Lecture Notes on Numerical Analysis

236 Pages·2016·4.93 MB·English

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Lecture Notes on Numerical Analysis
Virginia Tech · MATH/CS 5466 · Spring 2016
We model our world withcontinuousmathematics. Whetherour
interestisnaturalscience,engineering,evenfinanceandeconomics,
themodelswemostoftenemployarefunctionsofrealvariables. The
equationscanbelinearornonlinear,involvederivatives,integrals,
combinationsoftheseandbeyond. Thetricksandtechniquesone
learnsinalgebraandcalculusforsolvingsuchsystemsexactlycan-
nottacklethecomplexitiesthatariseinseriousapplications. Exact
solutionmayrequireanintractableamountofwork;worse,formany
problems,itisimpossibletowriteanexactsolutionusingelementary
functionslikepolynomials,roots,trigfunctions,andlogarithms.
Thiscoursetellsamarveloussuccessstory. Throughtheuseof
cleveralgorithms,carefulanalysis,andspeedycomputers,wecan ImagefromJohannesKepler’sAstrono-
mianova,1609,(ETHBibliothek).Inthis
constructapproximatesolutionstotheseotherwiseintractableprob-
textKeplerderiveshisfamousequation
lemswithremarkablespeed. NickTrefethendefinesnumericalanalysis thatsolvestwo-bodyorbitalmotion,
tobe‘thestudyofalgorithmsfortheproblemsofcontinuousmath- M=E−esinE,
ematics’. Thiscoursetakesatourthroughmanysuchalgorithms,
whereM(themeananomaly)ande
samplingavarietyoftechniquessuitableacrossmanyapplications. (theeccentricity)areknown,andone
solvesforE(theeccentricanomaly).
Weaimtoassessalternativemethodsbasedonbothaccuracyand
Thisvitalproblemspurredthede-
efficiency,todiscernwell-posedproblemsfromill-posedones,andto velopmentofalgorithmsforsolving
seethesemethodsinactionthroughcomputerimplementation. nonlinearequations.
Perhapstheimportanceofnumericalanalysiscanbebestappre-
ciatedbyrealizingtheimpactitsdisappearancewouldhaveonour
WehighlyrecommendTrefethen’s
world. Thespaceprogramwouldevaporate;aircraftdesignwould essay,‘TheDefinitionofNumerical
Analysis’,(reprintedonpages321–327
behobbled;weatherforecastingwouldagainbecomethestuffof
ofTrefethen&Bau,NumericalLinear
soothsayingandalmanacs. Theultrasoundtechnologythatuncov- Algebra),whichinspiresourpresent
erscancerandilluminatesthewombwouldvanish. Googlecouldn’t manifesto.
rankwebpages. Eventhelettersyouarereading,whoseshapesare
specifiedbypolynomialcurves,wouldsuffer. (Severalimportantex-
ceptionsinvolvediscrete,notcontinuous,mathematics: combinatorial
optimization,cryptographyandgenesequencing.)
Ononehand,weareinterestedincomplexity: wewantalgorithms
thatminimizethenumberofcalculationsrequiredtocomputeasolu-
tion. Butwearealsointerestedinthequalityofapproximation: since
wedonotobtainexactsolutions,wemustunderstandtheaccuracy
ofouranswers. Discrepanciesarisefromapproximatingacompli-
catedfunctionbyapolynomial,acontinuumbyadiscretegridof
points,ortherealnumbersbyafinitesetoffloatingpointnumbers.
Differentalgorithmsforthesameproblemwilldifferinthequalityof
theiranswersandthelaborrequiredtoobtainthoseanswers;wewill
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learnhowtoevaluatealgorithmsaccordingtothesecriteria.
Numericalanalysisformstheheartof‘scientificcomputing’or
‘computationalscienceandengineering,’fieldsthatalsoencompass
thehigh-performancecomputingtechnologythatmakesouralgo-
rithmspracticalforproblemswithmillionsofvariables,visualization
techniquesthatilluminatethedatasetsthatemergefromthesecom-
putations,andtheapplicationsthatmotivatethem.
Thoughnumericalanalysishasflourishedinthepastseventy
years,itsrootsgobackcenturies,whereapproximationswereneces-
saryincelestialmechanicsand,moregenerally,‘naturalphilosophy’.
Science,commerce,andwarfaremagnifiedtheneedfornumerical
analysis,somuchsothattheearlytwentiethcenturyspawnedthe
professionof‘computers,’peoplewhoconductedcomputationswith
hand-crankdeskcalculators. Butnumericalanalysishasalwaysbeen
morethanmerenumber-crunching,asobservedbyAlstonHouse-
holderintheintroductiontohisPrinciplesofNumericalAnalysis,pub-
lishedin1953,theendofthehumancomputerera:
Thematerialwasassembledwithhigh-speeddigitalcomputation
alwaysinmind,thoughmanytechniquesappropriateonlyto“hand”
computationarediscussed....Howotherwisethecontinueduseof
thesemachineswilltransformthecomputer’sartremainstobeseen.
Butthismuchcansurelybesaid,thattheireffectiveusedemandsa
moreprofoundunderstandingofthemathematicsoftheproblem,and
amoredetailedacquaintancewiththepotentialsourcesoferror,than
iseverrequiredbyacomputationwhosedevelopmentcanbewatched,
stepbystep,asitproceeds.
Thustheanalysiscomponentof‘numericalanalysis’isessential. We
relyontoolsofclassicalrealanalysis,suchascontinuity,differentia-
bility,Taylorexpansion,andconvergenceofsequencesandseries.
Matrixcomputationsplayafundamentalroleinnumericalanaly-
sis. Discretizationofcontinuousvariablesturnscalculusintoalgebra.
Algorithmsforthefundamentalproblemsinlinearalgebraarecov-
eredinMATH/CS5465. Ifyouhavemissedthisbeautifulcontent,
yourlifewillbepoorerforit;whenthemethodswediscussthis
semesterconnecttomatrixtechniques,wewillprovidepointers.
Theselecturenotesweredevelopedalongsidecoursesthatwere
supportedbytextbooks,suchasAnIntroductiontoNumericalAnalysis
bySüliandMayers,NumericalAnalysisbyGautschi,andNumerical
AnalysisbyKincaidandCheney. Thesenoteshavebenefitedfromthis
pedigree,andreflectcertainhallmarksofthesebooks. Wehavealso
beensignificantlyinfluencedbyG.W.Stewart’sinspiringvolumes,
AfternotesonNumericalAnalysisandAfternotesGoestoGraduateSchool.
Iamgratefulforcommentsandcorrectionsfrompaststudents,and
welcomesuggestionsforfurtherrepairandamendment.
—MarkEmbree
1
Interpolation
lecture 1
: Polynomial Interpolation in the Monomial Basis
Among the most fundamental problemsinnumericalanalysis
istheconstructionofapolynomialthatapproximatesacontinuous
realfunction f : [a,b] → IR. Oftheseveralwayswemightdesign
suchpolynomials,webeginwithinterpolation: wewillconstructpoly-
nomialsthatexactlymatch f atcertainfixedpointsintheinterval
[a,b] ⊂IR.
1.1 Polynomial interpolation: definitions and notation
Definition1.1. Thesetofcontinuousfunctionsthatmap [a,b] ⊂ IRto
IRisdenotedbyC[a,b]. Thesetofcontinuousfunctionswhosefirstr
derivativesarealsocontinuouson [a,b] isdenotedbyCr[a,b]. (Note
thatC0[a,b] ≡C[a,b].)
Definition1.2. Thesetofpolynomialsofdegree n orlessisdenoted
byP .
n
NotethatC[a,b],Cr[a,b] (forany a < b,r ≥ 0)andP arelinear
n
spacesoffunctions(sincelinearcombinationsofsuchfunctionsmain- Wefreelyusetheconceptofvector
taincontinuityandpolynomialdegree). Furthermore,notethatP is spaces.AsetoffunctionsVisareal
n
vectorspaceitisclosedundervector
an n+1dimensionalsubspaceofC[a,b].
additionandmultiplicationbyareal
Thepolynomialinterpolationproblemcanbestatedas: number:forany f,g ∈ V, f +g ∈ V,
andforany f ∈Vandα∈IR,αf ∈V.
Formoredetails,consultatexton
Given f ∈C[a,b] and n+1points {x }n satisfying linearalgebraorfunctionalanalysis.
j j=0
a ≤ x < x < ··· < x ≤ b,
0 1 n
determinesome p ∈P suchthat
n n
p (x ) = f(x ) for j =0,...,n.
n j j
4
Itshallbecomeclearwhywerequire n+1points x ,...,x ,andno
0 n
more,todetermineadegree-n polynomial p . (Youknowthe n = 1
n
casewell: twopointsdetermineauniqueline.) Ifthenumberofdata
pointsweresmaller,wecouldconstructinfinitelymanydegree-n
interpolatingpolynomials. Wereitlarger,therewouldingeneralbe
nodegree-n interpolant.
Asnumericalanalysts,weseekanswerstothefollowingquestions:
• Doessuchapolynomial p ∈P exist?
n n
• Ifso,isitunique?
• Does p ∈ P behavelike f ∈ C[a,b] atpoints x ∈ [a,b] when
n n
x (cid:54)= x for j =0,...,n?
j
• Howcanwecompute p ∈P efficientlyonacomputer?
n n
• Howcanwecompute p ∈ P accuratelyonacomputer(with
n n
floatingpointarithmetic)?
• Ifwewanttoaddanewinterpolationpoint xn+1,canweeasily
adjust pn togiveaninterpolatingpolynomial pn+1 ofonehigher
degree?
• Howshouldtheinterpolationpoints {x } bechosen?
j
Regardingthislastquestion,weshouldnotethat,inpractice,we
arenotalwaysabletochoosetheinterpolationpointsasfreelyas
wemightlike. Forexample,our‘continuousfunction f ∈ C[a,b]’
couldactuallybeadiscretelistofpreviouslycollectedexperimental
data,andwearestuckwiththevalues {x }n atwhichthedatawas
j j=0
measured.
1.2 Constructing interpolants in the monomial basis
Ofcourse,anypolynomial p ∈P canbewrittenintheform
n n
p (x) = c +c x+c x2+···+c xn
n 0 1 2 n
forcoefficients c ,c ,...,c . Wecanviewthisformulaasanexpres-
0 1 n
sionfor p asalinearcombinationofthebasisfunctions1, x, x2,...,
n
xn;thesebasisfunctionsarecalledmonomials.
Toconstructthepolynomialinterpolantto f,wemerelyneedto
determinethepropervaluesforthecoefficients c ,c ,...,c inthe
0 1 n
aboveexpansion. Theinterpolationconditions p (x ) = f(x ) for
n j j
5
j =0,...,n reducetotheequations
c +c x +c x2+···+c xn = f(x )
0 1 0 2 0 n 0 0
c +c x +c x2+···+c xn = f(x )
0 1 1 2 1 n 1 1
.
.
.
c +c x +c x2 +···+c xn = f(x ).
0 1 n 2 n n n n
Notethatthese n+1equationsarelinearinthe n+1unknown
parameters c ,...,c . Thus,ourproblemoffindingthecoefficients
0 n
c ,...,c reducestosolvingthelinearsystem
0 n
    
1 x x2 ··· xn c f(x )
0 0 0 0 0
 1 x1 x12 ··· x1n  c1   f(x1) 
(1.1)  1 x2 x22 ··· x2n  c2  =  f(x2) ,
    
 ... ... ... ... ...  ...   ... 
    
1 x x2 ··· xn c f(x )
n n n n n
whichwedenoteas Ac = f. Matricesofthisform,calledVander-
1
mondematrices,ariseinawiderangeofapplications. Providedall 1Highampresentsmanyinteresting
theinterpolationpoints {x } aredistinct,onecanshowthatthisma- propertiesofVandermondematrices
j
2 andalgorithmsforsolvingVander-
trixisinvertible. Hence,fundamentalpropertiesoflinearalgebra mondesystemsinChapter21of
allowustoconfirmthatthereisexactlyonedegree-n polynomialthat AccuracyandStabilityofNumerical
Algorithms,2nded.,(SIAM,2002).Van-
interpolates f atthegiven n+1distinctinterpolationpoints.
dermondematricesariseoftenenough
thatMATLABhasabuilt-incommand
Theorem1.1. Given f ∈ C[a,b] anddistinctpoints {xj}nj=0, a ≤ forcreatingthem.Ifx= [x0,...,xn]T,
x0 < x1 < ··· < xn ≤ b,thereexistsaunique pn ∈ Pn suchthat thenA=fliplr(vander(x)).
p (x ) = f(x ) for j =0,1,...,n.
n j j 2Infact,thedeterminanttakesthe
simpleform
Todeterminethecoefficients {c },wecouldsolvetheabovelinear
j
n n
systemwiththeVandermondematrixusingsomevariantofGaus- det(A)=∏ ∏ (xk−xj).
sianelimination(e.g.,usingMATLAB’s\command);thiswilltake j=0k=j+1
O(n3) floatingpointoperations. Alternatively,wecould(andshould) Thisisevidentforn = 1;wewill
notproveifforgeneraln,aswewill
useaspecializedalgorithmthatexploittheVandermondestructure
havemoreelegantwaystoestablish
todeterminethecoefficients {c } inonlyO(n2) operations,avast existenceanduniquenessofpolynomial
j
3 interpolants.Foracleverproof,see
improvement.
p.193ofBellman,IntroductiontoMatrix
Analysis,2nded.,(McGraw-Hill,1970).
1.2.1 Potential pitfalls of the monomial basis
3SeeHigham’sbookfordetailsand
Thoughitisstraightforwardtoseehowtoconstructinterpolating stabilityanalysisofspecializedVander-
polynomialsinthemonomialbasis,thisprocedurecangiveriseto mondealgorithms.
someunpleasantnumericalproblemswhenweactuallyattemptto
determinethecoefficients {c } onacomputer. Theprimarydifficulty
j
isthatthemonomialbasisfunctions1, x, x2,..., xn lookincreasingly
alikeaswetakehigherandhigherpowers. Figure1.1illustratesthis
behaviorontheinterval [a,b] = [0,1] with n =5and x = j/5.
j
6
Figure1.1: Thesixmonomialbasis
1 vectorsforP5,basedontheinterval
1
[a,b]=[0,1]withxj =j/5(redcircles).
Notethatthebasisvectorsincreasingly
0.8 alignasthepowerincreases:thisbasis
becomesill-conditionedasthedegreeof
theinterpolantgrows.
0.6 x
0.4 x2
x3
0.2 x4
x5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Becausethesebasisvectorsbecomeincreasinglyalike,onefinds
thattheexpansioncoefficients {c } inthemonomialbasiscanbe-
j
comeverylargeinmagnitudeevenifthefunction f(x) remainsof
modestsizeon [a,b].
Considerthefollowinganalogyfromlinearalgebra. Thevectors
(cid:34) (cid:35) (cid:34) (cid:35)
1 1
,
10−10 0
formabasisforIR2. However,bothvectorspointinnearlythesame
direction,thoughofcoursetheyarelinearlyindependent. Wecanwrite
thevector [1, 1]T asauniquelinearcombinationofthesebasisvec-
tors:
(cid:34) (cid:35) (cid:34) (cid:35) (cid:34) (cid:35)
1 1 1
(1.2) =10,000,000,000 −9,999,999,999 .
1 10−10 0
Althoughthevectorweareexpandingandthebasisvectorsthem-
selvesareallhavemodestsize(norm),theexpansioncoefficientsare
enormous. Furthermore,smallchangestothevectorweareexpand-
ingwillleadtohugechangesintheexpansioncoefficients. Thisisa
recipefordisasterwhencomputingwithfinite-precisionarithmetic.
Thissamephenomenoncanoccurwhenweexpresspolynomials
inthemonomialbasis. Asasimpleexample,considerinterpolating
f(x) = 2x+xsin(40x) atuniformlyspacedpoints(x = j/n, j =
j
0,...,n)intheinterval [0,1]. Notethat f ∈ C∞[0,1]: this f isa‘nice’
functionwithinfinitelymanycontinuousderivatives. Asseenin
Figures1.2–1.3, f oscillatesmodestlyontheinterval [0,1],butit
7
7
f(x)=2x+xsin(40x) Figure1.2: Degreen=10interpolant
6 pn(x)=interpolating polynomial, n=10 p10(x)to f(x) = 2x+xsin(40x)at
theuniformlyspacedpointsx0,...,x10
5 forxj = j/10over[a,b] = [0,1].
Eventhoughp10(xj) = f(xj)atthe
4 elevenpointsx0,...,xn(redcircles),the
interpolantgivesapoorapproximation
to f attheendsoftheinterval.
3
2
1
0
-1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
7
f(x)=2x+xsin(40x) Figure1.3: RepetitionofFigure1.2,but
6 p (x)=interpolating polynomial, n=30 nowwiththedegreen=30interpolant
n
atuniformlyspacedpointsxj=j/30on
5 [0,1].Thepolynomialstillovershoots f
nearx = 0andx = 1,thoughbyless
thanforn = 10;forthisexample,the
4
overshootgoesawayasnisincreased
further.
3
2
1
0
-1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
certainlydoesnotgrowexcessivelylargeinmagnitudeorexhibitany
nastysingularities.
Comparingtheinterpolantswith n = 10and n = 30betweenthe
twofigures,itappearsthat,insomesense, p → f as n increases.
n
Indeed,thisisthecase,inamannerweshallmakepreciseinfuture
lectures.
However,wemustaddressacrucialquestion:
Canweaccuratelycomputethecoefficientsc ,...,c
0 n
thatspecifytheinterpolatingpolynomial?
8
UseMATLAB’sbasicGaussianeliminationalgorithmtosolvethe
Vandermondesystem Ac = f for c viathecommandc = A\f,then
evaluate
n
p (x) = ∑c xj
n j
j=0
e.g.,usingMATLAB’spolyvalcommand.
Since p wasconstructedtointerpolate f atthepoints x ,...,x ,
n 0 n
wemight(attheveryleast!) expect
f(x )−p (x ) =0, j =0,...,n.
j n j
Sincewearedealingwithnumericalcomputationswithafinitepreci-
sionfloatingpointsystem,weshouldinsteadbewellsatisfiedifour
numericalcomputationsonlyachieve |f(xj)− pn(xj)| = O(εmach), Moreprecisely,wemightexpect
where εmach denotestheprecisionofthefloatingpointarithmetic |f(xj)−pn(xj)|≈εmach(cid:107)f(cid:107)L∞,
4
system.
where|f(cid:107)L∞ := max |f(x)|.
x∈[a,b]
Instead,theresultsofournumericalcomputationsareremarkably
inaccurateduetothemagnitudeofthecoefficients c0,...,cn andthe 4Forthedouble-precisionarithmetic
ill-conditioningoftheVandermondematrix. usedbyMATLAB,ε ≈2.2×10−16.
mach
Recallfromnumericallinearalgebrathattheaccuracyofsolving
thesystem Ac = f dependsontheconditionnumber (cid:107)A(cid:107)(cid:107)A−1(cid:107) of A.5 5Forinformationonconditioningand
Figure1.4showsthatthisconditionnumbergrowsexponentiallyas n theaccuracyofsolvinglinearsystems,
6 see,e.g..,Lecture12ofTrefethenand
increases. Thus,weshouldexpectthecomputedvalueof c tohave
Bau,NumericalLinearAlgebra(SIAM,
errorsthatscalelike (cid:107)A(cid:107)(cid:107)A−1(cid:107)ε . Moreover,considertheentries 1997).
mach
in c. For n =10(atypicalexample),wehave
6Thecurvehasveryregularbehavior
j cj upuntilaboutn = 20;beyondthat
0 0.00000 point,where(cid:107)A(cid:107)(cid:107)A−1(cid:107) ≈ 1/εmach,
thecomputationissufficientlyunstable
1 363.24705 thattheconditionnumberisnolonger
2 −10161.84204 computedaccurately!Forn > 20,
takeallthecurvesinFigure1.4witha
3 113946.06962
healthydoseofsalt.
4 −679937.11016
5 2411360.82690
6 −5328154.95033
7 7400914.85455
8 −6277742.91579
9 2968989.64443
10 −599575.07912
Theentriesin c growinmagnitudeandoscillateinsign,akintothe
simpleIR2 vectorexamplein(1.2). Thesign-flipsandmagnitudeof
thecoefficientswouldmake
n
p (x) = ∑c xj
n j
j=0
9
1030 Figure1.4: Illustrationofsomepitfalls
ofworkingwithinterpolantsinthe
1025 monomialbasisforlargen:(a)thecon-
conditionnumber (cid:107)A(cid:107)(cid:107)A−1(cid:107) ditionnumberofAgrowslargewithn;
1020 (b)asaresult,somecoefficientscjare
largeinmagnitude(blueline)andin-
1015 accuratelycomputed;(c)consequently,
1010 maxj|cj| thecomputed‘interpolant’pnisfar
from f attheinterpolationpoints(red
105 line):thisredcurveshouldbezero!
100
10-5 maxj|f(xj)−pn(xj)|
10-10
10-15
10-20
0 5 10 15 20 25 30 35 40
n
difficulttocomputeaccuratelyforlarge n,evenifallthecoefficients
c ,...,c weregivenexactly. Figure1.4showshowthelargestcom-
0 n
putedvaluein c growswith n. Finally,thisfigurealsoshowsthe
quantitywebegandiscussing,
max |f(x )−p (x )|.
j n j
0≤j≤n
Ratherthanbeingnearlyzero,thisquantitygrowswith n,untilthe
computed‘interpolating’polynomialdiffersfrom f atsomeinterpo-
lationpointbyroughly1/10forthelargervaluesof n: wemusthave
higherstandards!
Thisisanexamplewhereasimpleproblemformulationquickly
yieldsanalgorithm,butthatalgorithmgivesunacceptablenumerical
results.
Perhapsyouarenowtroubledbythisentirelyreasonablequestion:
Ifthecomputationsof p areasunstableasFigure1.4suggests,why
n
shouldweputanyfaithintheplotsofinterpolantsfor n = 10and,
especially, n =30inFigures1.2–1.3?
YoushouldtrustthoseplotsbecauseIcomputedthemusinga
muchbetterapproach,aboutwhichweshallnextlearn.
10
lecture 2
: Superior Bases for Polynomial Interpolants
1.3 Polynomial interpolants in a general basis
The monomial basis mayseemlikethemostnaturalwaytowrite
downtheinterpolatingpolynomial,butitcanleadtonumerical
problems,asseeninthepreviouslecture. Toarriveatmorestable
expressionsfortheinterpolatingpolynomial,wewillderiveseveral
differentbasesforP thatgivesuperiorcomputationalproperties:
n
theexpansioncoefficients {c } willtypicallybesmaller,anditwill
j
besimplertodeterminethosecoefficients. Thisisaninstanceof
ageneralprincipleofappliedmathematics: topromotestability,
expressyourprobleminawell-conditionedbasis.
Supposewehavesomebasis {bj}nj=0 forPn. Weseekthepolyno- Recallthat{bj}nj=0isabasisifthe
mial p ∈ Pn thatinterpolates f at x0,...,xn. Write p inthebasis functionsspanPnandarelinearly
independent.Thefirstrequirement
as meansthatforanypolynomialp∈Pn
p(x) = c b (x)+c b (x)+···+c b (x). wecanfindconstantsc0,...,cnsuch
0 0 1 1 n n that
p=c0b0+···+cnbn,
Weseekthecoefficients c ,...,c thatexpresstheinterpolant p in
0 n whilethesecondrequirementmeans
thisbasis. Theinterpolationconditionsare thatif
0=c0b0+···+cnbn
p(x ) = c b (x )+c b (x )+···+c b (x ) = f(x )
0 0 0 0 1 1 0 n n 0 0 thenwemusthavec0=···=cn=0.
p(x ) = c b (x )+c b (x )+···+c b (x ) = f(x )
1 0 0 1 1 1 1 n n 1 1
.
.
.
p(x ) = c b (x )+c b (x )+···+c b (x ) = f(x ).
n 0 0 n 1 1 n n n n n
Againwehave n+1equationsthatarelinearinthe n+1unknowns
c ,...,c ,hencewecanarrangetheseinthematrixform
0 n
    
b (x ) b (x ) ··· b (x ) c f(x )
0 0 1 0 n 0 0 0
 b (x ) b (x ) ··· b (x )  c   f(x ) 
(1.3)  0 ... 1 1 ... 1 ... n ... 1  ...1  =  ...1 ,
    
b (x ) b (x ) ··· b (x ) c f(x )
0 n 1 n n n n n
whichcanbesolvedviaGaussianeliminationfor c ,...,c .
0 n
Noticethatthelinearsystemforthemonomialbasisin(1.1)isa
specialcaseofthesystemin(1.3),withthechoice b (x) = xj. Next
j
wewilllookattwosuperiorbasesthatgivemorestableexpressions
fortheinterpolant. Weemphasizethatwhenthebasischanges,soto
dothevaluesof c ,...,c ,buttheinterpolatingpolynomial premainsthe
0 n
same,regardlessofthebasisweusetoexpressit.