{"id":"https://openalex.org/W2084568126","doi":"https://doi.org/10.1109/tc.1987.1676886","title":"Algorithmic Phase Diagrams","display_name":"Algorithmic Phase Diagrams","publication_year":1987,"publication_date":"1987-02-01","ids":{"openalex":"https://openalex.org/W2084568126","doi":"https://doi.org/10.1109/tc.1987.1676886","mag":"2084568126"},"language":"en","primary_location":{"id":"doi:10.1109/tc.1987.1676886","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tc.1987.1676886","pdf_url":null,"source":{"id":"https://openalex.org/S157670870","display_name":"IEEE Transactions on Computers","issn_l":"0018-9340","issn":["0018-9340","0016-9340","1557-9956","2326-3814"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Computers","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/A5077508846","display_name":"Hockney","orcid":null},"institutions":[{"id":"https://openalex.org/I71052956","display_name":"University of Reading","ror":"https://ror.org/05v62cm79","country_code":"GB","type":"education","lineage":["https://openalex.org/I71052956"]},{"id":"https://openalex.org/I1319063186","display_name":"Langley Research Center","ror":"https://ror.org/0399mhs52","country_code":"US","type":"facility","lineage":["https://openalex.org/I1319063186","https://openalex.org/I4210124779"]}],"countries":["GB","US"],"is_corresponding":true,"raw_author_name":"Hockney","raw_affiliation_strings":["Department of Computer Science, Reading University","Institute for Computer Applications in Science and Engineering, NASA-Langley Research Center, Hampton, VA, USA","Department of Computer Science, Reading University, Reading, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Reading University","institution_ids":["https://openalex.org/I71052956"]},{"raw_affiliation_string":"Institute for Computer Applications in Science and Engineering, NASA-Langley Research Center, Hampton, VA, USA","institution_ids":["https://openalex.org/I1319063186"]},{"raw_affiliation_string":"Department of Computer Science, Reading University, Reading, UK","institution_ids":["https://openalex.org/I71052956"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5077508846"],"corresponding_institution_ids":["https://openalex.org/I1319063186","https://openalex.org/I71052956"],"apc_list":null,"apc_paid":null,"fwci":0.645,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.7568935,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"C-36","issue":"2","first_page":"231","last_page":"233"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12292","display_name":"Graph Theory and Algorithms","score":0.949400007724762,"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"}},"topics":[{"id":"https://openalex.org/T12292","display_name":"Graph Theory and Algorithms","score":0.949400007724762,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/tridiagonal-matrix","display_name":"Tridiagonal matrix","score":0.816219687461853},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6717467308044434},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.5678719878196716},{"id":"https://openalex.org/keywords/phase-diagram","display_name":"Phase diagram","score":0.5393147468566895},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5225557684898376},{"id":"https://openalex.org/keywords/diagram","display_name":"Diagram","score":0.5036959052085876},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4885190725326538},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.39294180274009705},{"id":"https://openalex.org/keywords/phase","display_name":"Phase (matter)","score":0.2936873137950897},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2576589286327362}],"concepts":[{"id":"https://openalex.org/C51647924","wikidata":"https://www.wikidata.org/wiki/Q1755277","display_name":"Tridiagonal matrix","level":3,"score":0.816219687461853},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6717467308044434},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5678719878196716},{"id":"https://openalex.org/C85906118","wikidata":"https://www.wikidata.org/wiki/Q186693","display_name":"Phase diagram","level":3,"score":0.5393147468566895},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5225557684898376},{"id":"https://openalex.org/C186399060","wikidata":"https://www.wikidata.org/wiki/Q959962","display_name":"Diagram","level":2,"score":0.5036959052085876},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4885190725326538},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.39294180274009705},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.2936873137950897},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2576589286327362},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"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/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tc.1987.1676886","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tc.1987.1676886","pdf_url":null,"source":{"id":"https://openalex.org/S157670870","display_name":"IEEE Transactions on Computers","issn_l":"0018-9340","issn":["0018-9340","0016-9340","1557-9956","2326-3814"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Computers","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W645445933","https://openalex.org/W1480666649","https://openalex.org/W1629601946","https://openalex.org/W1985401272","https://openalex.org/W2010623798","https://openalex.org/W2066446185","https://openalex.org/W2396272875"],"related_works":["https://openalex.org/W4212973497","https://openalex.org/W1980786482","https://openalex.org/W2360550119","https://openalex.org/W3177208105","https://openalex.org/W3165035967","https://openalex.org/W2389963024","https://openalex.org/W2353849135","https://openalex.org/W3085591969","https://openalex.org/W1966138193","https://openalex.org/W2022416154"],"abstract_inverted_index":{"Algorithmic":[0],"phase":[1],"diagrams":[2,60],"are":[3,110],"a":[4,95],"neat":[5],"and":[6,37,71,74],"compact":[7],"representation":[8],"of":[9,12,17,23,35,43,47,52,57,76],"the":[10,14,21,24,32,41,50,68,72,77,85,89,103,120],"results":[11,34],"comparing":[13],"execution":[15],"time":[16],"several":[18],"algorithms":[19],"for":[20,94,106],"solution":[22,42],"same":[25],"problem.":[26],"As":[27],"an":[28,63],"example":[29],"we":[30],"show":[31],"recent":[33],"Gannon":[36],"Van":[38],"Rosendale":[39],"on":[40],"multiple":[44],"tridiagonal":[45,78],"systems":[46],"equations":[48],"in":[49,88],"form":[51],"such":[53],"diagrams.":[54],"The":[55],"act":[56],"preparing":[58],"these":[59],"has":[61],"revealed":[62],"unexpectedly":[64],"complex":[65,123],"relationship":[66],"between":[67],"best":[69,104],"algorithm":[70,105],"number":[73],"size":[75],"systems,":[79],"which":[80],"was":[81],"not":[82],"evident":[83],"from":[84,119],"algebraic":[86],"formulae":[87],"original":[90],"paper.":[91],"Even":[92],"so,":[93],"particular":[96],"computer,":[97],"one":[98],"diagram":[99,121],"suffices":[100],"to":[101,112],"predict":[102],"all":[107],"problems":[108],"that":[109],"likely":[111],"be":[113],"encountered-the":[114],"prediction":[115],"being":[116],"read":[117],"directly":[118],"without":[122],"calculation.":[124]},"counts_by_year":[{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-07T14:30:12.667765","created_date":"2025-10-10T00:00:00"}