6.5--复化求积公式

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1、6.5EEEzzz¦¦¦ÈÈÈúúúªªª6.5.1EEEzzzFFF///úúúªªªHomePageTitlePageJJII6.5.2EEEzzzSimpsonúúúªªªJIPage1of1006.5.3EEEzzzCotesúúúªªªGoBackFullScreenCloseQuit6.5.1EEEzzzFFF///úúúªªª½½½nnn6.5.1ef∈C2[a,b]§½n§b−ah=,xi=a+ih,i=0,1,···,nnK3η∈(a,b)§¦Z"#bhXn−1b−a200HomePa

2、gef(x)dx=f(a)+2f(xi)+f(b)−hf(η),(6.5.1)a212TitlePagei=1JJIIÙ¥"#Xn−1JIhTn=f(a)+2f(xi)+f(b)(6.5.2)Page2of1002i=1GoBack¡EEEzzzFFF///úúúªªª"?kFullScreenZCloseblimT=f(x)dx.(6.5.3)Quitnh→0a6.5.2EEEzzzSimpsonúúúªªª½½½nnn6.5.2ef∈C4[a,b]§½n§b−ah=,xi=a+ih,

3、i=0,1,···,n,nxi−1´f«m[xi−1,xi]¥::2xi−1+xi1xi−1==a+i−h,i=1,2,···,n,222K3η∈(a,b)§¦HomePageZ"nn−1#4bhXXb−ahTitlePage(4)f(x)dx=f(a)+4f(xi−1)+2f(xi)+f(b)−f(η),a621802JJIIi=1i=1(6.5.4)JI"#Page3of100hXnXn−1GoBackSn=f(a)+4f(xi−1)+2f(xi)+f(b)(6.5.5)

4、62i=1i=1FullScreen¡EEEzzzSimpsonúúúªªª§?kCloseZbQuitlimSn=f(x)dx.(6.5.6)h→0a6.5.3EEEzzzCotesúúúªªªHomePageTitlePageJJIIJIPage4of100GoBackFullScreenCloseQuit½½½nnn6.5.3ef∈C6[a,b]§½n§b−ah=,xi=a+ih,i=0,1,···,n,n321xi−3=a+i−h,xi−2=a+i−h,xi−1=a+i−h,444444i=1,2,··

5、·,n.K3η∈(a,b)§¦Z"bhXnXnXnf(x)dx=7f(a)+32f(xi−3)+12f(xi−1)+32f(xi−1)90424ai=1i=1i=1#HomePageXn−12(b−a)h6+14f(x)+7f(b)−f(6)(η),(6.5.7)TitlePagei9456i=1JJIIJI"#nnnn−1hXXXXPage5of100Cn=7f(a)+32f(xi−3)+12f(xi−1)+32f(xi−1)+14f(xi)+7f(b)90424GoBacki=

6、1i=1i=1i=1(6.5.8)FullScreen¡EEEzzzCotesúúúªªª"?kCloseZbQuitlimCn=f(x)dx.(6.5.9)h→0a~~~6.5.1Z1sinxI=dx(I=0.946083070367183···)0x(1)^EzF/úªTnOICq§¦

7、I−Tn

8、≤0.5×10−5§nAõº¿OAT.n(2)^EzSimpsonúªSnOICq§¦

9、I−Sn

10、≤0.5×10−5§nAõº¿OAS.nHomePage)))ÏZsinx1TitlePag

11、ef(x)==cos(xt)dt,JJIIx0JIék≥1§kZZPage6of1001dk1kπf(k)(x)=[cos(xt)]dt=tkcosxt+dt.GoBackdxk200FullScreenÏdCloseZ1Z1Quitf(k)(x)≤tkcosxt+kπdt≤tkdt=1.020k+1(1)d½n6.5.1§3η∈(0,1)§¦Z"#1sinxhXn−1sinxsin11−0i200dx=1+2+−hf(η).0x2xi112i=1âK¿§k

12、I−T

13、≤0.5×10−5§=n1−

14、0h21(1−0)21h2f00(η)≤·==≤0.5×10−5.12122