{"id":"https://openalex.org/W1977204226","doi":"https://doi.org/10.1145/1145768.1145775","title":"Hybrid symbolic-numeric computation","display_name":"Hybrid symbolic-numeric computation","publication_year":2006,"publication_date":"2006-07-09","ids":{"openalex":"https://openalex.org/W1977204226","doi":"https://doi.org/10.1145/1145768.1145775","mag":"1977204226"},"language":"en","primary_location":{"id":"doi:10.1145/1145768.1145775","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1145768.1145775","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 2006 international symposium on Symbolic and algebraic computation","raw_type":"proceedings-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/A5056874049","display_name":"Erich Kaltofen","orcid":"https://orcid.org/0000-0003-2739-3230"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Erich Kaltofen","raw_affiliation_strings":["Massachusetts Institute of Technology, Cambridge, Massachusetts","Massachusetts Institute of Technology, Cambridge, Massachusetts#TAB#"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, Massachusetts","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, Massachusetts#TAB#","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5026679766","display_name":"Lihong Zhi","orcid":"https://orcid.org/0000-0001-9973-2217"},"institutions":[{"id":"https://openalex.org/I4210120485","display_name":"Academy of Mathematics and Systems Science","ror":"https://ror.org/02jkmyk67","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lihong Zhi","raw_affiliation_strings":["Academy of Mathematics and Systems Science, Beijing, China"],"affiliations":[{"raw_affiliation_string":"Academy of Mathematics and Systems Science, Beijing, China","institution_ids":["https://openalex.org/I4210120485"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5056874049"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":0.327,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.57260987,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"7","last_page":"7"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9986000061035156,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9986000061035156,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.98580002784729,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9857000112533569,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/symbolic-computation","display_name":"Symbolic computation","score":0.8187428712844849},{"id":"https://openalex.org/keywords/overdetermined-system","display_name":"Overdetermined system","score":0.5448226928710938},{"id":"https://openalex.org/keywords/symbolic-numeric-computation","display_name":"Symbolic-numeric computation","score":0.5428694486618042},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5074562430381775},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.48481693863868713},{"id":"https://openalex.org/keywords/linear-algebra","display_name":"Linear algebra","score":0.4484342932701111},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4276597797870636},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.4251534640789032},{"id":"https://openalex.org/keywords/system-of-polynomial-equations","display_name":"System of polynomial equations","score":0.41469836235046387},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.38900354504585266},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.36314675211906433},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3335971236228943},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.14308124780654907}],"concepts":[{"id":"https://openalex.org/C110812573","wikidata":"https://www.wikidata.org/wiki/Q175515","display_name":"Symbolic computation","level":2,"score":0.8187428712844849},{"id":"https://openalex.org/C81901731","wikidata":"https://www.wikidata.org/wiki/Q331348","display_name":"Overdetermined system","level":2,"score":0.5448226928710938},{"id":"https://openalex.org/C167324811","wikidata":"https://www.wikidata.org/wiki/Q12207781","display_name":"Symbolic-numeric computation","level":3,"score":0.5428694486618042},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5074562430381775},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.48481693863868713},{"id":"https://openalex.org/C139352143","wikidata":"https://www.wikidata.org/wiki/Q82571","display_name":"Linear algebra","level":2,"score":0.4484342932701111},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4276597797870636},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.4251534640789032},{"id":"https://openalex.org/C105335568","wikidata":"https://www.wikidata.org/wiki/Q7663735","display_name":"System of polynomial equations","level":3,"score":0.41469836235046387},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.38900354504585266},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36314675211906433},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3335971236228943},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.14308124780654907},{"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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1145768.1145775","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1145768.1145775","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 2006 international symposium on Symbolic and algebraic computation","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":4,"referenced_works":["https://openalex.org/W32371573","https://openalex.org/W1985565156","https://openalex.org/W2060143865","https://openalex.org/W2127391761"],"related_works":["https://openalex.org/W2363922099","https://openalex.org/W89774232","https://openalex.org/W4246795820","https://openalex.org/W2913172947","https://openalex.org/W4386576538","https://openalex.org/W2952511264","https://openalex.org/W2090112195","https://openalex.org/W2912098470","https://openalex.org/W1977204226","https://openalex.org/W2015689303"],"abstract_inverted_index":{"Several":[0],"standard":[1],"problems":[2,110,167,193,196],"in":[3,39,114,162,197],"symbolic":[4,55,176,191,412],"computation,":[5],"such":[6],"as":[7,130,194],"greatest":[8],"common":[9],"divisor":[10],"and":[11,77,87,147,156,159,200,202,264,274,329,359,365,370,414,440],"factorization":[12,334],"of":[13,23,65,96,102,135,144,181,239,258,317,323,331,351,380,389,394,421,436],"polynomials,":[14],"sparse":[15],"interpolation,":[16],"or":[17,47,175,232,276,368],"computing":[18,309],"solutions":[19,28,420],"to":[20,43,104,140,164,187,221,301,308,362,373,407],"overdetermined":[21],"systems":[22],"polynomial":[24,261,313,423],"equations":[25,398],"have":[26],"non-trivial":[27,70],"only":[29],"if":[30],"the":[31,40,53,66,97,115,120,124,142,222,237,240,259,303,310,332,340,352,376,422,433,437],"input":[32,243],"coefficients":[33,41,67,238,244],"satisfy":[34],"certain":[35],"algebraic":[36],"constraints.":[37],"Errors":[38],"due":[42],"floating":[44],"point":[45],"round-off":[46],"through":[48],"phsical":[49],"measurement":[50],"thus":[51],"render":[52],"exact":[54],"algorithms":[56,76,138,345],"unusable.":[57],"By":[58],"symbolic-numeric":[59,137],"methods":[60,177,286],"one":[61],"computes":[62],"minimal":[63],"deformations":[64],"that":[68,168,205],"yield":[69],"results.":[71],"We":[72,298,385],"will":[73,253,266],"present":[74,267],"hybrid":[75,136],"benchmark":[78],"computations":[79],"based":[80,269,347],"on":[81,126,185,203,270,348],"Gauss-Newton":[82],"optimization,":[83,153],"singular":[84,312],"value":[85],"decomposition(SVD)":[86],"structure-preserving":[88],"total":[89],"least":[90,320],"squares":[91],"(STLS)":[92],"fitting":[93],"for":[94,226,280,339],"several":[95,189,227],"above":[98],"problems.A":[99],"significant":[100],"body":[101],"results":[103],"solve":[105,165],"those":[106],"\"approximate":[107],"computer":[108],"algebra\"":[109],"has":[111],"been":[112],"discovered":[113],"past":[116],"10":[117],"years.":[118],"In":[119,235,249],"Computer":[121],"Algebra":[122],"Handbook":[123],"section":[125],"\"Hybrid":[127],"Methods\"":[128],"concludes":[129],"follows":[131,336],"[2]:":[132],"\"The":[133],"challenge":[134],"is":[139,184],"explore":[141],"effects":[143],"imprecision,":[145],"discontinuity,":[146],"algorithmic":[148],"complexity":[149],"by":[150,173,356,411,427],"applying":[151],"mathematical":[152,166],"perturbation":[154],"theory,":[155],"inexact":[157],"arithmetic":[158],"other":[160],"tools":[161],"order":[163],"today":[169],"are":[170,346,405,425],"not":[171],"solvable":[172],"numerical":[174,195,282],"alone.\"":[178],"The":[179,284,403,419],"focus":[180],"our":[182,250,337],"tutorial":[183,251],"how":[186,300],"formulate":[188],"approximate":[190,223,260,305,333,341,377],"computation":[192],"linear":[198,395],"algebra":[199],"optimization":[201,272,372],"software":[204],"realizes":[206],"their":[207],"solutions.Approximate":[208],"Greatest":[209],"Common":[210],"Divisors":[211],"[3].":[212],"Our":[213,327,344],"paper":[214],"at":[215,319],"this":[216],"conference":[217],"presents":[218],"a":[219,255,315,349,387,392],"solution":[220,328],"GCD":[224,262,306,342],"problem":[225,263,307,335],"multivariate":[228,378],"polynomials":[229,390],"with":[230,314,400],"real":[231],"complex":[233],"coefficients.":[234,402],"addition,":[236],"minimally":[241],"deformed":[242],"can":[245,287],"be":[246],"linearly":[247],"constrained.":[248],"we":[252,265],"give":[254],"precise":[256],"definition":[257],"techniques":[268],"parametric":[271],"(slow)":[273],"STLS":[275,369],"Gauss/Newton":[277,371],"iteration":[278],"(fast)":[279],"its":[281,441],"solution.":[283],"fast":[285],"compute":[288,375],"globally":[289],"optimal":[290],"solutions,":[291],"but":[292],"they":[293],"cannot":[294],"verify":[295],"global":[296],"optimality.":[297],"show":[299],"apply":[302],"constrained":[304],"nearest":[311],"root":[316],"multiplicity":[318],"k\u22652.Approximate":[321],"Factorization":[322],"Multivariate":[324],"Polynomials":[325],"[1].":[326],"implementation":[330],"approach":[338],"problem.":[343],"generalization":[350],"differential":[353,397],"forms":[354],"introduced":[355],"W.":[357],"Ruppert":[358],"S.":[360],"Gao":[361],"many":[363],"variables,":[364],"use":[366],"SVD":[367],"numerically":[374],"factors.Solutions":[379],"Zero-dimensional":[381],"Polynomial":[382],"Systems":[383],"[4].":[384],"translate":[386],"system":[388,393,424,439],"into":[391],"partial":[396],"(PDEs)":[399],"constant":[401],"PDEs":[404],"brought":[406],"an":[408,429],"involutive":[409,438],"form":[410],"prolongations":[413],"numeric":[415],"projections":[416],"via":[417],"SVD.":[418],"obtained":[426],"solving":[428],"eigen-problem":[430],"constructed":[431],"from":[432],"null":[434],"spaces":[435],"geometric":[442],"projections.":[443]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}