Improved Error Bounds for Inner Products in Floating-Point Arithmetic
Given two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for an inner product returns a floating-point number ${\widehat r}$ such that $|{\widehat r}- x^Ty| \leqslant nu|x|^T|y|$ with $u$ the unit...
Uložené v:
| Vydané v: | SIAM journal on matrix analysis and applications Ročník 34; číslo 2; s. 338 - 344 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Philadelphia
Society for Industrial and Applied Mathematics
01.01.2013
|
| Predmet: | |
| ISSN: | 0895-4798, 1095-7162 |
| On-line prístup: | Získať plný text |
| Tagy: |
Pridať tag
Žiadne tagy, Buďte prvý, kto otaguje tento záznam!
|
| Abstract | Given two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for an inner product returns a floating-point number ${\widehat r}$ such that $|{\widehat r}- x^Ty| \leqslant nu|x|^T|y|$ with $u$ the unit roundoff. This result, which holds for any radix and with no restriction on $n$, can be seen as a generalization of a similar bound given in [S. M. Rump, BIT, 52 (2012), pp. 201--220] for recursive summation in radix $2$, namely, $|{\widehat r}- x^Te| \leqslant (n-1)u|x|^Te$ with $e=[1,1,\ldots,1]^T$. As a direct consequence, the error bound for the floating-point approximation $\widehat{C}$ of classical matrix multiplication with inner dimension $n$ simplifies to $|\widehat{C}-AB|\leqslant nu|A||B|$. [PUBLICATION ABSTRACT] |
|---|---|
| AbstractList | Given two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for an inner product returns a floating-point number ${\widehat r}$ such that $|{\widehat r}- x theta y| \leqslant nu|x| theta |y|$ with $u$ the unit roundoff. This result, which holds for any radix and with no restriction on $n$, can be seen as a generalization of a similar bound given in [S. M. Rump, BIT, 52 (2012), pp. 201--220] for recursive summation in radix $2$, namely, $|{\widehat r}- x theta e| \leqslant (n-1)u|x| theta e$ with $e=[1,1,\ldots,1] theta $. As a direct consequence, the error bound for the floating-point approximation $\widehat{C}$ of classical matrix multiplication with inner dimension $n$ simplifies to $|\widehat{C}-AB|\leqslant nu|A||B|$. Given two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for inner product returns a floating-point number $\hat r$ such that $|{\hat r}-x^Ty| \le nu|x|^T|y|$ with $u$ the unit roundoff. This result, which holds for any radix and with no restriction on $n$, can be seen as a generalization of a similar bound given in~\cite{Rump12} for recursive summation in radix $2$, namely $|{\hat r}- x^Te| \le (n-1)u|x|^Te$ with $e=[1,1,\ldots,1]^T$. As a direct consequence, the error bound for the floating-point approximation $\hat C$ of classical matrix multiplication with inner dimension $n$ simplifies to $|\hat{C}-AB|\le nu|A||B|$. Given two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for an inner product returns a floating-point number ${\widehat r}$ such that $|{\widehat r}- x^Ty| \leqslant nu|x|^T|y|$ with $u$ the unit roundoff. This result, which holds for any radix and with no restriction on $n$, can be seen as a generalization of a similar bound given in [S. M. Rump, BIT, 52 (2012), pp. 201--220] for recursive summation in radix $2$, namely, $|{\widehat r}- x^Te| \leqslant (n-1)u|x|^Te$ with $e=[1,1,\ldots,1]^T$. As a direct consequence, the error bound for the floating-point approximation $\widehat{C}$ of classical matrix multiplication with inner dimension $n$ simplifies to $|\widehat{C}-AB|\leqslant nu|A||B|$. [PUBLICATION ABSTRACT] |
| Author | Rump, Siegfried M. Jeannerod, Claude-Pierre |
| Author_xml | – sequence: 1 givenname: Claude-Pierre surname: Jeannerod fullname: Jeannerod, Claude-Pierre – sequence: 2 givenname: Siegfried M. surname: Rump fullname: Rump, Siegfried M. |
| BackLink | https://inria.hal.science/hal-00840926$$DView record in HAL |
| BookMark | eNplkU9PAjEQxRuDiYAe_AabeNHDSrvt9s8RCQgJiRz03JTdrpQsLbZdEr-9JSgmeprJy28m780MQM86qwG4RfARIcxGqIBcEML5BegjKMqcIVr0QD-pZU6Y4FdgEMIWQkSJQH0wXez23h10nU29dz57cp2tQ9akdmGt9tnKu7qrYsiMzWatU9HY93zljI3Z2Ju42eloqmtw2ag26JvvOgRvs-nrZJ4vX54Xk_Eyr0hRxpxSgbGoRMMxEargUOB13SgFtS4ZImyt6ZqVVNBGFYo2jcKspiJJpFKYqwIPwcNp70a1cu_NTvlP6ZSR8_FSHjUIOYGioAeU2PsTm_J9dDpEuTOh0m2rrHZdkKikDDLGOE3o3R906zpvUxKJcIEJhZCUiRqdqMq7ELxuZGViuoez0SvTSgTl8QXy_IJfu-eJH8__2S-XOYTS |
| CitedBy_id | crossref_primary_10_1137_20M1334796 crossref_primary_10_1137_19M1257780 crossref_primary_10_1016_j_amc_2018_02_004 crossref_primary_10_1587_nolta_11_279 crossref_primary_10_1007_s10915_025_02978_y crossref_primary_10_1088_1742_6596_1218_1_012011 crossref_primary_10_1007_s00211_023_01370_y crossref_primary_10_1137_24M1685109 crossref_primary_10_1137_130927231 crossref_primary_10_1177_10943420241313064 crossref_primary_10_1007_s10589_025_00676_x crossref_primary_10_1137_18M1226312 crossref_primary_10_1109_TC_2015_2441714 crossref_primary_10_1587_nolta_6_360 crossref_primary_10_15748_jasse_12_44 crossref_primary_10_1080_10236198_2023_2236728 crossref_primary_10_1007_s00211_015_0759_9 crossref_primary_10_1007_s13160_024_00677_z crossref_primary_10_1365_s13291_016_0138_1 crossref_primary_10_1587_nolta_6_364 crossref_primary_10_1007_s13042_022_01550_6 crossref_primary_10_1007_s11075_021_01186_7 crossref_primary_10_1016_j_cam_2015_04_010 crossref_primary_10_1088_1742_6596_1490_1_012062 crossref_primary_10_1016_j_parco_2019_102571 crossref_primary_10_1007_s11075_015_9967_8 crossref_primary_10_1088_1742_6596_1821_1_012022 crossref_primary_10_1587_nolta_5_339 crossref_primary_10_1109_TC_2019_2924005 crossref_primary_10_1137_22M1522619 crossref_primary_10_1007_s13160_024_00644_8 |
| Cites_doi | 10.1007/s10543-011-0342-4 10.1007/BF02165411 10.1007/PL00009321 10.1137/050645671 10.1145/227699.227701 10.1016/S0747-7171(08)80013-2 10.1090/S0025-5718-07-01931-X |
| ContentType | Journal Article |
| Copyright | 2013, Society for Industrial and Applied Mathematics Distributed under a Creative Commons Attribution 4.0 International License |
| Copyright_xml | – notice: 2013, Society for Industrial and Applied Mathematics – notice: Distributed under a Creative Commons Attribution 4.0 International License |
| DBID | AAYXX CITATION 3V. 7WY 7WZ 7X2 7XB 87Z 88A 88F 88I 88K 8AL 8FE 8FG 8FH 8FK 8FL 8G5 ABJCF ABUWG AFKRA ARAPS ATCPS AZQEC BBNVY BENPR BEZIV BGLVJ BHPHI CCPQU D1I DWQXO FRNLG F~G GNUQQ GUQSH HCIFZ JQ2 K60 K6~ K7- KB. L.- L6V LK8 M0C M0K M0N M1Q M2O M2P M2T M7P M7S MBDVC P5Z P62 PATMY PDBOC PHGZM PHGZT PKEHL PQBIZ PQBZA PQEST PQGLB PQQKQ PQUKI PRINS PTHSS PYCSY Q9U 7SC 8FD L7M L~C L~D 1XC VOOES |
| DOI | 10.1137/120894488 |
| DatabaseName | CrossRef ProQuest Central (Corporate) ABI/INFORM Collection ABI/INFORM Global (PDF only) Agricultural Science Collection ProQuest Central (purchase pre-March 2016) ABI/INFORM Collection Biology Database (Alumni Edition) Military Database (Alumni Edition) Science Database (Alumni Edition) Telecommunications (Alumni Edition) Computing Database (Alumni Edition) ProQuest SciTech Collection ProQuest Technology Collection ProQuest Natural Science Collection ProQuest Central (Alumni) (purchase pre-March 2016) ABI/INFORM Collection (Alumni) Research Library (Alumni) Materials Science & Engineering Collection (subscription) ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection Agricultural & Environmental Science Collection ProQuest Central Essentials Biological Science Collection ProQuest Central Business Premium Collection Technology Collection Natural Science Collection ProQuest One Community College ProQuest Materials Science Collection ProQuest Central Business Premium Collection (Alumni) ABI/INFORM Global (Corporate) ProQuest Central Student Research Library Prep SciTech Premium Collection ProQuest Computer Science Collection ProQuest Business Collection (Alumni Edition) ProQuest Business Collection Computer Science Database Materials Science Database ABI/INFORM Professional Advanced ProQuest Engineering Collection Biological Sciences ABI/INFORM Global Agricultural Science Database Computing Database Military Database Research Library Science Database Telecommunications Database Biological Science Database Engineering Database (subscription) Research Library (Corporate) AAdvanced Technologies & Aerospace Database (subscription) ProQuest Advanced Technologies & Aerospace Collection Environmental Science Database Materials Science Collection ProQuest Central Premium ProQuest One Academic ProQuest One Academic Middle East (New) ProQuest One Business ProQuest One Business (Alumni) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection (ProQuest) Environmental Science Collection ProQuest Central Basic Computer and Information Systems Abstracts Technology Research Database Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Hyper Article en Ligne (HAL) Hyper Article en Ligne (HAL) (Open Access) |
| DatabaseTitle | CrossRef Agricultural Science Database ProQuest Business Collection (Alumni Edition) Research Library Prep Computer Science Database ProQuest Central Student ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection SciTech Premium Collection ProQuest Military Collection ProQuest Central China ABI/INFORM Complete ProQuest Telecommunications ProQuest One Applied & Life Sciences Natural Science Collection Biological Science Collection ProQuest Central (New) Engineering Collection Advanced Technologies & Aerospace Collection Business Premium Collection ABI/INFORM Global Engineering Database ProQuest Science Journals (Alumni Edition) ProQuest Biological Science Collection ProQuest One Academic Eastern Edition Agricultural Science Collection ProQuest Technology Collection ProQuest Telecommunications (Alumni Edition) Biological Science Database ProQuest Business Collection Environmental Science Collection ProQuest One Academic UKI Edition Environmental Science Database ProQuest One Academic ProQuest One Academic (New) ABI/INFORM Global (Corporate) ProQuest One Business Technology Collection ProQuest One Academic Middle East (New) Materials Science Collection ProQuest Central (Alumni Edition) ProQuest One Community College Research Library (Alumni Edition) ProQuest Natural Science Collection ProQuest Biology Journals (Alumni Edition) ProQuest Central ABI/INFORM Professional Advanced ProQuest Engineering Collection ProQuest Central Korea Agricultural & Environmental Science Collection Materials Science Database ProQuest Research Library ABI/INFORM Complete (Alumni Edition) ProQuest Materials Science Collection ProQuest Computing ABI/INFORM Global (Alumni Edition) ProQuest Central Basic ProQuest Science Journals ProQuest Computing (Alumni Edition) ProQuest Military Collection (Alumni Edition) ProQuest SciTech Collection Advanced Technologies & Aerospace Database Materials Science & Engineering Collection ProQuest One Business (Alumni) ProQuest Central (Alumni) Business Premium Collection (Alumni) Computer and Information Systems Abstracts Technology Research Database Computer and Information Systems Abstracts – Academic Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional |
| DatabaseTitleList | Computer and Information Systems Abstracts Agricultural Science Database |
| Database_xml | – sequence: 1 dbid: BENPR name: ProQuest Central url: https://www.proquest.com/central sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics Computer Science |
| EISSN | 1095-7162 |
| EndPage | 344 |
| ExternalDocumentID | oai:HAL:hal-00840926v1 2935824121 10_1137_120894488 |
| GroupedDBID | -~X .4S .DC 123 186 4.4 7WY 7X2 7XC 88I 8CJ 8FE 8FG 8FH 8FL 8G5 8V8 AALVN AASXH AAYXX ABDBF ABDPE ABJCF ABKAD ABMZU ABUWG ACGFO ACGOD ACIWK ACPRK ACUHS ADBBV AEMOZ AENEX AFFHD AFFNX AFKRA AFRAH AHQJS AKVCP ALMA_UNASSIGNED_HOLDINGS ANXRF ARAPS ARCSS ATCPS AZQEC BBNVY BENPR BEZIV BGLVJ BHPHI BPHCQ CCPQU CITATION CS3 CZ9 D0L D1I D1J D1K DQ2 DU5 DWQXO EAP EBR EBS EBU ECS EDO EJD EMK EST ESX FA8 FRNLG GNUQQ GUQSH H13 HCIFZ H~9 I-F K1G K6- K60 K6V K6~ K7- KB. KC. L6V LK5 LK8 M0C M0K M1Q M2O M2P M7P M7R M7S P1Q P2P P62 PATMY PDBOC PHGZM PHGZT PQBIZ PQBZA PQGLB PQQKQ PROAC PTHSS PYCSY RJG RNS RSI TH9 TN5 TUS YNT ZKB ZY4 3V. 7XB 88A 88K 8AL 8FK JQ2 L.- M0N M2T MBDVC PKEHL PQEST PQUKI PRINS Q9U 7SC 8FD L7M L~C L~D PUEGO 1XC VOOES |
| ID | FETCH-LOGICAL-c425t-669339c9f8349a28093bdfaa0ee57147be6b75696fa2a6ffa37d69b754ca38a23 |
| IEDL.DBID | K7- |
| ISICitedReferencesCount | 42 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000321043700003&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0895-4798 |
| IngestDate | Sat Nov 29 15:13:24 EST 2025 Fri Sep 05 09:48:30 EDT 2025 Fri Jul 25 12:20:17 EDT 2025 Sat Nov 29 02:43:41 EST 2025 Tue Nov 18 21:23:58 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 2 |
| Keywords | rounding error analysis unit in the first place floating-point inner product |
| Language | English |
| License | Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c425t-669339c9f8349a28093bdfaa0ee57147be6b75696fa2a6ffa37d69b754ca38a23 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23 |
| OpenAccessLink | https://inria.hal.science/hal-00840926 |
| PQID | 1323460045 |
| PQPubID | 666305 |
| PageCount | 7 |
| ParticipantIDs | hal_primary_oai_HAL_hal_00840926v1 proquest_miscellaneous_1567077786 proquest_journals_1323460045 crossref_citationtrail_10_1137_120894488 crossref_primary_10_1137_120894488 |
| PublicationCentury | 2000 |
| PublicationDate | 2013-01-01 |
| PublicationDateYYYYMMDD | 2013-01-01 |
| PublicationDate_xml | – month: 01 year: 2013 text: 2013-01-01 day: 01 |
| PublicationDecade | 2010 |
| PublicationPlace | Philadelphia |
| PublicationPlace_xml | – name: Philadelphia |
| PublicationTitle | SIAM journal on matrix analysis and applications |
| PublicationYear | 2013 |
| Publisher | Society for Industrial and Applied Mathematics |
| Publisher_xml | – name: Society for Industrial and Applied Mathematics |
| References | atypb9 atypb8 atypb1 atypb3 atypb2 atypb10 atypb7 |
| References_xml | – ident: atypb7 doi: 10.1007/s10543-011-0342-4 – ident: atypb10 doi: 10.1007/BF02165411 – ident: atypb9 doi: 10.1007/PL00009321 – ident: atypb8 doi: 10.1137/050645671 – ident: atypb3 doi: 10.1145/227699.227701 – ident: atypb2 doi: 10.1016/S0747-7171(08)80013-2 – ident: atypb1 doi: 10.1090/S0025-5718-07-01931-X |
| SSID | ssj0016491 |
| Score | 2.2539868 |
| Snippet | Given two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order... |
| SourceID | hal proquest crossref |
| SourceType | Open Access Repository Aggregation Database Enrichment Source Index Database |
| StartPage | 338 |
| SubjectTerms | Approximation Arithmetic Computer Arithmetic Computer Science Constrictions Errors Floating point arithmetic Mathematical analysis Numerical Analysis Product returns Rounding Vectors (mathematics) |
| Title | Improved Error Bounds for Inner Products in Floating-Point Arithmetic |
| URI | https://www.proquest.com/docview/1323460045 https://www.proquest.com/docview/1567077786 https://inria.hal.science/hal-00840926 |
| Volume | 34 |
| WOSCitedRecordID | wos000321043700003&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVPQU databaseName: ABI/INFORM Collection customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: 7WY dateStart: 19800301 isFulltext: true titleUrlDefault: https://www.proquest.com/abicomplete providerName: ProQuest – providerCode: PRVPQU databaseName: ABI/INFORM Global customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: M0C dateStart: 19800301 isFulltext: true titleUrlDefault: https://search.proquest.com/abiglobal providerName: ProQuest – providerCode: PRVPQU databaseName: Advanced Technologies & Aerospace Database customDbUrl: eissn: 1095-7162 dateEnd: 20241207 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: P5Z dateStart: 19880101 isFulltext: true titleUrlDefault: https://search.proquest.com/hightechjournals providerName: ProQuest – providerCode: PRVPQU databaseName: Agricultural Science Database customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: M0K dateStart: 19800301 isFulltext: true titleUrlDefault: https://search.proquest.com/agriculturejournals providerName: ProQuest – providerCode: PRVPQU databaseName: Biological Science Database customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: M7P dateStart: 19800301 isFulltext: true titleUrlDefault: http://search.proquest.com/biologicalscijournals providerName: ProQuest – providerCode: PRVPQU databaseName: Computer Science Database customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: K7- dateStart: 19800301 isFulltext: true titleUrlDefault: http://search.proquest.com/compscijour providerName: ProQuest – providerCode: PRVPQU databaseName: Engineering Database customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: M7S dateStart: 19800301 isFulltext: true titleUrlDefault: http://search.proquest.com providerName: ProQuest – providerCode: PRVPQU databaseName: Environmental Science Database customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: PATMY dateStart: 19800301 isFulltext: true titleUrlDefault: http://search.proquest.com/environmentalscience providerName: ProQuest – providerCode: PRVPQU databaseName: Materials Science Database customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: KB. dateStart: 19800301 isFulltext: true titleUrlDefault: http://search.proquest.com/materialsscijournals providerName: ProQuest – providerCode: PRVPQU databaseName: Military Database customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: M1Q dateStart: 19800301 isFulltext: true titleUrlDefault: https://search.proquest.com/military providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest Central customDbUrl: eissn: 1095-7162 dateEnd: 20241207 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: BENPR dateStart: 19880101 isFulltext: true titleUrlDefault: https://www.proquest.com/central providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest Research Library customDbUrl: eissn: 1095-7162 dateEnd: 20241207 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: M2O dateStart: 19880101 isFulltext: true titleUrlDefault: https://search.proquest.com/pqrl providerName: ProQuest – providerCode: PRVPQU databaseName: Science Database customDbUrl: eissn: 1095-7162 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0016491 issn: 0895-4798 databaseCode: M2P dateStart: 19800301 isFulltext: true titleUrlDefault: https://search.proquest.com/sciencejournals providerName: ProQuest |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1Lb9QwEB6xLQc48EYslFVAHLiYJn7GJ9StdlXUdglQoOUSeRNbXakkJUn7-_F4s0EgxIXLp8i2rCgzfs043wfwKta08OuOJcxIQzg1khjOJBHWMimUcXHQIvhypBaL9PRUZ33Are2vVW7mxDBRl3WBMfJdf2piXOIO5O3lD4KqUZhd7SU0RrCdUJqgnx8qMmQRJF8r5qVaYAQp7ZmFEqZ2E-pL_dEk_W09Gp3jbcg_JuWw0szv_u873oM7_R4z2ls7xX24YasHcPt4IGhtH8JsHUywZTRrmrqJpqiu1EZ-Bxu9QzGuKFszwbbRqormF7XB29Ekq1dV5_tddeff8efHR_B5PjvZPyC9ogIp_NjsiJSaMV1olzKuDU1jzZalMya2VqiEq6WVSyWkls54qzlnmCql9kW8MCw1lD2Graqu7BOIjMOUpNM2KTUXghrBbGkKzbjgrnB0DK833zUverpxVL24yMOxg6l8MMEYXg5NL9ccG39t5I0z1CMr9sHeUY5lqAngfU5eJ2PY2Rgm78dim_-yyhheDNV-FGFqxFS2vvJthFSxQi69p__u4hncokEQA4MwO7DVNVf2OdwsrrtV20xgpL6eTWB7OltkHyfBGRGnbzwex_sBDxGTD4j0fcAs4AmiCs_qk8dMfPsJLevrvQ |
| linkProvider | ProQuest |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1Nb9QwEB2VggQc-EZdKGAQSFyibuKv-IBQgV3tqtvVHgq0p-BNbHWlNilJWsSf4jcyk2SDQIhbD1xty0rilxnPjP0ewMuhiVL0Oy7gVtlARFYFVnAVSOe4ktr6YaNF8Gmm5_P48NAsNuDH-i4MHatc28TGUGdFSjnyHYyauFC0A3l79jUg1Siqrq4lNFpY7Lnv3zBkq95MP-D6voqi8ejg_SToVAWCFPFZB0phDG9S42MujI1iDOmXmbd26JzUodBLp5ZaKqO8xSf33nKdKYNNIrU8tkR0gCb_KtpxTVIR-vNRX7VQolXoi42kjFXcMRmFXO-EEbZiKBT_5v-uHNPpyz-cQOPZxrf_t29yB251e2i224L-Lmy4_B7c3O8JaKv7MGqTJS5jo7IsSvaO1KMqhjt0NiWxMbZomW4rtsrZ-KSwdPo7WBSrvMZ5V_XxKV3ufAAfL-U9HsJmXuRuC5j1VHL1xoWZEVJGVnKX2dRwIYVPfTSA1-t1TNKOTp1UPU6SJqziOumXfAAv-qFnLYfIXwchGPp-Yv2e7M4SaiPNA_yn1EU4gO01EJLO1lTJLxQM4HnfjVaCSj82d8U5jpFKDzVxBT769xTP4PrkYH-WzKbzvcdwI2rEPyjhtA2bdXnunsC19KJeVeXTBvoMvlw2rn4CP949QQ |
| 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=Improved+Error+Bounds+for+Inner+Products+in+Floating-Point+Arithmetic&rft.jtitle=SIAM+journal+on+matrix+analysis+and+applications&rft.au=Jeannerod%2C+Claude-Pierre&rft.au=Rump%2C+Siegfried+M.&rft.date=2013-01-01&rft.issn=0895-4798&rft.eissn=1095-7162&rft.volume=34&rft.issue=2&rft.spage=338&rft.epage=344&rft_id=info:doi/10.1137%2F120894488&rft.externalDBID=n%2Fa&rft.externalDocID=10_1137_120894488 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0895-4798&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0895-4798&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0895-4798&client=summon |