{"id":"https://openalex.org/W1979379291","doi":"https://doi.org/10.1016/0925-7721(95)00048-8","title":"Rational orthogonal approximations to orthogonal matrices","display_name":"Rational orthogonal approximations to orthogonal matrices","publication_year":1997,"publication_date":"1997-01-01","ids":{"openalex":"https://openalex.org/W1979379291","doi":"https://doi.org/10.1016/0925-7721(95)00048-8","mag":"1979379291"},"language":"en","primary_location":{"id":"doi:10.1016/0925-7721(95)00048-8","is_oa":true,"landing_page_url":"https://doi.org/10.1016/0925-7721(95)00048-8","pdf_url":null,"source":{"id":"https://openalex.org/S165542689","display_name":"Computational Geometry","issn_l":"0925-7721","issn":["0925-7721","1879-081X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Geometry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1016/0925-7721(95)00048-8","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039907465","display_name":"Victor Milenkovic","orcid":"https://orcid.org/0000-0001-8524-9742"},"institutions":[{"id":"https://openalex.org/I145608581","display_name":"University of Miami","ror":"https://ror.org/02dgjyy92","country_code":"US","type":"education","lineage":["https://openalex.org/I145608581"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Victor J. Milenkovic","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA","Engineering Consultant, 1222 Highway 42, Sturgeon Bay, WI 54235, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA","institution_ids":["https://openalex.org/I145608581"]},{"raw_affiliation_string":"Engineering Consultant, 1222 Highway 42, Sturgeon Bay, WI 54235, USA","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008553891","display_name":"Veljko Milenkovic","orcid":null},"institutions":[{"id":"https://openalex.org/I145608581","display_name":"University of Miami","ror":"https://ror.org/02dgjyy92","country_code":"US","type":"education","lineage":["https://openalex.org/I145608581"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Victor J. Milenkovic","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA","Engineering Consultant, 1222 Highway 42, Sturgeon Bay, WI 54235, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA","institution_ids":["https://openalex.org/I145608581"]},{"raw_affiliation_string":"Engineering Consultant, 1222 Highway 42, Sturgeon Bay, WI 54235, USA","institution_ids":[]}]},{"author_position":"middle","author":{"id":null,"display_name":"Veljko Milenkovic","orcid":null},"institutions":[{"id":"https://openalex.org/I145608581","display_name":"University of Miami","ror":"https://ror.org/02dgjyy92","country_code":"US","type":"education","lineage":["https://openalex.org/I145608581"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Veljko Milenkovic","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA","Engineering Consultant, 1222 Highway 42, Sturgeon Bay, WI 54235, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA","institution_ids":["https://openalex.org/I145608581"]},{"raw_affiliation_string":"Engineering Consultant, 1222 Highway 42, Sturgeon Bay, WI 54235, USA","institution_ids":[]}]},{"author_position":"last","author":{"id":null,"display_name":"Veljko Milenkovic","orcid":null},"institutions":[{"id":"https://openalex.org/I145608581","display_name":"University of Miami","ror":"https://ror.org/02dgjyy92","country_code":"US","type":"education","lineage":["https://openalex.org/I145608581"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Veljko Milenkovic","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA","Engineering Consultant, 1222 Highway 42, Sturgeon Bay, WI 54235, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA","institution_ids":["https://openalex.org/I145608581"]},{"raw_affiliation_string":"Engineering Consultant, 1222 Highway 42, Sturgeon Bay, WI 54235, USA","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5039907465"],"corresponding_institution_ids":["https://openalex.org/I145608581"],"apc_list":{"value":2180,"currency":"USD","value_usd":2180},"apc_paid":{"value":2180,"currency":"USD","value_usd":2180},"fwci":1.0332,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.74310595,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"7","issue":"1-2","first_page":"25","last_page":"35"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9962999820709229,"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":0.9962999820709229,"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/T12923","display_name":"Digital Image Processing Techniques","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9937999844551086,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/mathematics","display_name":"Mathematics","score":0.6500120162963867},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5810319185256958},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.49091941118240356},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.451813280582428},{"id":"https://openalex.org/keywords/approximation-algorithm","display_name":"Approximation algorithm","score":0.428947389125824},{"id":"https://openalex.org/keywords/approximation-error","display_name":"Approximation error","score":0.4254823625087738},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36741524934768677},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.32807856798171997},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07149285078048706}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6500120162963867},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5810319185256958},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.49091941118240356},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.451813280582428},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.428947389125824},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.4254823625087738},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36741524934768677},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32807856798171997},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07149285078048706},{"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1016/0925-7721(95)00048-8","is_oa":true,"landing_page_url":"https://doi.org/10.1016/0925-7721(95)00048-8","pdf_url":null,"source":{"id":"https://openalex.org/S165542689","display_name":"Computational Geometry","issn_l":"0925-7721","issn":["0925-7721","1879-081X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Geometry","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1016/0925-7721(95)00048-8","is_oa":true,"landing_page_url":"https://doi.org/10.1016/0925-7721(95)00048-8","pdf_url":null,"source":{"id":"https://openalex.org/S165542689","display_name":"Computational Geometry","issn_l":"0925-7721","issn":["0925-7721","1879-081X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W1592882366","https://openalex.org/W1980119849","https://openalex.org/W2028263615","https://openalex.org/W2074843875","https://openalex.org/W2081768503","https://openalex.org/W2102738679","https://openalex.org/W2125512233","https://openalex.org/W2133740833","https://openalex.org/W2133857440","https://openalex.org/W2141040012","https://openalex.org/W2162294210","https://openalex.org/W2479395228","https://openalex.org/W2946128076","https://openalex.org/W3038928148","https://openalex.org/W3123589846","https://openalex.org/W4234862676","https://openalex.org/W6635415891","https://openalex.org/W6678750293","https://openalex.org/W6679882314","https://openalex.org/W6679993972","https://openalex.org/W6683424660"],"related_works":["https://openalex.org/W2364505198","https://openalex.org/W2355835716","https://openalex.org/W2365142053","https://openalex.org/W2361792563","https://openalex.org/W2110817230","https://openalex.org/W3091355678","https://openalex.org/W1606488336","https://openalex.org/W1988569331","https://openalex.org/W4255193494","https://openalex.org/W189995875"],"abstract_inverted_index":{"Several":[0],"algorithms":[1],"are":[2,152,162],"presented":[3],"for":[4,74,157,164],"approximating":[5],"an":[6,15,25,64,120],"orthogonal":[7,16],"rotation":[8],"matrix":[9,17],"M":[10],"in":[11,145,168],"three":[12],"dimensions":[13,147],"by":[14,87],"with":[18,29,68,102,122],"rational":[19],"entries.":[20],"The":[21,56,134],"first":[22],"algorithm":[23,58,118],"generates":[24,98,119],"approximation":[26,65,121,166],"M2(M,":[27],"\u03b5)":[28,67,101],"accuracy":[30,69,103],"\u03b5":[31,104],"and":[32,37,71,89,105,126,159],"(2b":[33,40],"+":[34,41,46],"2)-bit":[35,42],"numerators":[36],"a":[38],"common":[39],"denominator":[43],"(bit-size":[44],"2b":[45],"2),":[47],"where":[48],"b":[49],"=":[50],"\u2308\u2212":[51],"1g\u03b5\u2309":[52],"(\u03b5":[53],"\u2248":[54,124],"2\u2212b).":[55],"second":[57,117],"uses":[59],"basis":[60],"reduction":[61],"to":[62,108],"generate":[63],"M\u03bd(M,":[66],"\u03b5\u03bd1.5":[70],"bit-size":[72,106],"\u03bdb":[73],"some":[75],"1.5":[76,125],"\u2264":[77,79],"\u03bd":[78,82,123],"6":[80],"(but":[81],"cannot":[83],"be":[84,109],"controlled":[85],"except":[86],"trial":[88],"error).":[90],"A":[91],"third":[92,132],"algorithm,":[93],"based":[94],"on":[95],"integer":[96],"programming,":[97],"optimal":[99],"Mopt(M,":[100],"proven":[107],"no":[110],"more":[111,153],"than":[112,130,154],"1.5b.":[113],"In":[114],"practice,":[115],"the":[116,131,165,169],"is":[127],"much":[128],"faster":[129],"algorithm.":[133],"best":[135],"bit-sizes":[136],"which":[137],"one":[138],"could":[139],"obtain":[140],"using":[141],"previously":[142],"known":[143],"results":[144],"two":[146],"(Canny":[148],"et":[149],"al.,":[150],"1992)":[151],"3b":[155],"bits":[156],"numerator":[158],"denominator.":[160],"Applications":[161],"described":[163],"functions":[167],"area":[170],"of":[171],"solid":[172],"modeling.":[173]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2026-04-05T06:14:27.290980","created_date":"2025-10-10T00:00:00"}