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1、、l/2丿X=maxx.co!
2、•,})]nllxll2=V30,llxll,=4b.Supposex=(2,1,一3,4)丁•Thenc.Suppose-1ThenIIAII,=4.IIAIL=4IK=^ZK-I】J=1d.Suppose2-1A=-12Thenthespectralradius(谱半径)p(A)=3IIAll2=_3[/?(A)=max
3、2jand&isaneigenvalueofAforeachi.H4=Vp(ata),AtAforeachi.]nn~~wherep(AA)=maxand2・isaneigenvalueofia.T
4、heTrapezoidalrule(梯形求积公式)appliedto£f(x)dxgivesthevalue4,andSimpson'srulegivesthevalue2.Then/(l)=1/2Infact.r=^[/(0)+/(2)]=4=>/(0)+/⑵=4,5=^[/(0)+4/(l)+/(2)]=2=>/(l)=^-[6-/(0)-/(2)]=i[6-4]=6442b.Thefunction/(x)=eAisapproximatedontheinterval[-1,1]bythesecondMaclaurin(麦
5、克劳林)polynomialP2(x)=l+x+x2/2.Thenthelinearpolynomial片(x)thatbestuniformlyapproximates(最佳一致逼近)P2(x)on[-1,1]is兀+5/4.兀2(Solution.Since£(x)=l+x,thelinearpolynomialP^x)thatbestuniformlyapproximatesP2(x)on[-1,1JisP^x)=P2(x)-aj2(x)(y215=l+x+———:(2x2-1)=x+—.I2)2*x2!4%(兀)=1
6、,7]U)=x,T2(x)=2x-7j(x)-7^(x)=2x-x-=2x2-1)2.a)ShowthatthesequenceisgeneratedbyNewton^smethodforfindingtherootofequationx2-3=0・b)Thesequence{}convergesto5/3wheneverg号,3.c)Usex0=L5tocomputex3with6significantdigits(有效数字)•(15%)Proof,a)Let/(x)=x2-3.ThenbyNewton'siterati
7、onformula,wehave^-1-Xn-l~32^-1兀=XH-
8、/(•VJ/g)13b)Letg(x)=—x+一・Theng(x)wC[3/2,3]•22x13Sinceg'(x)=ismonotonicallyincreasingandg'(V^)=O,weder2ivethatg(x)ismonotonicallydecreasingontheinterval[3/2,V3]andmonotonicallyincreasingontheinterval[命，3].Thereforeminxe[3/2,3]gM=g
9、(f3)=y/3,maxg⑴=max{g(3/2),g(3)}=max{7/4,2}=2,xw⑶2.3]―7ismonotonicallyincreasingmeansthatforwhichimpliesg(兀)g[V3,2]c[3/2,3].Thefactthatgf(x)=——2anyxg[3/2,3],—10、onthatthesequencex=g(x_))=—%+convergesto，，2，2⑺x（）w•
11、,3x=2(x)=—x+—,ie，x=[3whenever22xc)Substitutingx0=1.5intoxngives13137similarly2°2x0224岀+Z“732
