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1、NumericalAnalysisLectureNotesPeterJ.Olver12.MinimizationInthispart,wewillintroduceandsolvethemostbasicmathematicaloptimizationproblem:minimizeaquadraticfunctiondependingonseveralvariables.Thiswillrequireashortintroductiontopositivedefinitematrices.Assumingthecoefficientmatrixofthequadratictermsisp
2、ositivedefinite,theminimizercanbefoundbysolvinganassociatedlinearalgebraicsystem.Withthesolutioninhand,weareabletotreatawiderangeofapplications,includingleastsquaresfittingofdata,interpolation,aswellasthefiniteelementmethodforsolvilngboundaryvalueproblemsfordifferentialequations.12.1.PositiveDefinit
3、eMatrices.Minimizationoffunctionsofseveralvariablesreliesonanextremelyimportantclassofsymmetricmatrices.Definition12.1.Ann×nmatrixKiscalledpositivedefiniteifitissymmetric,KT=K,andsatisfiesthepositivityconditionxTKx>0forallvectors06=x∈Rn.(12.1)WewillsometimeswriteK>0tomeanthatKisasymmetric,positive
4、definitematrix.Warning:TheconditionK>0doesnotmeanthatalltheentriesofKarepositive.Therearemanypositivedefinitematricesthathavesomenegativeentries;seeExample12.2below.Conversely,manysymmetricmatriceswithallpositiveentriesarenotpositivedefinite!Remark:Althoughsomeauthorsallownon-symmetricmatricestobe
5、designatedaspositivedefinite,wewillonlysaythatamatrixispositivedefinitewhenitissymmetric.But,tounderscoreourconventionandremindthecasualreader,wewilloftenincludethesuperfluousadjective“symmetric”whenspeakingofpositivedefinitematrices.GivenanysymmetricmatrixK,thehomogeneousquadraticpolynomialXnq(x)=
6、xTKx=kxx,(12.2)ijiji,j=15/18/08210c2008PeterJ.OlverisknownasaquadraticformonRn.Thequadraticformiscalledpositivedefiniteifq(x)>0forall06=x∈Rn.(12.3)Thus,aquadraticformispositivedefiniteifandonlyifitscoefficientmatrixis.4−2Example12.2.EventhoughthesymmetricmatrixK=hastwo−23negativeentries,itis,neve
7、rtheless,apositivedefinitematrix.Indeed,thecorrespondingquadraticformq(x)=xTKx=4x2−4xx+3x2=(2x−x)2+2x2≥01122122isasumoftwonon-negativequantities.Moreover,q(x)=0ifandonlyifboth2x1−x2=0andx2=0,whichimpliesx1=0also.Thisprovesq(x)>0for
