Table Of ContentLecture Notes in Mathematics 2091
Miguel A. Sainz · Joaquim Armengol
Remei Calm · Pau Herrero
Lambert Jorba · Josep Vehi
Modal Interval
Analysis
New Tools for Numerical Information
Lecture Notes in Mathematics 2091
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis(Zurich)
MarioDiBernardo(Bristol)
AlessioFigalli(Pisa/Austin)
DavarKhoshnevisan(SaltLakeCity)
IoannisKontoyiannis(Athens)
GaborLugosi(Barcelona)
MarkPodolskij(Heidelberg)
SylviaSerfaty(ParisandNY)
CatharinaStroppel(Bonn)
AnnaWienhard(Heidelberg)
Forfurthervolumes:
http://www.springer.com/series/304
Miguel A. Sainz • Joaquim Armengol
Remei Calm (cid:129) Pau Herrero (cid:129) Lambert Jorba
Josep Vehi
Modal Interval Analysis
New Tools for Numerical Information
123
MiguelA.Sainz JoaquimArmengol
InformáticaMatemática EnginyeriaEleJctrica
AplicadayEstadística ElectroJnicaiAutomaJtica
EscolaPolitecnicaSuperior EscolaPolitecnicaSuperior
UniversityofGirona UniversityofGirona
Girona,Spain Girona,Spain
RemeiCalm PauHerrero
InformáticaMatemática ImperialCollegeLondon
AplicadayEstadística London,UnitedKingdom
EscolaPolitecnicaSuperior
UniversityofGirona
Girona,Spain
LambertJorba JosepVehi
MatemáticaEconómica EnginyeriaEleJctrica
FinancierayActuarial ElectroJnicaiAutomaJtica
FacultaddeEconomiayEmpresa EscolaPolitecnicaSuperior
UniversitatdeBarcelona UniversityofGirona
Barcelona,Spain Girona,Spain
ISBN978-3-319-01720-4 ISBN978-3-319-01721-1(eBook)
DOI10.1007/978-3-319-01721-1
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
MathematicsSubjectClassification(2010):65G40,65G50,03B10,26A24,65F10
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Preface
ThebasicideaofIntervalMathematicsisthatordinaryset-theoreticalintervalsI.R/
providea consistentsupportfornumericalcomputing.The set-theoreticalformof
IntervalAnalysishasproducedalargeamountofworksinceitsinitiationinthelate
1950s,[1,60,61].Therearesomelaterpapersdevotedtothestructuralanalysisof
themethodoritscompletion[13,14,49,66,72,87,96,97],butwithnofundamental
departurefromitsinitialset-theoreticalfoundations.
Thisbookpresentsanewintervaltheory,theModalIntervalAnalysis(MIA),as
astructural,algebraic,andlogicalcompletionoftheclassicalintervals.Thestarting
pointofMIA isquitesimple:todefinea modalintervalattachingaquantifierto a
classical interval, and to introduce the basic relation of inclusion between modal
intervals by means of the inclusion between the sets of predicates they accept.
So a modal interval consists in a classical interval, which defines its domain, and
a quantifier, which defines its modality. This modal approach introduces interval
extensions of the real continuous functions, gives equivalences between logical
formulas and interval inclusions, and provides the semantic theorems that justify
theseequivalencesandguidelinestogettheseinclusions.
The significant change of perspective in the treatment of information, coming
from this new approach, makes Modal Interval Analysis more a new tool for
the general practice of Numerical Applied Mathematics than a contribution to
the previous Interval Theory. It supposes a complete philosophy of numerical
information which is, or can be, its best virtue and produces at each stage of its
development not only one body of solutions, but also questions leading to the
constructionofthenextstageofthetheory.Modalintervalssystemisnotabreaking-
off with the classic intervals, but a algebraic, structural, and logic completion of
themthatopensawaytonewformsofnumericalinformationtreatment.
This book summarizes the most relevant results and features of MIA and also
provides several application examples that illustrate the use of them in different
problemsanddomains.Thebookcontainsthedetaileddevelopmentofthetheory,
the main concepts and results, together with several examples to clarify their
meaningandtobalancethemathematicalitemsandproofs.Definitionscontainthe
v
vi Preface
concepts,theoremsprovidethe main results, and corollariesand lemmasembrace
detailedlogicaldevelopments.
Implementation of arithmetics, computation rules, and some algorithms is in
thesoftwareMISOdevelopedbytheresearchgroupMiceLaboftheUniversityof
Girona(Spain)andavailableathttp://www.cs.utep.edu/interval-comp/intsoft.html
After the introductory Chap.1, about real, digital numbers and intervals, and
limitationsofclassicalintervaltheory,Chap.2givesanaccountofthefundamental
definitionsandstructureswhichsupportthesemanticallyorientedsystemofmodal
intervals I(cid:2).R/. Basic concepts such as predicates, canonical coordinates, modal
inclusion,andequality,duality,intervalpredicatesandco-predicates,rounding,and
latticeoperatorsarepresentedindetail.ThesetI(cid:2).R/ofmodalintervalsturnsout
tobeacompletionofI.R/,inasimilarwaytothatinwhichthecomplexnumbers
are a completion of the real numbers. So, a subset of I(cid:2).R/, the “proper” modal
interval(Œa;b(cid:2)witha (cid:2) b)isidentifiablewithaclassicalintervalŒa;b(cid:2)andallthe
resultsofClassicalIntervalAnalysisarealsoresultsofModalIntervalAnalysis.
InaccordancewiththesenseofthetermanalysisinMathematics,asadiscipline
inwhichtheobjectsofstudyare,firstandforemost,functions,ModalIntervaltheory
can be considered, indeed, as an analysis because it studies a numerical field, the
modalintervals,andthefunctionsdefinedonit.So,Chap.3dealswiththeinterval
extensionoftherealcontinuousfunctions.Thehistoricreasonforthetheoryofthese
extensionsofcontinuousrealfunctionsistoovercomethelimitedcharacterofthe
classicalset-theoreticalapproach.Thegeometricalsemanticsof.nC1/-dimensional
realspace,RnC1,isbasicallydefinedbythecontinuousfunctionsf fromRn toR.
The semantic interval functionsf(cid:2) and f(cid:2)(cid:2) from I(cid:2).Rn/ to I(cid:2).R/, consistently
referringto the continuousfunctionsf from Rn to R, are obtained by translating
tomodaltermstheset-theoreticaldefinitionofasimpleintervalextensionofareal
continuousfunction.Whenthecontinuousrealfunctionisconsideredasasyntactic
tree,itcanalsobeextendedtoarationalintervalfunctionfRfromRntoR,byusing
thecomputingprogramimplicitlydefinedbythesyntaxoftheexpressiondefining
the function.The idea of interpretability is givenas a definite formulationvia the
cornerstoneswhicharetheSemanticTheorems.
Chapter 4 is devoted to characterizing the existence of optimal computations
for a semantic function. Both modal extensions f(cid:2) and f(cid:2)(cid:2) are semantically
interpretable, but not computable in general. When f(cid:2) and f(cid:2)(cid:2) are computed
through the modal rational extension fR, a null, partial, or complete loss of
informationisgenerated.ThepointistofindfunctionsforwhichtheprogramfR
isoptimal,thatis,forwhichfR.X/equalsf(cid:2).X/andf(cid:2)(cid:2).X/.
Modal interval arithmetic operators and metric functions are considered in
Chap.5. The modal arithmetic coincides, certainly, with the arithmetic of the
Kaucher’sExtendedIntervalSpaceIR[49]withanimportantdifference:inI(cid:2).R/
interval results provided by the arithmetic have a logical meaning related to the
pointsoftheoperandintervalsdomains,thankstothesemanticandinterpretability
theorems. Thus, unlike Extended Interval Space, which is a formal and algebraic
completion of the Classic Intervals Space I.R/, Gardenyes’ Modal Intervals are
alsoasemanticcompletionofI.R/.
Preface vii
Chapter6containsproceduresforsolvingintervallinearequationsandsystems.
The Jacobi method is adapted to interval systems together with convergence and
non-convergenceconditions.Animportantpointistoprovidealogicalmeaningto
thesolutionusingthesemantictheorems.
The definitionofthe semanticextensionofa realcontinuousfunctiondoesnot
provide any indication about how to compute it. Some conditions under which
f(cid:2) can be computed through the syntactic extension are given in Chap.3, but
in the most general case it is obtained by means of an algorithm developed in
Chap.7,referredasf(cid:2)-algorithm.First,someconsiderationsabouttwins(intervals
ofintervals)aregiventoprovideabackgroundforthisf(cid:2)-algorithm.
The matter of the necessary rounding is introduced in Chap.1 and dealt with
in the following chapters. Nevertheless, a shortcoming of the modal theory is
managingtheroundingofanintervalwhenitappearsbothasitisanddualizedin
thesamecomputation,forexampleinthesolutionofalinearsystem.Toovercome
this difficulty, in Chap.8 a new object based on modal intervals is introduced:
marks. Definitions, relations, the extension of a continuousfunction to a function
of marks, operators of marks, and the corresponding semantic results are given
in detail together with examples, not only to illustrate the different concepts and
results, but also to show that marks can be used in a very practical way to aware
aboutillcomputationswhichcanappearintheuseofalgorithmswithrealnumbers.
Chapter 9 closes the loop opened in Chap.2 dealing with intervals and modal
intervals of marks, following a parallel developmentto the one started in Chap.2
for modalintervalsof realnumbersI(cid:2).R/. Predicates, relations, lattice operators,
semanticandsyntacticfunctionalextensionstointervalsofmarks,andthesemantic
theorem,togetherwiththearithmeticoperators,areoutlinedthroughoutthechapter.
Finally, Chap.10 is devoted to showing some applications of modal intervals.
Specifically, they are used to deal with three problems:minimax,characterization
of solution sets of quantified constraint satisfaction problems, and statement of
problemsin controlengineeringorprocesscontrolfroma semantic pointofview.
Algorithmsandproceduresaboutthesetopicsarepresented,togetherwithexamples
toillustratetheprocedures.
The beginningsof MIAcan besituated in the SIGLAproject,developedatthe
University of Barcelona in the late 1970s. In the 1980s and 1990s it was further
continued by the SIGLA/X group (University of Barcelona and Polytechnical
University of Catalonia in Spain), some of whose results can be found in [20–
27,86,90]. The kernel of its main applications has been developed inside the
MiceLaboftheUniversityofGirona(Spain)fromthe1990stothepresent.
We,theauthors,areindebtedtomanypeoplewhohaveplayedsignificantrolesin
thedevelopmentofModalIntervalTheory.Firstofall,toDr.E.Gardenyes,founder
oftheModalIntervalAnalysissince hisfirst worksinthe 1980suntilearly2000s
together with a set of coworkers, Dr. H. Mielgo, Dr. A. Trepat, Dr. J.M. Janer,
Dra. R. Estela, and some of us. With this book we want to render him tribute.
Along the hardcore of the theory, we have wanted to preserve, in some way, his
conceptualiststyleandnotation,exceptforsomeadaptationstothestandards.Also,
we wish to thank several colleagues for their valuable comments and criticisms,
viii Preface
Dr.V.Kreinovich,Dr.A.Neumaier,Dr.L.Jaulin,Dr.A.Goldsztein,Dr.S.Ratschan
and, in a very special way, we thank Dr. S.P. Shary and Dr. E. Walter, for the
patient reading of the manuscript. Nevertheless, any error, omission, or obscurity
areentirelyourresponsibility.
Girona,Spain MiguelA.Sainz
Girona,Spain JoaquimArmengol
Girona,Spain RemeiCalm
London,UK PauHerrero
Barcelona,Spain LambertJorba
Girona,Spain JosepVehi
Notations
Inordertomakeclearenoughthemainmathematicalsubjectsputtoworkalongthe
text,wehaveusedthefollowingtypefacesandnotations:
(cid:129) Lowercaseanditalicxforarealnumber.
(cid:129) Lowercaseanditalicf forarealfunctionofonevariable.
(cid:129) Lowercase,boldanditalicxforavectorwithrealcomponents
xD.x ;x ;:::;x /:
1 2 m
(cid:129) Lowercase,boldanditalicf foravectorialfunctionwithfunctionalcomponents
f D.f ;f ;:::;f /:
1 2 m
(cid:129) Uppercase,italicAandapostropheforaclassicalintervalwithrealbounds
A0 DŒa ;a (cid:2)0 or A0 DŒa;a(cid:2)0:
1 2
(cid:129) UppercaseanditalicAforamodalintervalwithrealbounds
ADŒa ;a (cid:2) or ADŒa;a(cid:2):
1 2
(cid:129) Uppercase, bold and italic for an modal interval vector with modal interval
components
AD.A ;A ;:::;A /:
1 2 m
(cid:129) Uppercaseandboldforarealmatrixwithrealelements
AD.a /:
ij
ix
