Control-Theoretic Forward Error Analysis of Iterative Numerical Algorithms

It has been known for at least five decades that control theory can be used to study iterative algorithms. However, little work can be found in the control systems literature on numerical algorithms, especially on the study of finite precision effects. In this technical note, we consider numerical i...

Full description

Saved in:
Bibliographic Details
Published in:IEEE transactions on automatic control Vol. 58; no. 6; pp. 1524 - 1529
Main Authors: Hasan, A., Kerrigan, Eric C., Constantinides, George A.
Format: Journal Article
Language:English
Published: IEEE 01.06.2013
Subjects:
Algorithm design and analysis
Control systems
Error analysis
Finite precision
forward error analysis
Hardware
Heuristic algorithms
input-to-state stability (ISS)
Iterative methods
numerical algorithms as dynamical systems
Quantization
ISSN:0018-9286, 1558-2523
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Abstract It has been known for at least five decades that control theory can be used to study iterative algorithms. However, little work can be found in the control systems literature on numerical algorithms, especially on the study of finite precision effects. In this technical note, we consider numerical iterative algorithms in finite precision as dynamical systems and study the effects of finite precision using control theory. By using the control tools of input-to-state stability and results from the study of quantization in control systems, we present new systematic ways to find bounds on the forward error for iterative algorithms. The advantages of the proposed schemes are shown by applying them to find bounds for the classical iterative methods for solving a system of linear equations.
AbstractList It has been known for at least five decades that control theory can be used to study iterative algorithms. However, little work can be found in the control systems literature on numerical algorithms, especially on the study of finite precision effects. In this technical note, we consider numerical iterative algorithms in finite precision as dynamical systems and study the effects of finite precision using control theory. By using the control tools of input-to-state stability and results from the study of quantization in control systems, we present new systematic ways to find bounds on the forward error for iterative algorithms. The advantages of the proposed schemes are shown by applying them to find bounds for the classical iterative methods for solving a system of linear equations.
Author Kerrigan, Eric C.
Hasan, A.
Constantinides, George A.
Author_xml – sequence: 1
  givenname: A.
  surname: Hasan
  fullname: Hasan, A.
  email: ammar.hasan07@imperial.ac.uk
  organization: Dept. of Electr. & Electron. Eng., Imperial Coll. London, London, UK
– sequence: 2
  givenname: Eric C.
  surname: Kerrigan
  fullname: Kerrigan, Eric C.
  email: e.kerrigan@imperial.ac.uk
  organization: Dept. of Aeronaut., Imperial Coll. London, London, UK
– sequence: 3
  givenname: George A.
  surname: Constantinides
  fullname: Constantinides, George A.
  email: g.constantinides@imperial.ac.uk
  organization: Dept. of Electr. & Electron. Eng., Imperial Coll. London, London, UK
BookMark eNp9kE1LAzEQhoNUsK3eBS_5A1vzvdnjsrRaKXqp5yVNJzay3UgSlf57t7R48OBlhoF5hneeCRr1oQeEbimZUUqq-3XdzBihbMYYk5LyCzSmUuqCScZHaEwI1UXFtLpCk5Teh1EJQcfoqQl9jqEr1jsIEbK3eBHit4lbPI8xRFz3pjskn3BweJkhmuy_AD9_7iF6azpcd28h-rzbp2t06UyX4Obcp-h1MV83j8Xq5WHZ1KvCMsVzIZ011VZIrWQJVGvGteOuFEBKtzHMKFYC2O1GViBKJhw3SpoKhgpEGLHhU6ROd20MKUVwrfV5iHV8xPiupaQ9GmkHI-3RSHs2MoDkD_gR_d7Ew3_I3QnxAPC7rjhnYgj-A16nbno
CODEN IETAA9
CitedBy_id crossref_primary_10_1016_j_cnsns_2019_104995
crossref_primary_10_1016_j_conengprac_2015_12_005
crossref_primary_10_1016_j_ifacol_2017_08_1253
crossref_primary_10_1109_TAC_2016_2566880
crossref_primary_10_1109_TNNLS_2020_3041364
crossref_primary_10_1016_j_amc_2022_127207
crossref_primary_10_1109_LCSYS_2024_3406943
crossref_primary_10_1016_j_amc_2018_02_020
crossref_primary_10_1016_j_ifacol_2020_12_1764
crossref_primary_10_1016_j_amc_2018_04_063
Cites_doi 10.1109/TPDS.2010.125
10.1137/1.9780898719185
10.1137/1.9780898718027
10.1017/S0962492906340019
10.1063/1.3241491
10.1137/0704053
10.1016/j.automatica.2006.12.028
10.1016/0024-3795(91)90021-N
10.1137/1.9780898718669
10.1007/b83677
10.1109/SASP.2008.4570793
10.1109/TAC.2007.899109
10.1007/s11075-005-1523-5
10.1080/00207170600611265
10.1137/0710026
10.1109/9.1283
10.1016/S0005-1098(01)00028-0
10.1049/ic.2010.0314
ContentType Journal Article
DBID 97E
RIA
RIE
AAYXX
CITATION
DOI 10.1109/TAC.2012.2225513
DatabaseName IEEE Xplore (IEEE)
IEEE All-Society Periodicals Package (ASPP) 1998–Present
IEEE Electronic Library (IEL)
CrossRef
DatabaseTitle CrossRef
DatabaseTitleList
Database_xml – sequence: 1
  dbid: RIE
  name: IEEE Electronic Library (IEL)
  url: https://ieeexplore.ieee.org/
  sourceTypes: Publisher
DeliveryMethod fulltext_linktorsrc
Discipline Engineering
EISSN 1558-2523
EndPage 1529
ExternalDocumentID 10_1109_TAC_2012_2225513
6332482
Genre orig-research
GroupedDBID -~X
.DC
0R~
29I
3EH
4.4
5GY
5VS
6IK
97E
AAJGR
AARMG
AASAJ
AAWTH
ABAZT
ABQJQ
ABVLG
ACGFO
ACGFS
ACIWK
ACNCT
AENEX
AETIX
AGQYO
AGSQL
AHBIQ
AI.
AIBXA
AKJIK
AKQYR
ALLEH
ALMA_UNASSIGNED_HOLDINGS
ASUFR
ATWAV
BEFXN
BFFAM
BGNUA
BKEBE
BPEOZ
CS3
DU5
EBS
EJD
F5P
HZ~
H~9
IAAWW
IBMZZ
ICLAB
IDIHD
IFIPE
IFJZH
IPLJI
JAVBF
LAI
M43
MS~
O9-
OCL
P2P
RIA
RIE
RNS
TAE
TN5
VH1
VJK
~02
AAYXX
CITATION
ID FETCH-LOGICAL-c263t-5fca9d458657e188238f3f74e07fba2a627eecdb59e4724f3a65a9ea65e04a4b3
IEDL.DBID RIE
ISICitedReferencesCount 13
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000319471200013&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 0018-9286
IngestDate Sat Nov 29 02:25:59 EST 2025
Tue Nov 18 21:20:27 EST 2025
Wed Aug 27 02:52:00 EDT 2025
IsPeerReviewed true
IsScholarly true
Issue 6
Language English
License https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c263t-5fca9d458657e188238f3f74e07fba2a627eecdb59e4724f3a65a9ea65e04a4b3
PageCount 6
ParticipantIDs ieee_primary_6332482
crossref_citationtrail_10_1109_TAC_2012_2225513
crossref_primary_10_1109_TAC_2012_2225513
PublicationCentury 2000
PublicationDate 2013-06-01
PublicationDateYYYYMMDD 2013-06-01
PublicationDate_xml – month: 06
  year: 2013
  text: 2013-06-01
  day: 01
PublicationDecade 2010
PublicationTitle IEEE transactions on automatic control
PublicationTitleAbbrev TAC
PublicationYear 2013
Publisher IEEE
Publisher_xml – name: IEEE
References ref12
ref14
golub (ref23) 1996
haimovich (ref18) 2006
ref2
ref1
ref17
ref16
ref19
barrowes (ref27) 2010
hasan (ref25) 2012
constantinides (ref13) 2009
ref24
helmke (ref10) 2006
ref20
strang (ref26) 2007
ref21
grne (ref11) 2002
ref8
lopes (ref15) 2009
ref7
michel (ref22) 2008
ref9
ref4
ref3
ref6
ref5
References_xml – year: 2006
  ident: ref18
  publication-title: Quantisation issues in feedback control
– year: 2007
  ident: ref26
  publication-title: Computational Science and Engineering
– ident: ref14
  doi: 10.1109/TPDS.2010.125
– year: 2008
  ident: ref22
  publication-title: Stability of Dynamical Systems Continuous Discontinuous and Discrete Systems
– year: 1996
  ident: ref23
  publication-title: Matrix Computations
– ident: ref17
  doi: 10.1137/1.9780898719185
– year: 2010
  ident: ref27
  publication-title: Multiple Precision Toolbox
– ident: ref24
  doi: 10.1137/1.9780898718027
– ident: ref9
  doi: 10.1017/S0962492906340019
– ident: ref7
  doi: 10.1063/1.3241491
– year: 2006
  ident: ref10
  article-title: A control theory approach to linear equation solvers
  publication-title: Proc 17th Int Symp Math Theory Netw Syst
– ident: ref1
  doi: 10.1137/0704053
– year: 2009
  ident: ref15
  article-title: More flops or more precision? Accuracy parameterizable linear equations solvers for model-predictive control
  publication-title: Proc IEEE Int Symp Field-Programmable Custom Computing Machines (FCCM)
– year: 2012
  ident: ref25
  publication-title: Control Theoretic Analysis and Design of Numerical Algorithms
– ident: ref5
  doi: 10.1016/j.automatica.2006.12.028
– ident: ref6
  doi: 10.1016/0024-3795(91)90021-N
– ident: ref4
  doi: 10.1137/1.9780898718669
– year: 2002
  ident: ref11
  publication-title: Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization
  doi: 10.1007/b83677
– ident: ref12
  doi: 10.1109/SASP.2008.4570793
– ident: ref3
  doi: 10.1109/TAC.2007.899109
– ident: ref8
  doi: 10.1007/s11075-005-1523-5
– ident: ref20
  doi: 10.1080/00207170600611265
– ident: ref2
  doi: 10.1137/0710026
– start-page: 138
  year: 2009
  ident: ref13
  article-title: Tutorial paper: Parallel architectures for model predictive control
  publication-title: Proc Eur Control Conf
– ident: ref19
  doi: 10.1109/9.1283
– ident: ref16
  doi: 10.1016/S0005-1098(01)00028-0
– ident: ref21
  doi: 10.1049/ic.2010.0314
SSID ssj0016441
Score 2.179136
Snippet It has been known for at least five decades that control theory can be used to study iterative algorithms. However, little work can be found in the control...
SourceID crossref
ieee
SourceType Enrichment Source
Index Database
Publisher
StartPage 1524
SubjectTerms Algorithm design and analysis
Control systems
Error analysis
Finite precision
forward error analysis
Hardware
Heuristic algorithms
input-to-state stability (ISS)
Iterative methods
numerical algorithms as dynamical systems
Quantization
Title Control-Theoretic Forward Error Analysis of Iterative Numerical Algorithms
URI https://ieeexplore.ieee.org/document/6332482
Volume 58
WOSCitedRecordID wos000319471200013&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: PRVIEE
  databaseName: IEEE Electronic Library (IEL)
  customDbUrl:
  eissn: 1558-2523
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0016441
  issn: 0018-9286
  databaseCode: RIE
  dateStart: 19630101
  isFulltext: true
  titleUrlDefault: https://ieeexplore.ieee.org/
  providerName: IEEE
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1LS8NAEF5q8aAHX1WsL_bgRTBtss_ssZQW9VA8VOgtbDazWtBG0tbf726ShgoieAkhbJbwbWZmZ2fmG4RuY5JJz3viVsBkAUtBOZHSIjBRGoHSmeFEl80m5GQSz2bquYXum1oYACiTz6Dnb8tYfpabtT8q6wvqzH_sFO6OlKKq1WoiBt6uV1rXCTCJm5BkqPrTwdDncJGe9214RH-YoK2eKqVJGR_-72OO0EG9dcSDaq2PUQsWJ2h_i1Cwg56GVeZ5MN3UJ-JxXvjEWDwqirzAGw4SnFv8WBIqO22HJ-sqbuMmf3_Ni_nq7WN5il7Go-nwIai7JQSGCLoKuDVaZYzHgkuI3MaZxpZaySCUNtVECyIBTJZyBUwSZqkWXCtwVwiZZik9Q-1FvoBzhJ3X4MRYy0xTxqjQSmrLIjertERwS7qovwEwMTWVuO9o8Z6ULkWoEgd54iFPasi76K5547Oi0fhjbMej3Yyrgb74_fEl2iNVh4ogjK5Qe1Ws4Rrtmq_VfFnclP_INzuBuQM
linkProvider IEEE
linkToHtml http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV3dS8MwEA9jCuqDX1Ocn33wRbBbm8_mcQzHprP4MGFvJU0THcxVus6_36TtygQRfCmlpKH80rvL5e5-B8BtABNmeU_MCsjExbHiRqQEdaUf-4qLRBIoimYTLAyD6ZS_NMB9XQujlCqSz1TH3hax_CSVK3tU1qXImP_AKNwt2zmrqtaqYwbWspd614gwDOqgpMe7k17fZnHBjvVuiI9-GKGNriqFURkc_O9zDsF-tXl0euVqH4GGWhyDvQ1KwRZ47Je55-5kXaHoDNLMpsY6D1mWZs6ahcRJtTMqKJWNvnPCVRm5MZPP39Jslr9_LE_A6-Bh0h-6Vb8EV0KKcpdoKXiCSUAJU77ZOqNAI82w8piOBRQUMqVkEhOuMINYI0GJ4MpclYcFjtEpaC7ShToDjvEbjCALlgiEMaKCM6Gxb2ZlGlKiYRt01wBGsiITtz0t5lHhVHg8MpBHFvKogrwN7uo3PksijT_Gtiza9bgK6PPfH9-AneHkeRyNR-HTBdiFZb8K1_MvQTPPVuoKbMuvfLbMrov_5RsOMLxM
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=Control-Theoretic+Forward+Error+Analysis+of+Iterative+Numerical+Algorithms&rft.jtitle=IEEE+transactions+on+automatic+control&rft.au=Hasan%2C+Ammar&rft.au=Kerrigan%2C+Eric+C.&rft.au=Constantinides%2C+George+A.&rft.date=2013-06-01&rft.issn=0018-9286&rft.eissn=1558-2523&rft.volume=58&rft.issue=6&rft.spage=1524&rft.epage=1529&rft_id=info:doi/10.1109%2FTAC.2012.2225513&rft.externalDBID=n%2Fa&rft.externalDocID=10_1109_TAC_2012_2225513
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0018-9286&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0018-9286&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0018-9286&client=summon