15],"-":[16,47],"\\lambda":[17,48,130],"I)":[18,49],"x":[19],"=":[20,51,87],"b$":[21],"is":[22,27,34,63,94,101],"given":[23],"where":[24,85],"each":[25,105],"$A^{(i)}$":[26,106],"an":[28],"$n_i$\u2010by\u2010$n_i$":[29],"matrix.":[30],"The":[31,110],"first":[32],"step":[33],"to":[35,39,103,107,136],"convert":[36,104],"problem":[38],"triangular":[40],"$(T^{(p)}":[42,125],"T^{(1)}":[46],"y":[50],"c$":[52],"by":[53,65],"computing":[54],"(complex)":[56],"Schur":[57,99],"decompositions":[58],"$A^{(i)}$.":[61],"This":[62],"followed":[64],"recursive":[67],"back\u2010substitution":[68,135],"process":[69],"that":[70],"fully":[71],"exploits":[72],"Kronecker":[74],"structure":[75],"and":[76,132],"requires":[77],"just":[78],"$O(N(n_1":[79],"+":[80,82],"n_p))$":[83],"flops":[84],"$N":[86],"n_1":[88],"n_p$.":[90],"similar":[92],"employed":[95],"when":[96],"real":[98],"decomposition":[100],"used":[102,133],"quasi\u2010triangular":[108],"form.":[109],"numerical":[111],"properties":[112],"these":[114],"new":[115],"methods":[116],"are":[117],"same":[119],"as":[120],"if":[121],"we":[122],"explicitly":[123],"formed":[124],"T^{(1)}-":[129],"I)$":[131],"conventional":[134],"solve":[137],"y.":[139]},"counts_by_year":[{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
