Interval Quadratic Equations: A Review

In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an...

Full description

Saved in:
Bibliographic Details
Published inAppliedMath Vol. 3; no. 4; pp. 909 - 956
Main Authors Elishakoff, Isaac, Yvain, Nicolas
Format Journal Article
LanguageEnglish
Published MDPI AG 01.12.2023
Subjects
interval coefficients
interval quadratic equations
interval variables
real roots
uncertain variables
uncertainty
Online AccessGet full text
ISSN2673-9909
2673-9909
DOI10.3390/appliedmath3040048

Cover

Abstract In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an interval variable. The four methods reviewed here in order to solve this problem are: (i) the method of classic interval analysis used by Elishakoff and Daphnis, (ii) the direct method based on minimizations and maximizations also used by the same authors, (iii) the method of quantifier elimination used by Ioakimidis, and (iv) the interval parametrization method suggested by Elishakoff and Miglis and again based on minimizations and maximizations. We will also compare the results yielded by all these methods by using the computer algebra system Mathematica for computer evaluations (including quantifier eliminations) in order to conclude which method would be the most efficient way to solve problems relevant to interval quadratic equations.
AbstractList In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an interval variable. The four methods reviewed here in order to solve this problem are: (i) the method of classic interval analysis used by Elishakoff and Daphnis, (ii) the direct method based on minimizations and maximizations also used by the same authors, (iii) the method of quantifier elimination used by Ioakimidis, and (iv) the interval parametrization method suggested by Elishakoff and Miglis and again based on minimizations and maximizations. We will also compare the results yielded by all these methods by using the computer algebra system Mathematica for computer evaluations (including quantifier eliminations) in order to conclude which method would be the most efficient way to solve problems relevant to interval quadratic equations.
Author Yvain, Nicolas
Elishakoff, Isaac
Author_xml – sequence: 1
  givenname: Isaac
  surname: Elishakoff
  fullname: Elishakoff, Isaac
– sequence: 2
  givenname: Nicolas
  surname: Yvain
  fullname: Yvain, Nicolas
BookMark eNqNkMFKw0AQhhepYK19AU85eYtOMrvdrLdSqhYKouh5mexuNCVN4iZt6du7tSKCF2HgHwa-j-E_Z4O6qR1jlwlcIyq4obatSmfX1L8jcACenbBhOpEYKwVq8Gs_Y-OuWwFAmgmJMhuyq0XdO7-lKnrakPXUlyaaf2xCNnV3G02jZ7ct3e6CnRZUdW78nSP2ejd_mT3Ey8f7xWy6jA1C1scmSbgkweUELSlUKU6sVcYRCSSlCg4cFYJ0goinwhauyB2QMEWaBy7HEVscvbahlW59uSa_1w2V-uvQ-DdNPvxYOR3oMLksuLAchMgFopTW2EMDXEBw4dG1qVva76iqfoQJ6ENz-m9zgUqPlPFN13lX_Af6BAvrdhI
Cites_doi 10.1016/j.mechrescom.2012.04.004
10.1119/1.2344330
10.1137/1.9781611970906
10.3390/app122110725
10.1023/A:1014797921296
10.1137/1.9780898717716
10.1016/S0165-0114(95)00412-2
10.1016/S0747-7171(88)80004-X
10.1007/3-540-07407-4_17
10.1007/s00419-015-1086-4
10.25079/ukhjse.v5n2y2021.pp81-89
ContentType Journal Article
DBID AAYXX
CITATION
ADTOC
UNPAY
DOA
DOI 10.3390/appliedmath3040048
DatabaseName CrossRef
Unpaywall for CDI: Periodical Content
Unpaywall
DOAJ Directory of Open Access Journals
DatabaseTitle CrossRef
DatabaseTitleList CrossRef
Database_xml – sequence: 1
  dbid: DOA
  name: DOAJ Directory of Open Access Journals
  url: https://www.doaj.org/
  sourceTypes: Open Website
– sequence: 2
  dbid: UNPAY
  name: Unpaywall
  url: https://proxy.k.utb.cz/login?url=https://unpaywall.org/
  sourceTypes: Open Access Repository
DeliveryMethod fulltext_linktorsrc
EISSN 2673-9909
EndPage 956
ExternalDocumentID oai_doaj_org_article_5df5dfb7f45d4055b53377dcd4004450
10.3390/appliedmath3040048
10_3390_appliedmath3040048
GroupedDBID AAYXX
AFZYC
ALMA_UNASSIGNED_HOLDINGS
AMVHM
ARCSS
CITATION
GROUPED_DOAJ
MODMG
M~E
OK1
ADTOC
UNPAY
ID FETCH-LOGICAL-c308t-c1147a54763da939236dd9ceaa53a99f40439307e5aa425dfefbe0a5cf2b7a5b3
IEDL.DBID UNPAY
ISSN 2673-9909
IngestDate Fri Oct 03 12:52:02 EDT 2025
Tue Aug 19 17:55:48 EDT 2025
Thu Oct 16 04:25:20 EDT 2025
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 4
Language English
License cc-by
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c308t-c1147a54763da939236dd9ceaa53a99f40439307e5aa425dfefbe0a5cf2b7a5b3
OpenAccessLink https://proxy.k.utb.cz/login?url=https://www.mdpi.com/2673-9909/3/4/48/pdf?version=1701409724
PageCount 48
ParticipantIDs doaj_primary_oai_doaj_org_article_5df5dfb7f45d4055b53377dcd4004450
unpaywall_primary_10_3390_appliedmath3040048
crossref_primary_10_3390_appliedmath3040048
PublicationCentury 2000
PublicationDate 2023-12-01
PublicationDateYYYYMMDD 2023-12-01
PublicationDate_xml – month: 12
  year: 2023
  text: 2023-12-01
  day: 01
PublicationDecade 2020
PublicationTitle AppliedMath
PublicationYear 2023
Publisher MDPI AG
Publisher_xml – name: MDPI AG
References Mamehrashi (ref_4) 2021; 5
ref_14
ref_13
Wyrzykowski (ref_5) 2008; Volume 4967
Alolyan (ref_10) 2007; 1
ref_11
ref_18
ref_17
ref_16
ref_15
Landowski (ref_6) 2017; 93
Hansen (ref_9) 2002; 8
Davenport (ref_25) 1988; 5
Elishakoff (ref_19) 2016; 86
ref_24
ref_23
ref_21
Newburgh (ref_22) 1996; 34
ref_20
ref_1
Buckley (ref_3) 1997; 86
ref_2
ref_26
Elishakoff (ref_12) 2015; 271
ref_7
Elishakoff (ref_8) 2012; 44
References_xml – volume: 44
  start-page: 1
  year: 2012
  ident: ref_8
  article-title: Novel parameterized intervals may lead to sharp bounds
  publication-title: Mech. Res. Commun.
  doi: 10.1016/j.mechrescom.2012.04.004
– volume: 271
  start-page: 1024
  year: 2015
  ident: ref_12
  article-title: Exact enclosures of roots of interval quadratic equations by Sridhara’s and Fagnano’s or modified Fagnano’s formulas
  publication-title: Appl. Math. Comput.
– volume: 34
  start-page: 23
  year: 1996
  ident: ref_22
  article-title: Real, imaginary, and complex numbers: Where does the physics hide?
  publication-title: Phys. Teach.
  doi: 10.1119/1.2344330
– ident: ref_24
– ident: ref_7
  doi: 10.1137/1.9781611970906
– ident: ref_26
– ident: ref_11
  doi: 10.3390/app122110725
– ident: ref_16
– ident: ref_14
– ident: ref_1
– ident: ref_18
– volume: 8
  start-page: 115
  year: 2002
  ident: ref_9
  article-title: Sharp bounds on interval polynomial roots
  publication-title: Reliab. Comput.
  doi: 10.1023/A:1014797921296
– ident: ref_23
– ident: ref_21
– ident: ref_17
  doi: 10.1137/1.9780898717716
– volume: 93
  start-page: 65
  year: 2017
  ident: ref_6
  article-title: RDM interval method for solving quadratic interval equation
  publication-title: Prz. Elektrotechniczny
– volume: 86
  start-page: 289
  year: 1997
  ident: ref_3
  article-title: Neural net solutions to fuzzy problems: The quadratic equation
  publication-title: Fuzzy Sets Syst.
  doi: 10.1016/S0165-0114(95)00412-2
– ident: ref_2
– volume: 5
  start-page: 29
  year: 1988
  ident: ref_25
  article-title: Real quantifier elimination is doubly exponential
  publication-title: J. Symb. Comput.
  doi: 10.1016/S0747-7171(88)80004-X
– ident: ref_13
  doi: 10.1007/3-540-07407-4_17
– ident: ref_15
– volume: 86
  start-page: 1203
  year: 2016
  ident: ref_19
  article-title: Generalized Galileo Galilei problem in interval setting for functionally related loads
  publication-title: Arch. Appl. Mech.
  doi: 10.1007/s00419-015-1086-4
– volume: 5
  start-page: 81
  year: 2021
  ident: ref_4
  article-title: A new method for solving interval and fuzzy quadratic equations of dual form
  publication-title: UKH J. Sci. Eng.
  doi: 10.25079/ukhjse.v5n2y2021.pp81-89
– volume: Volume 4967
  start-page: 1392
  year: 2008
  ident: ref_5
  article-title: Fuzzy solution of interval linear equations
  publication-title: Parallel Processing and Applied Mathematics, Proceedings of the 7th International Conference on Parallel Processing and Applied Mathematics (PPAM 2007), Gdansk, Poland, 9–12 September 2007
– ident: ref_20
– volume: 1
  start-page: 1041
  year: 2007
  ident: ref_10
  article-title: Real roots of quadratic interval polynomials
  publication-title: Int. J. Math. Anal.
SSID ssj0002857378
Score 2.2420666
SecondaryResourceType review_article
Snippet In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose...
SourceID doaj
unpaywall
crossref
SourceType Open Website
Open Access Repository
Index Database
StartPage 909
SubjectTerms interval coefficients
interval quadratic equations
interval variables
real roots
uncertain variables
uncertainty
SummonAdditionalLinks – databaseName: DOAJ Directory of Open Access Journals
  dbid: DOA
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LS8NAEB6kF_Ugior1RQ7iRYO1m81mvVVpKYKCYKG3MLuzi0iJrbaI_97ZbC0VD3oQcgoJm8yD-WZ25xuAE8RCOI5sqSuI0gxzyy7l2-llYTjaoc9Rh37nu_u8P8huh3K4NOornAmL9MBRcBeSPF9G-UwSgwtpGJ8oRZaC8WUxW28VeimZeq5LRlIJVcQuGcF5fdwPdsQw8ElEw_0WiWrC_nVYnVVj_HjH0WgpyvQ2YWMOD5NO_KwtWHHVNpzWZTs2ieRhhhRUZpPuJHJ0v10lnSQW-Hdg0Os-3vTT-XyD1IpWMU0t5yIKZcYuTqgZqIicSFuHKAVq7QPxjWYfdBKRXYu888a1UFrfNvyeEbvQqF4qtweJJkYWuTAMRzhEe28syTBDI7Slole6CWdf_1qOI41FyfA_SKb8KZkmXAdxLJ4MFNT1DVZMOVdM-ZtimnC-EOYf1tz_jzUPYC2Mg4_HTQ6hMX2duSMGDVNzXNvHJ215v60
  priority: 102
  providerName: Directory of Open Access Journals
Title Interval Quadratic Equations: A Review
URI https://www.mdpi.com/2673-9909/3/4/48/pdf?version=1701409724
https://doaj.org/article/5df5dfb7f45d4055b53377dcd4004450
UnpaywallVersion publishedVersion
Volume 3
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVAON
  databaseName: DOAJ Directory of Open Access Journals
  customDbUrl:
  eissn: 2673-9909
  dateEnd: 99991231
  omitProxy: true
  ssIdentifier: ssj0002857378
  issn: 2673-9909
  databaseCode: DOA
  dateStart: 20210101
  isFulltext: true
  titleUrlDefault: https://www.doaj.org/
  providerName: Directory of Open Access Journals
– providerCode: PRVEBS
  databaseName: EBSCOhost Mathematics Source - trial do 30.11.2025
  customDbUrl:
  eissn: 2673-9909
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0002857378
  issn: 2673-9909
  databaseCode: AMVHM
  dateStart: 20220601
  isFulltext: true
  titleUrlDefault: https://www.ebsco.com/products/research-databases/mathematics-source
  providerName: EBSCOhost
– providerCode: PRVHPJ
  databaseName: ROAD: Directory of Open Access Scholarly Resources
  customDbUrl:
  eissn: 2673-9909
  dateEnd: 99991231
  omitProxy: true
  ssIdentifier: ssj0002857378
  issn: 2673-9909
  databaseCode: M~E
  dateStart: 20210101
  isFulltext: true
  titleUrlDefault: https://road.issn.org
  providerName: ISSN International Centre
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LS8NAEB5qPYgHH6hYUclBvGhM7WaTrAhSpSJCi4IFPUiYfakotWqr6MHf7mwexcdFhZBD2E12JzM73-zufAuwhpgwQ57NN4nWfoiRIpOyDX87keTt0EYoXL5zuxMddcPjc35egd0yF8Ztq6RQ_CYbpBtRzHwaLkXAgjAIk6Cv7d5zMY_kiMQdWVMjHIPxiBMSr8J4t3PSvHDnyZU18zwZRpF9viJsNAHBa5ar7hdflFH2T8LEsNfH1xe8u_vkZw6n4bJsYb695HZrOJBb6u0beeN_uzADUwUA9Zq5xsxCxfTmYD2bGCSl806HqJ1SKK_1kLOAP-14TS9fQpiH7mHr7ODIL05Q8BWrJwNfUbQTIw9pENEoCAqxSGuhDCJnKIR11DqCrNxwRDJebY2Vpo5c2YakepItQLV33zOL4AlN2CVikgAPgQBrpdLcndLhEl_RxqIGG6Us035OlJFSgOEkn_6UfA32nbhHJR3Jdfbg_vEqLWwmpfbQJWMbck24kkuCpnGslXavCHm9Bpujn_WLby79rfgyVAePQ7NCmGMgV7NYne7t99ZqoWYfrwjXhA
linkProvider Unpaywall
linkToUnpaywall http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LS8NAEB5qexAPPlCxopKDeNG0tZtNsiJIlZYiWBQs1IOE2ZeKpa21VfTXO9ukxcdFhZzCbrKZzOx8s7vzDcAuYswMeTbfxFr7AYaKTMpW_cNYkrdDG6Jw-c4XrbDZDs47vJOD42kujDtWSaH4w2SSroYR82m6FGVWDspBXB5oe_KSrSM5InFH1lQN5qAQckLieSi0W5e1G1dPbtozzZNhFNmnO8JGExC8Z6nqfvFFE8r-BZgf9wb49ord7ic_01iC2-kI0-Mlj6XxSJbU-zfyxv9-wjIsZgDUq6UaswI501uFvcnCICmddzVG7ZRCefWnlAX8-cireekWwhq0G_Xrs6afVVDwFavEI19RtBMhD2gS0SgICrFQa6EMImcohHXUOoKs3HBEMl5tjZWmglzZqqR-kq1DvtfvmQ3whCbsEjJJgIdAgLVSae6qdLjEV7SRKML-VJbJICXKSCjAcJJPfkq-CKdO3LOWjuR6cqM_vEsym0loPHTJyAZcE67kkqBpFGml3SMCXinCwexn_eKdm39rvgX50XBstglzjOROplofpgnVXg
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Interval+Quadratic+Equations%3A+A+Review&rft.jtitle=AppliedMath&rft.au=Elishakoff%2C+Isaac&rft.au=Yvain%2C+Nicolas&rft.date=2023-12-01&rft.issn=2673-9909&rft.eissn=2673-9909&rft.volume=3&rft.issue=4&rft.spage=909&rft.epage=956&rft_id=info:doi/10.3390%2Fappliedmath3040048&rft.externalDBID=n%2Fa&rft.externalDocID=10_3390_appliedmath3040048
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2673-9909&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2673-9909&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2673-9909&client=summon