C H A P T E R 3 Interpolation andPolynomial Approximation 3.1 Introduction Engineers andscientists commonly assume thatrelationships between variables inaphysical problem canbeapproximately reproduced from data given bytheproblem .The ultimate goal might betodetermine thevalues atintermediate points ,toapproximate theintegral or derivative oftheunderlying function ,ortosimply giveasmooth orcontinuous representation ofthevariables intheproblem . Interpolation refers todetermining afunction thatexactly represents acollection ofdata. The most elementary type ofinterpolation consists offitting apolynomial toacollection ofdata points.Polynomials have derivatives andintegrals thatarethemselves polynomials , sothey areanatural choice forapproximating derivatives andintegrals .Inthischapter wewillseethatpolynomials toapproximate continuous functions areeasily constructed . The following result implies thatthere arepolynomials thatarearbitrarily close toany continuous function . Weierstrass Approximation Theorem Suppose that/isdefined andcontinuous on [a,b].Foreach e>0,there exists a polynomial P(x)defined on [a,b\,with theproperty that(seeFigure 3.1) 1/00-P(x)\besupprcsxd tan theeBook and/oreChacxcnul .Editorial review has deemed thatany vupprc'-cdcontent decs notaiaxtlaXy afTect theoverall learning experience ('engage Leaning rociscv therightnrerrx'ccadditional contort atanytime iJsubsequent nglrs restrictions require it.64 CHAPTER 3 Interpolation andPolynomial Approximation Karl Weierstrass (1815-1897 )is often referred toasthefather of modem analysis because ofhis insistence onrigor inthe demonstration ofmathematical results .Hewasinstrumental in developing tests forconvergence ofscries ,anddetermining ways torigorously define irrational numbers .Hewasthefirstto demonstrate thatafunction could beeverywhere continuous but nowhere differentiable ,aresult thatshocked some ofhis contemporaries . Very little ofWcierstrass ’s work waspublished during his lifetime ,buthislectures , particularly onthetheory of functions ,hadsignificant influence onanentire generation ofstudents .TheTaylor polynomials were introduced inChapter 1,where theywere described asone ofthefundamental building blocks ofnumerical analysis .Given thisprominence ,youmight assume thatpolynomial interpolation makes heavy useofthese functions .However ,thisis notthecase.TheTaylor polynomials agree asclosely aspossible with agiven function ata specific point,butthey concentrate their accuracy only near thatpoint Agood interpolation polynomial needs toprovide arelatively accurate approximation overanentire interval ,and Taylor polynomials donotdothat.Forexample ,suppose wecalculate thefirstsixTaylor polynomials about*o=0forf(x)=e x.Since thederivatives off(x)arealle x,which evaluated atx0=0gives 1,theTaylor polynomials are x2X2*3 ftW=l.P\(x)=l+X t Pl(x)=l+*+y,P)(x)=l+X+y+— , X2X3 i>4(x)=l+x+y+-+^,a n d P5(X)=l+x+y+^+^+^Thegraphs ofthese Taylor polynomials arcshown inFigure 3.2.Notice thattheerror becomes progressively worse aswemove away from zero. Figure 3.2 y, 20 15VII ViVi IIII *0"0 *.--A ss ///,y=p}(*)/’// // 10 ///Jp/y=PM 5PM 1~v— PM y «0v*/ -i 1 2 3* Although better approximations areobtained forthisproblem ifhigher -degree Taylor polynomials areused ,thissituation isnotalways true.Consider ,asanextreme example , using Taylor polynomials ofvarious degrees forf(x)=l/xexpanded about X Q=1to approximate /(3)= Since /(x)=x_ 1,/'(x)=-x-2,/"(x)=(— 1)22  x-3. Copyright 2012 Cengagc Learning .AIRights Reserved May notbecopied ,canned ,orimplicated ,inwhole orinpar.Doctoelectronic rights.xvnc third pur.ycontent may besupplied ftem theeBook and/oreChaptcnnl .Editorial review h*> deemed Cutanysuppressed content does nottnaxrUXy alTcct theoverall learning experience .('engage l.cammureserves theright*>remove additional conteatatanytime ifsubsequent nghts restrictions require It3.2 Lagrange Polynomials 65 and,ingeneral , /<">(*)=(-1)"«!*—', theTaylor polynomials forn>0are AW=£ -D*=£(-l)‘(*-1)*. When weapproximate /(3)=^byP„ (3)forlarger values ofn,theapproximations become increasingly inaccurate ,asshown Table 3.1. Table 3.1 n 0 1 2 3 4 5 6 7 PnO ) 1-1 3-5 11 -21 43 -85 TheTaylor polynomials have theproperty thatalltheinformation used intheapprox - imation isconcentrated atthesingle point XQ,soitisnotuncommon forthese polynomials togive inaccurate approximations aswemove away from xo-This limits Taylor polynomial approximation tothesituation inwhich approximations areneeded only atpoints close to XQ.Forordinary computational purposes ,itismore efficient tousemethods thatinclude information atvarious points ,which wewillconsider intheremainder ofthischapter .The primary useofTaylor polynomials innumerical analysis isnotforapproximation purposes ; instead itisforthederivation ofnumerical techniques . 3.2 Lagrange Polynomials Theinterpolation formula named forJoseph Louis Lagrange (1736-1813 )waslikely known byIsaac Newton around 1675, butitappears tohave been published firstin1779 byEdward Waring (1736-1798 ).Lagrange wrote extensively onthesubject ofinterpolation andhiswork had significant influence onlater mathematicians .Hepublished thisresult in1795.Intheprevious section wediscussed thegeneral unsuitability ofTaylor polynomials for approximation .These polynomials areuseful only over small intervals forfunctions whose derivatives exist andareeasily evaluated .Inthissection wefindapproximating polynomials thatcanbedetermined simply byspecifying certain points ontheplane through which they must pass. Lagrange Interpolating Polynomials Determining apolynomial ofdegree 1thatpasses through thedistinct points (x0,yo)and (xi,yi)isthesame asapproximating afunction /forwhich f(xo)=yoand f(x\)=yi bymeans ofafirst-degree polynomial interpolating ,oragreeing with,thevalues of/atthe given points .Wefirstdefine thefunctions L0(x)=— — — and L\(x)=— — — , *0-*1 x\-XQ andnote thatthese definitions imply that £o(*o)=— — — =1,£o(*i)=— — — =0,L\(xo)=0,and L\(xi)=1. *0— *1 Xo-X\ Copyright 2012 Cc«£»fcLearn in*.AIR.(huReversed Mayr*xbecopied ,canned ,o*duplicated.»whole oempan .Doctoelectronic rifhu.vomc third pony content may besupposed ftem theeBook and/orcCh deemed Cutanyvu«ucv*cdcontent dee>notimxttaly alTect theoverall Icamir .itexperience .C'criitape[.camme rexxvev therljtlu 10remote additional conceal atanytime i ivutoeqjroi nghtv roirictionv require It.66 C H A P T E R 3 Interpolation andPolynomial Approximation Wethen define P(x)=L0(x)f(x0)+L,(JT)/(XI)=- f(x0)+-— — f(xi). Xo X\ XQ This gives P(xo)=1 fix0)+0 fix i)=fix0)=y0 and PM =0 fixo)+1 /(*,)=/(*,)=y,. So,Pistheunique linear function passing through (*o,yo)and(ATJ,y\). Example 1Determine thelinear Lagrange interpolating polynomial thatpasses through thepoints (2,4) and(5,1). Solution Inthiscase wehave £o(*)= =-I(*-5)and £|(x)=j— |=-2), SO 1 1 4 20 1 2P(x)=— -(x-5) 4+~ix— 2) 1=— -x+— +-x— -=— x+6. Thegraph ofy=Pix )isshown inFigure 3.3. Figure 3.3 y, \J2,4)4- 3-\ 2 1-y=P{x)=-x-6\.^^ 1 1 1 1 1 12 345* Togeneralize the concept oflinear interpolation tohigher -degree polynomials , consider theconstruction ofapolynomial ofdegree atmost n,shown inFigure 3.4, thatpasses through then+1points C*0,fixo )),(*1,/(*i))f. . ..ixn,fixn )). Copyright 2012 Cenfajc Lcarni #*.AIRights Reversed May notbecopied .scanned .ordedicated .inwhole orinpar.Doc toelectronic rights .>onic third pur.ycontent may besuppressed (tarn theeBook andfcr eChaptcnnl .Editorial review h*> deemed Cut M>suppre-edcontent does nottnaxrUXy alTcct themcra’.llearning experience .Ccagagc [.camonreserves therightto remove additional conteat*anytime ifsubsequent rights rcstrictiona require It3.2 Lagrange Polynomials 67 Figure 3.4 Figure 3.5y- y=mr sa,II* : *oX, x2 X nX Inthiscase,weconstruct ,foreach k=0, apolynomial ofdegree ny which wewilldenote byL„ ,*(x),with theproperty that£„ .*(*«)=0when i^kand IMW =1-Tosatisfy Ln k {Xi)=0foreach i#kythenumerator ofLn>*(x)must contain the term (x-X0)(x-XJ)" ""(x-X*_ I)(X-X*+i)   (x-Xn). Tosatisfy L„ ,*(x*)=1,thedenominator ofLn deemed Cutanysuppressed content does notmaterial yalTcct theioera .1learning experience .('engage [.camonreserves theright K>remove additional conteatatanytime ifsubvcqjem rights restrictions require IL68 C H A P T E R 3 Interpolation andPolynomial Approximation nthLagrange Interpolating Polynomial Pn(x)=f(Xo)L„'0(x)+   +f(Xn)Lntn{x)= f(xk)L„ ,*(*), *=0 where Ln.kW=(x-x0)(x-X\)---(x-x*-i)(x-xk+i) (**-*o)U*-Xi)   (x*-x*_ i)(x*-x*+i)(x-xn)_  (**-xn) foreach &=0,1,...,n. I fxo,x\,...,xnare (n+1)distinct numbers and/isafunction whose values are given atthese numbers ,then Pn(x)istheunique polynomial ofdegree atmost nthat agrees with/(x)atx<>,xi,...,xn.The notation fordescribing theLagrange interpolating polynomial Pn(x)israther complicated because Pn{x)isthesum ofthen+1polynomials f(xk)Ln %k(x),fork=0,1,...,n,each ofwhich isofdegree nyprovided f(xk)#0. Toreduce thenotational complication ,wewill write Lnk (x)simply asL*(x)when there should benoconfusion thatitsdegree isn. Example 2 (a)Usethenumbers (called nodes )xo=2,x\=2.75,andx2=4tofindthesecond Lagrange interpolating polynomial for/(x)=1/jt. (b)Usethispolynomial toapproximate /(3)=1/3. Solution (a)Wefirstdetermine thecoefficient polynomials Lo(x),L\(x),andL2(x).They are Lo(x)= Ldx )=(x-2.75 )(x-4) (2— 2/75)(2— 4)“ (x-2)(x-4) (2.75— 2)(2.75— 4)|(oc-2.75)(*-4), =-~(^-2)(^-4), an d L2{X)(x-2)(x-2.75) (4-2)(4-2.75)I-2)(x-2.75). Also ,f{xo)=/(2)=1/2,/(x,)=/(2.75)=4/11,and/(x2)=/(4)=1/4,so 2 ^(*)=£/(**)£*(*) *=0 =\(X-2.75)(x-4)-^(x-2)(x-4)+^(x-2)(x-2.75 ) 1,35 49=— x~ x4-— .22 88 44 (b)Anapproximation to/(3)=1/3(seeFigure 3.6)is /(3)%P{3)= — +— =— %0.32955 . JK )22 88 44 88 Recall thatintheSection 3.1(seeTable 3.1)wefound thatnoTaylor polynomial expanded about xo=1could beused toreasonably approximate f(x)=1/xatx=3. Copyright 2012 Cengagc Learn in*.AIRights Reversed May notbecopied ,canned ,ocduplicated .inwhole orinpar.Doc toelectronic rights .some third pur.ycontetr .may besuppreved rrom theeBook and/oreChaptcnnl .Editorial review h*> deemed Cutany suppic-cdcontent does notmaterialy alTcct theioera .1learning experience .('engage [.camon reserves theright K>remove additional conceal atanytimeiisubsequent rights restrictions require IL3.2 Lagrange Polynomials 69 Figure 3.6 yt 4 3- 2-\y=/W 1^ y= - 1 2 3 4 5 * The Lagrange polynomials have remainder terms thatarereminiscent ofthose for theTaylor polynomials .ThenthTaylor polynomial about XQconcentrates alltheknown information atXQandhasanerror term oftheform (n+1)!-x0)n+\ where §(JC)isbetween xand JCO.The nthLagrange polynomial uses information atthe distinct numbers JCO,x\ x„.Inplace of(JC-JCO)"+i,itserror formula usesaproduct of then+1terms (JC JCO),(x— JCJ),...,(JC JC ),andthenumber £(JC)canlieanywhere in theinterval thatcontains thepoints JCO,X\,...,JC,and JC.Otherwise ithasthesame form astheerror formula fortheTaylor polynomials . Lagrange Polynomial Error Formula (JC)) f(x)=Pn(x)+3 3 CX-x0)(x-X\)   (x-JC„ ),(n+1)! forsome (unknown )number f(x)that lies inthesmallest interval that contains JCO,JCJ,...,jcnandjc. This error formula isanimportant theoretical result ,because Lagrange polynomials areused extensively forderiving numerical differentiation andintegration methods .Enror bounds forthese techniques areobtained from theLagrange polynomial error formula .The specific useofthiserror formula ,however ,isrestricted tothose functions whose derivatives have known bounds .The next Illustration shows interpolation techniques forasituation in which theLagrange error formula cannot beused.This shows thatweshould look fora more efficient way toobtain approximations viainterpolation . Illustration Table 3.2lists values ofafunction /atvarious points .The approximations to/(1.5) obtained byvarious Lagrange polynomials that usethisdata will becompared totryto determine theaccuracy oftheapproximation . Copyright 2012 Cengagc Learn in*.AIRights Reversed Mayr*xbecopied.scanned .ocimplicated ,inwhole orinpar.Doctocjectronie ilghtv.some third pur.ycontent may besupplied rican theeBook and/orcChaptcnM .Editorial review h*> deemed Cutanysuppressed cement does notmamlaXy alTcct theioera .1learning experience .('engage [.camonreserves theright 10remove additional eonteatatanytime ifsubsequent nghts restrictions require It70 C H A P T E R 3 Interpolation andPolynomial Approximation Table 3.2 x /(*) 1.0 0.7651977 1.3 0.6200860 1.6 0.4554022 1.9 0.2818186 2.2 0.1103623The most appropriate linear polynomial uses XQ=1.3andx\=1.6because 1.5is between 1.3and1.6.The value oftheinterpolating polynomial at1.5is 7>i(1.5)=(1.5-1.6) (1.3-1.6)/(1.3)+(1.5-1.3) (1.6-1.3)/(1.6) (1.5-1.6) (1-3-1.6)(0.6200860 )+(16-13)(04554022 )=05102968 ' Two polynomials ofdegree 2canreasonably beused ,onewith x0=1.3,x\=1.6,and X2=1.9,which gives />2(1.5)=(1.5— 1.6)(1.5— (1.3-1.6)(1.3-1.9) 1.9)(0.6200860 )+(1.5— 1.3)(1.5— 1.9) (1.6— 1.3)(1.6— 1.9)(0.4554022 ) +(1.5— 1.3)(1.5— 1.6) (L9— 1.3)(1.9— 1.6)(0.2818186 )=0.5112857 , andonewithxo=1.0,x\=1.3,and*2=1.6,which gives/*2(1-5)=0.5124715 . Inthethird -degree case ,there arealso tworeasonable choices forthepolynomial .One with*0=1.3,x\=1.6,X2=1.9,andx$=2.2,which gives />3(1.5)=0.5118302 .The second third -degree approximation isobtained with*0=1.0,X\=1.3,*2=1.6,and *3=1.9,which gives />3(1.5)=0.5118127 . Thefourth -degree Lagrange polynomial uses alltheentries inthetable .With xo=1.0, x\=1.3,X2=1.6,*3=1.9,andx4=2.2,theapproximation is/*4(1.5)=0.5118200 . Because />3(1.5),/*3(1.5),and/*4(1.5)allagree towithin 2x10-5units ,weexpect thisdegree ofaccuracy forthese approximations .Wealso expect /*4(1.5)tobethemost accurate approximation because ituses more ofthegiven data . Thefunction weareapproximating isactually theBessel function ofthefirst kind of order zero ,whose value at1.5isknown tobe0.5118277 .Therefore ,thetrueaccuracies of theapproximations areasfollows : |/*i(1.5)— /(1.5)|»1.53xltT3, I/*2(1-5)— /(1.5)|5.42 x10~4, |£2(1.5)-/(1.5)|«6.44x10-4, |/*j(1.5)-/(1.5)|2.5x10-6, |P3(1.5)-/(1.5)|ss1.50x10-s, |/*4(1.5)-/(1.5)|«7.7x10"6. Although /*3(1-5)isthemost accurate approximation ,ifwehad noknowledge of theactual value of/(1.5),wewould accept /*4(1.5)asthebest approximation because it includes themost data about thefunction .The Lagrange error term cannot beapplied here because wehave noknowledge ofthefourth derivative of/.Unfortunately ,thisisgenerally thecase . Neville 'sMethod Apractical difficulty with Lagrange interpolation isthatbecause theerror term isdifficult toapply ,thedegree ofthepolynomial needed forthedesired accuracy isgenerally not Copyright 2012 Cengagc Learn in*.AIRights Reserved May n«becopied ,canned ,ocduplicated.»whole oempan .Doc 10electronic rights .some third pony content may besuppressed ftem sheeBook and/orcClupccm !.Editorial roiew h*> deemed Cut artysuppressed content dees notmaterial yafTcet theoverall learning experience .('engage [.cannon reserve*theright Mremove additional conceal atanytime i!subsequent right*restrictions require it3.2 Lagrange Polynomials 71 known until thecomputations aredetermined .Theusual practice istocompute theresults given from various polynomials until appropriate agreement isobtained ,aswasdone inthe previous example .However ,thework done incalculating theapproximation bythesecond polynomial does notlessen thework needed tocalculate thethird approximation ;noristhe fourth approximation easier toobtain once thethird approximation isknown ,andsoon.To derive these approximating polynomials inamanner thatuses theprevious calculations to advantage ,weneed tointroduce some newnotation . Let/beafunction defined atXQ,x\,*2,.. ,x„andsuppose thatm1,m2,...,mkare kdistinct integers with 0 deemed Cut artysuppressed content does notmaterial yaffect theiocra’1Ieamir .itexperience .('engage Learn xigrocne*theright toremove additional contort atanytime ifsufcseqjcni rights restrictions require it72 C H A P T E R 3 Interpolation andPolynomial Approximation Moreover , PM.g<*>-<«-«> <*», ./M. X i-X j (X i-x,) andsimilarly ,T^x,)=/(x;).Butthere isonly onepolynomial ofdegree atmost kthat agrees with/(x)atxo,XI,...,x*,andthispolynomial bydefinition isPo.i*(x).Hence , POA hW= =(*-XJ)PQAJ-IJ+I*(*)~(*-Xi)Po.i i-i,.+1kW (X i-X j) Table 3.3 Program NEVLLE 31 implements theNeville ’s method . EricHarold Neville (1889-1961 ) gave thismodification ofthe Lagrange formula inapaper published in1932 fN|.This result implies thattheapproximations from theinterpolating polynomials can begenerated recursively inthemanner shown inTable 3.3.Therow-by-rowgeneration is performed tomove across therows asrapidly aspossible ,because these entries aregiven bysuccessively higher -degree interpolating polynomials .This procedure iscalled Neville ’s method . Xo Po *1 Pi POA X2 Pi PI.2 POA.2 X3 P'S P2.3 P1.2.3 Po.1.2,3 x4p4 PSA PlAA Pl.2,3,4 PD.1.2,3.4 The Pnotation used inTable 3.3iscumbersome because ofthenumber ofsubscripts used torepresent theentries .Note,however ,thatasanarray isbeing constructed ,only two subscripts areneeded .Proceeding down thetable corresponds tousing consecutive points Xiwith larger i,andproceeding totheright corresponds toincreasing thedegree ofthe interpolating polynomial .Since thepoints appear consecutively ineach entry ,weneed to describe only astarting point andthenumber ofadditional points used inconstructing the approximation .Toavoid thecumbersome subscripts weletQ ij {x),for0 deemed Cutanysuppressed content dees notmaterial yalTcct theoverall Ieamir .itexperience .Cette jgc[.camon roenn theright Mremove additional conceal atanytime i!subsequent right*restrictions require It.3.2 Lagrange Polynomials 73 and 1 0.07441 (22.1=QY[(-0.1)0.8329-(-0.2)0.7885 ]= = 0.7441 . The final approximation wecanobtain from thisdata is 1 0.2276 (22.2=Q3[(0.1)0.7441 -(-0.2)0.7410 ]= = 0.7420 . These values areshown inTable 3.6. Table 3.6  X i X— X j 0 deemed Cutany suppic-cdcontent does notmaterialy alTcct theoverall learning experience .('engage [.cammu roenet theright 10remote additional conceal atanytimeiisubsequent rights restrictions require It74 C H A P T E R 3 Interpolation andPolynomial Approximation 6. UseNeville ’smethod toapproximate V3with thefunction f i x )=Jxandthevalues XQ=0,x x=1, x2=2,X)=4,andx4=5.Compare theaccuracy with thatofExercise 5. 7. The data forExercise 3were generated using thefollowing functions .Usetheerror formula tofinda bound fortheerror andcompare thebound totheactual error forthecases n=1andn=2. a. f(x)=x l n x b.f(x)=x3+4.001*2+4.002*+1.101 c. f(x)=*cos*-2*2+3*-1 d./(*)=sin(e*— 2) 8. UsetheLagrange interpolating polynomial ofdegree 3orlessandfour -digit chopping arithmetic to approximate cos0.750 using thefollowing values .Find anerror bound fortheapproximation . cos0.698=0.7661 cos0.733 =0.7432 cos0.768 =0.7193 cos0.803 =0.6946 The actual value ofcos0.750 is0.7317 (tofour decimal places ).Explain thediscrepancy between the actual error andtheerror bound . 9. Use thefollowing values andfour -digit rounding arithmetic toconstruct athird Lagrange polyno - mial approximation to/(1.09).The function being approximated is/(*)=log10(tanx ).Use this knowledge tofindabound fortheerror intheapproximation . /(1.00)=0.1924 /(1.05)=0.2414 /(1.10)=0.2933 /(1.15 )=0.3492 10. Repeat Exercise 9using MATLAB inlong format mode . 11. LetP3(x)betheinterpolating polynomial forthedata (0,0).(0.5 ,y),(1,3),and(2,2).Find yifthe coefficient ofx3inP3(x)is6. 12. Neville ’smethod isused toapproximate /(0.5),giving thefollowing table . *o=0 P0=0 x,=0.4 P{=2.8 P01=3.5 *2=0-7Pi _fY2 P.M.2=f Determine P2=/(0.7). 13. Suppose youneed toconstruct eight -decimal -place tables forthecommon ,orbase -10,logarithm function from*=1to*=10insuch away that linear interpolation isaccurate towithin 10~6. Determine abound forthestepsize forthistable .What choice ofstepsizewould youmake toensure that*=10isincluded inthetable? 14. Suppose Xj=jforj=0,1,2,3anditisknown that P0il(x)=2*+1,F0.2C*)=x+1.and />,.2.3(2.5)=3. Find P0,I.2.3(2.5). 15. Neville ’smethod isused toapproximate /(0)using/(-2),/(-1),/(1),and/(2).Suppose /(— 1) wasoverstated by2and/(1)wasunderstated by3.Determine theerror intheoriginal calculation of thevalue oftheinterpolating polynomial toapproximate /(0). 16. Thefollowing table lists thepopulation oftheUnited States from 1960 to2010 . Year 1960 1970 1980 1990 2000 2010 Population (thousands )179,323 203,302 226,542 249 ,633 281,442 307,746 a.Find theLagrange polynomial ofdegree 5fitting thisdata ,andusethispolynomial toestimate thepopulation intheyears 1950 ,1975 ,and2020 . b.Thepopulation in1950 wasapproximately 151,326 ,000 .How accurate doyouthink your 1975 and2020 figures are? Copyright 2012 Cenpapc Learn in*.AIR.(huReversed May notbecopied ,canned ,o*duplicated.»whole oempan .Doctoelectronic rifhu.*wcthird pony contcac may besoppre ^tedftem theeBook and/orcCh deemed CutanyMpprcucd content dee>notnuxtiily alTect theoverall Icamir .itexperience .C'crtitape Learnmp rexxvei thertpltt 10renxwe additional conceal atanytime i isubvoyjem npht »rotrictionc require It3.3 Divided Differences 75 17. InExercise 15ofSection 1.2,aMaclaurin series wasintegrated toapproximate erf(l).where erf(x) isthenormal distribution error function defined by 2 f x 2erf(x)=^J o e'<*' a.UsetheMaclaurin series toconstruct atable forerf(x)thatisaccurate towithin 10-4forerf(x,), where x,=0.2i,for /=0,1,....5. b.Useboth linear interpolation andquadratic interpolation toobtain anapproximation toerf(\). Which approach seems more feasible ? — 3.3 Divided Differences Iterated interpolation wasused intheprevious section togenerate successively higher degree polynomial approximations ataspecific point .Divided -difference methods introduced in thissection areused tosuccessively generate thepolynomials themselves . Divided Differences Wefirstneed tointroduce thedivided -difference notation ,which should remind youofthe Aitkcn’sA2notation defined onpage 53.Suppose wearegiven then+1points (JCO,/(*o)). (JCI,/(*i)),. ..(JC,/(x„ )).There aren+1zeroth divided differences ofthefunction /. Foreach i=0,1,.. .,nwedefine/[*,]simply asthevalue of/at: /[*,]=/(*,) Theremaining divided differences aredefined inductively .There arenfirstdivided differ- ences of/,oneforeach i=0,1,...,n— 1.The firstdivided difference relative tox,and x i+iisdenoted /[*,,JC,+I]andisdefined by flXhXi +l]=f*±iWhl . X i+1 X j After the(A:— l)stdivided differences , f[x i yX i+i,x l+2,. ...X i+k-i]and f[x i+1,x1+2»   * *;+*], have been determined ,theArthdivided difference relative to xl+i,x<+2,.. .,x,+*is defined by /[*« X,+l,...,X|+*_ 1,*/+*]/"[Xi+i> 2,   »Xi+ifc]f[X j,Xj+l»...,X|+£_ i] X i+k-X i Asinsomany areas ,Isaac Newton isprominent inthestudy ofdifference equations .He developed interpolation formulas asearly as1675 ,using hisA notation intables ofdifferences . Hetook avery general approach tothedifference formulas ,so explicit examples thathe produced ,including Lagrange ’s formulas ,areoften known by other names .Theprocess ends with thesingle nthdivided difference , f\.X1 1X2t   *X f l] f[X Q$X i f. .|/[*0,*1 x„ ]= X n-X0 With thisnotation ,itcanbeshown thatthenthLagrange interpolation polynomial for/ with respect toX Q,X\,. ..,x ncanbeexpressed as P n(x)=f[x0]+/[*0,*l](*-X o)+f i x0,X Ux2)(x-X0)(x-Xj)+   +/[*0,*1*i»K*-*o)(*-*!)   (*-X n-i) which iscalled Newton 'sdivided -difference formula .Incompressed form wehave the following . Copyright 2012 Cc«£»fcLcarnin*.AIRighb Reversed May notbecopied ,canned ,ordnplicaied.»whole o tmpar.Doc 10electronic itfhu .vomc third pur.ycontent may besuppreved Tiom ihceBook and/orcCh deemed Cut arty vupprevsed content doev notmaterial yalTcct theiAcra .lteamireexperience .Ccngjgc [.camonroervev thenjht Mremove additional conceal atanytime i!vubveqjcnt nuhtv revtrictionv require It.