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0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 302 "Times" 1 12 103 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet \+ Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "Simpson's rule " }}{PARA 0 "" 0 " " {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 " " {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "lo ad numerical integration procedures including: " }{TEXT 0 4 "simp" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 271 6 "intg.m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 4 "simp " }{TEXT -1 1 " " }{TEXT -1 24 "used in this worksheet. " }}{PARA 0 " " 0 "" {TEXT -1 122 "It can be read into a Maple session by a command \+ similar to the one that follows, where the file path gives its locatio n. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "read \"K:\\\\Maple/pr ocdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 16 "Simpson's rule: " }{TEXT 0 16 "student[simpson]" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 29 "First consider three points (" }{XPPEDIT 18 0 "-h,y[0];" "6$,$% \"hG!\"\"&%\"yG6#\"\"!" }{TEXT -1 5 "), ( " }{XPPEDIT 18 0 "0,y[1];" " 6$\"\"!&%\"yG6#\"\"\"" }{TEXT -1 6 " ), ( " }{XPPEDIT 18 0 "h,y[2];" " 6$%\"hG&%\"yG6#\"\"#" }{TEXT -1 17 ") along a curve " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{GLPLOT2D 400 371 371 {PLOTDATA 2 "69-%'CURVESG6$7S7 $$!3++++++++:!#<$\"3B+D1++]7GF*7$$!3&*****\\P&3YV\"F*$\"3*=$=]ivUCHF*7 $$!3!***\\ivX6vg,$F*7$$!31++D;as88F*$\"3TvNt'R`H6$F*7$$! 3#****\\P\"\\J\\7F*$\"3s90gK41/KF*7$$!3'***\\7V0@&=\"F*$\"3q]py(fe&)G$ F*7$$!33+]i&exd7\"F*$\"3KF,u8ydhLF*7$$!3'***\\i+#QU1\"F*$\"3b\\VFjN(>V $F*7$$!3****\\i!3%f+5F*$\"3_qwf)y&R*\\$F*7$$!3;++D\"oS:P*!#=$\"3iQ'=C@ f8c$F*7$$!3h*****\\<#)*=()FX$\"3d)e)pc0&)>OF*7$$!3#*****\\(G3U9)FX$\"3 oem2p&Qrm$F*7$$!3Y*****\\-\\r\\(FX$\"3NCG;v#Her$F*7$$!3?+++vGVZoFX$\"3 3*))\\;A(3gPF*7$$!3_*****\\(4J@iFX$\"3uDM<\"R]&)z$F*7$$!37++D1Bt_cFX$ \"3vrB26[3IQF*7$$!3')*****\\FPm(\\FX$\"3A=;,1yejQF*7$$!3()*******4'*QS %FX$\"3!f!o@&RZ())QF*7$$!3?++Dc>mPPFX$\"3wo0*\\d1X\"RF*7$$!3'3+++&=$z9 $FX$\"3v.e<%>TV$RF*7$$!3N***\\iX/4]#FX$\"3=4cHy35`RF*7$$!3C***\\(o8y%) =FX$\"3%*eU#)[rBoRF*7$$!33****\\i:#>C\"FX$\"3!Q2y>f*R\")RF*7$$!3O!*** \\7ev:l!#>$\"3c#>_W-+8*RF*7$$!3uF++](o2[\"!#?$\"3O[Q/&pE)**RF*7$$\"3i( ***\\P>:mkFir$\"3h'H:NhMl+%F*7$$\"3d***\\iv&QA7FX$\"3_*[Zg8p2,%F*7$$\" 3j++]PPBW=FX$\"3%)=IJ7%\\Q,%F*7$$\"3%*)*****\\Nm'[#FX$\"3'4\\$[0Dg:SF* 7$$\"36****\\(yb^6$FX$\"3,HfrsO6;SF*7$$\"3')***\\PMaKs$FX$\"3Cj;@'Gjc, %F*7$$\"3a****\\7TW)R%FX$\"3938$pA&H9SF*7$$\"3*y*****\\@80]FX$\"3+zVpz G[7SF*7$$\"3_+++D6!Hl&FX$\"3P>$Q@jZ,,%F*7$$\"3j)**\\P4w)RiFX$\"3(G#*pO d_y+%F*7$$\"3s,++vZf\")oFX$\"3Bn*)yIgM0SF*7$$\"3'z**\\P/-a[(FX$\"3xB0+ mP<.SF*7$$\"3R++v=Yb;\")FX$\"3Q'**))z8\"H,SF*7$$\"3s)****\\i@Ot)FX$\"3 !)zC!G:C++%F*7$$\"3g)**\\PfL'z$*FX$\"3ia+++!*>=+5F*$ \"317b8QI++SF*7$$\"3-++DE&4Q1\"F*$\"3xGX$=))3<+%F*7$$\"3=+]P%>5p7\"F*$ \"3YEXEQO![+%F*7$$\"39+++bJ*[=\"F*$\"3E$[&=H@04SF*7$$\"33++Dr\"[8D\"F* $\"3;y.MTFy:SF*7$$\"3++++Ijy58F*$\"3s@p;I/nBSF*7$$\"31+]P/)fTP\"F*$\"3 U*p'QD%>U.%F*7$$\"31+]i0j\"[V\"F*$\"3;v`OtycYSF*7$$\"3++++++++:F*$\"3y *\\P*****\\iSF*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fb[lFa[l-F$6$7S7$ F($\"3++++++]iIF*7$F.$\"33G!e;XAo7$F*7$F3$\"3c-)>>i\\5=$F*7$F8$\"3#3>7 'G+5SKF*7$F=$\"3YdH8LVZ(H$F*7$FB$\"3[&3NjN;DN$F*7$FG$\"3HO)>0\"=r,MF*7 $FL$\"3Qlj#Hr*y]MF*7$FQ$\"3s-$)H^Vb*\\$F*7$FV$\"3'4%z6Yq9YNF*7$Ffn$\"3 g!30-#Q(>f$F*7$F[o$\"3pSqBrWdIOF*7$F`o$\"31athlJ0sOF*7$Feo$\"36B4oNef6 PF*7$Fjo$\"3Ihe+]aqZPF*7$F_p$\"3+W%GgB)zyPF*7$Fdp$\"3I,xqrnm8QF*7$Fip$ \"3-75NXoTTQF*7$F^q$\"3a`d%)eJjrQF*7$Fcq$\"3%*)Q>k,Gl*QF*7$Fhq$\"3TAb% 33T=#RF*7$F]r$\"32#GhPY**R%RF*7$Fbr$\"39%**=!Qg4lRF*7$Fgr$\"31ZNiL#\\E )RF*7$F]s$\"3/xMkf#H'**RF*7$Fcs$\"3.z<#=5?^,%F*7$Fhs$\"3eUNcwS#o-%F*7$ F]t$\"3ym6C[GgPSF*7$Fbt$\"35`Fs\\yqYSF*7$Fgt$\"3+8lt0%=O0%F*7$F\\u$\"3 y;JK,[UeSF*7$Fau$\"35NbGE`fhSF*7$Ffu$\"3q!f_T$**\\iSF*7$F[v$\"3oMCI+VV hSF*7$F`v$\"3Co.=onleSF*7$Fev$\"3!=tcF+\\O0%F*7$Fjv$\"3c3BqTp0ZSF*7$F_ w$\"3%)p3F=x@QSF*7$Fdw$\"3CJL!R<]w-%F*7$Fiw$\"3IT7'R-ZX,%F*7$F^x$\"375 *>U\\a***RF*7$Fcx$\"3WPu_/(HI)RF*7$Fhx$\"3wur/-fCkRF*7$F]y$\"3eO&e:MI_ %RF*7$Fby$\"3E?)*Q\")*o8#RF*7$Fgy$\"3w#Gr%F]]mQ\"1F`_mFdelFdbm-Fh^m6&7 $$\"$E\"F]]m$\"$A%F]]mQ\"2F`_mFdelFdbm-Fh^m6&7$$\"#;F[fl$\"#PF[flQ)y~= ~p(x)F`_mFidlFa_m-Fh^m6&7$$!$X\"F]]m$\"#FF[flQ)y~=~f(x)F`_mFjzFa_m-%*A XESTICKSG6$Fb[lFb[l-%+AXESLABELSG6%F\\amQ!F`_m-Fb_m6#%(DEFAULTG-%%VIEW G6$;$F]]mFb[l$\"\"#Fb[l;Fi_m$\"\"&Fb[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" }}{TEXT -1 3 " " }}{PARA 0 "" 0 " " {TEXT -1 5 "Let " }{XPPEDIT 18 0 "y=p(x)" "6#/%\"yG-%\"pG6#%\"xG" } {TEXT -1 7 " where " }{XPPEDIT 18 0 "p(x) = a*x^2+b*x+c;" "6#/-%\"pG6# %\"xG,(*&%\"aG\"\"\"*$F'\"\"#F+F+*&%\"bGF+F'F+F+%\"cGF+" }{TEXT -1 53 " be the parabola passing through these three points." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([ y[0] = a*h^2-b*h+c, ``],[y[1] = c ,`` ],[y[2] = a*h^2+b*h+c ,`` ])" " 6#-%*PIECEWISEG6%7$/&%\"yG6#\"\"!,(*&%\"aG\"\"\"*$%\"hG\"\"#F/F/*&%\"b GF/F1F/!\"\"%\"cGF/%!G7$/&F)6#F/F6F77$/&F)6#F2,(*&F.F/*$F1F2F/F/*&F4F/ F1F/F/F6F/F7" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 30 "The are a under the parabola is" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "Int(a*x^2+b*x+c,x = -h .. h) = a*x^3/3+b*x^2/2+c*x;" "6 #/-%$IntG6$,(*&%\"aG\"\"\"*$%\"xG\"\"#F*F**&%\"bGF*F,F*F*%\"cGF*/F,;,$ %\"hG!\"\"F4,(*(F)F**$F,\"\"$F*F9F5F**(F/F**$F,F-F*F-F5F**&F0F*F,F*F* " }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([h, ``],[-h, ``])" "6#-%*P IECEWISEG6$7$%\"hG%!G7$,$F'!\"\"F(" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(a*h^3/3+b*h ^2/2+c*h)-``(-a*h^3/3+b*h^2/2-c*h);" "6#/%!G,&-F$6#,(*(%\"aG\"\"\"*$% \"hG\"\"$F+F.!\"\"F+*(%\"bGF+*$F-\"\"#F+F3F/F+*&%\"cGF+F-F+F+F+-F$6#,( *(F*F+*$F-F.F+F.F/F/*(F1F+*$F-F3F+F3F/F+*&F5F+F-F+F/F/" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " \+ " }{XPPEDIT 18 0 "`` = 2/3*a*h^3+2*c*h;" "6#/%!G,&**\"\"#\"\"\"\"\"$ !\"\"%\"aGF(%\"hGF)F(*(F'F(%\"cGF(F,F(F(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 32 "From the equations above we have" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y[0]+y[2] = 2*a*h^2+2*c;" "6#/,& &%\"yG6#\"\"!\"\"\"&F&6#\"\"#F),&*(F,F)%\"aGF)%\"hGF,F)*&F,F)%\"cGF)F) " }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[0]+y[2]+4*y[1]=2*a*h^2+2*c+4*y[1 ]" "6#/,(&%\"yG6#\"\"!\"\"\"&F&6#\"\"#F)*&\"\"%F)&F&6#F)F)F),(*(F,F)% \"aGF)%\"hGF,F)*&F,F)%\"cGF)F)*&F.F)&F&6#F)F)F)" }{TEXT -1 1 " " }} {PARA 258 "" 0 "" {TEXT -1 10 "and since " }{XPPEDIT 18 0 "y[1]=c" "6# /&%\"yG6#\"\"\"%\"cG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y[0]+4*y[1]+y[2]=2*a*h^2+6*c" "6#/,(&%\"yG6#\"\"! \"\"\"*&\"\"%F)&F&6#F)F)F)&F&6#\"\"#F),&*(F0F)%\"aGF)%\"hGF0F)*&\"\"'F )%\"cGF)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h/3" "6#*&%\"hG\"\" \"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[y[0]+4*y[1]+y[2]] = h/3 ;" "6#/7#,(&%\"yG6#\"\"!\"\"\"*&\"\"%F*&F'6#F*F*F*&F'6#\"\"#F**&%\"hGF *\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[2*a*h^2+6*c];" "6#7#,&*( \"\"#\"\"\"%\"aGF'%\"hGF&F'*&\"\"'F'%\"cGF'F'" }{XPPEDIT 18 0 "``=2/3* a*h^3+2*c*h" "6#/%!G,&**\"\"#\"\"\"\"\"$!\"\"%\"aGF(%\"hGF)F(*(F'F(%\" cGF(F,F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Hence the " }{TEXT 262 23 "area under the parab ola" }{TEXT -1 29 " can be calculated using the " }{TEXT 296 1 "y" } {TEXT -1 33 " coordinates of the three points " }{XPPEDIT 18 0 "y[0],y [1],y[2]" "6%&%\"yG6#\"\"!&F$6#\"\"\"&F$6#\"\"#" }{TEXT -1 28 " and th e horizontal spacing " }{TEXT 298 1 "h" }{TEXT -1 13 " between the " } {TEXT 297 1 "x" }{TEXT -1 29 " coordinates by the formula: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "[y[0]+4*y[1]+y[2]];" "6#7# ,(&%\"yG6#\"\"!\"\"\"*&\"\"%F)&F&6#F)F)F)&F&6#\"\"#F)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "h/3;" "6#*&%\"hG\"\"\"\"\"$!\"\"" }{TEXT -1 15 " - ------ (i)." }}{PARA 0 "" 0 "" {TEXT -1 50 "The last expression provid es an approximation for " }{XPPEDIT 18 0 "Int(f(x),x = -h .. h);" "6#- %$IntG6$-%\"fG6#%\"xG/F);,$%\"hG!\"\"F-" }{TEXT -1 57 ", and this meth od of approximating an integral is called " }{TEXT 262 14 "Simpson's r ule" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 153 "Usually the rule \+ is used repeatedly. For example the following picture illustrates usin g the formula (i) three times to estimate an integral of the form " } {XPPEDIT 18 0 "Int(f(x),x=a..b)" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\" bG" }{TEXT -1 20 ". The interval from " }{TEXT 299 1 "a" }{TEXT -1 4 " to " }{TEXT 300 1 "b" }{TEXT -1 58 " is divided into six subintervals by seven equally spaced " }{TEXT 301 1 "x" }{TEXT -1 8 " values " } {XPPEDIT 18 0 "a = x[0],x[1],x[2],x[3],x[4],x[5],x[6] = b;" "6)/%\"aG& %\"xG6#\"\"!&F&6#\"\"\"&F&6#\"\"#&F&6#\"\"$&F&6#\"\"%&F&6#\"\"&/&F&6# \"\"'%\"bG" }{TEXT -1 12 ", such that " }{XPPEDIT 18 0 "x[i]-x[i-1]=h " "6#/,&&%\"xG6#%\"iG\"\"\"&F&6#,&F(F)F)!\"\"F-%\"hG" }{TEXT -1 5 " fo r " }{XPPEDIT 18 0 "i = 1,2,` . . . `,6;" "6&/%\"iG\"\"\"\"\"#%(~.~.~. ~G\"\"'" }{TEXT -1 2 ". 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Fi\\nFhhpFjhpF^hp-Ffgp6%7$$\"$s$FafnFgipFjhpF^hp-Ffgp6%7$$\"$u%Fafn$\" $$HFafnFjhpF^hp-Ffgp6%7$$\"#kFc[m$\"$x#FafnFjhpF^hp-Ffgp6%7$F][l$Fa[lF afnF\\hpF^hp-Ffgp6%7$F\\\\nFb[qF\\hpF^hp-Ffgp6%7$Fe[nFb[qF\\hpF^hp-Ffg p6%7$Fi\\nFb[qF\\hpF^hp-Ffgp6%7$F`]nFb[qF\\hpF^hp-Ffgp6%7$Fg]nFb[qF\\h pF^hp-Ffgp6%7$F`^nFb[qF\\hpF^hp-Ffgp6%7$$!#DFafn$\"#@Fc[mQ'0~~~~0F]hp- F_hp6$Fahp\"\")-Ffgp6%7$F\\\\n$\"#DFc[mQ'1~~~~1F]hpF]]q-Ffgp6%7$$\"$C# Fafn$\"#BFc[mQ'2~~~~2F]hpF]]q-Ffgp6%7$$\"$/$FafnFj\\qQ'3~~~~3F]hpF]]q- Ffgp6%7$$\"$w$FafnF[^qQ'4~~~~4F]hpF]]q-Ffgp6%7$$\"$y%Fafn$\"$)GFafnQ'5 ~~~~5F]hpF]]q-Ffgp6%7$$\"$W'Fafn$\"$s#FafnQ'6~~~~6F]hpF]]q-Ffgp6%7$$F_ ]qFafn$!#8FafnQ\"0F]hpF]]q-Ffgp6%7$$\"$3\"FafnF^`qQ\"1F]hpF]]q-Ffgp6%7 $$\"$3#FafnF^`qQ\"2F]hpF]]q-Ffgp6%7$$\"$3$FafnF^`qQ\"3F]hpF]]q-Ffgp6%7 $$\"$3%FafnF^`qQ\"4F]hpF]]q-Ffgp6%7$$\"$3&FafnF^`qQ\"5F]hpF]]q-Ffgp6%7 $$\"$3'FafnF^`qQ\"6F]hpF]]q-Ffgp6&7$F`^n$\"#JFc[mQ)y~=~f(x)F]hpFcgpF^h p-%+AXESLABELSG6%F\\hpQ!F]hp-F_hp6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%% VIEWG6$;$!\"&Fc[mFigp;$FafnFc[m$\"#KFc[m" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36 " "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "C urve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The corresponding estima te is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[y[0]+4*y[1 ]+y[2]]" "6#7#,(&%\"yG6#\"\"!\"\"\"*&\"\"%F)&F&6#F)F)F)&F&6#\"\"#F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "h/3+[y[2]+4*y[3]+y[4]]" "6#,&*&%\"hG\" \"\"\"\"$!\"\"F&7#,(&%\"yG6#\"\"#F&*&\"\"%F&&F,6#F'F&F&&F,6#F0F&F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "h/3+[y[4]+4*y[5]+y[6]]" "6#,&*&%\"hG\" \"\"\"\"$!\"\"F&7#,(&%\"yG6#\"\"%F&*&F.F&&F,6#\"\"&F&F&&F,6#\"\"'F&F& " }{TEXT -1 1 " " }{XPPEDIT 18 0 "h/3" "6#*&%\"hG\"\"\"\"\"$!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=[y[0]+4*y[1]+2*y[2]+4*y[3]+2*y[4]+4* y[5]+y[6]]" "6#/%!G7#,0&%\"yG6#\"\"!\"\"\"*&\"\"%F+&F(6#F+F+F+*&\"\"#F +&F(6#F1F+F+*&F-F+&F(6#\"\"$F+F+*&F1F+&F(6#F-F+F+*&F-F+&F(6#\"\"&F+F+& F(6#\"\"'F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h/3" "6#*&%\"hG\"\"\"\"\" $!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "More generally, if we are given an " }{TEXT 262 10 " odd number" }{TEXT -1 10 " of points" }{XPPEDIT 18 0 "``(x[0],y[0]),`` (x[1],y[1]),` . . . `,``(x[n],y[n]);" "6&-%!G6$&%\"xG6#\"\"!&%\"yG6#F) -F$6$&F'6#\"\"\"&F+6#F1%(~.~.~.~G-F$6$&F'6#%\"nG&F+6#F9" }{TEXT -1 15 " along a curve " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 22 ", with equally spaced " }{TEXT 273 1 "x" }{TEXT -1 13 " c oordinates " }{XPPEDIT 18 0 "a = x[0],x[1],` . . . `,x[n] = b;" "6&/% \"aG&%\"xG6#\"\"!&F&6#\"\"\"%(~.~.~.~G/&F&6#%\"nG%\"bG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "y[i]=f(x[i])" "6#/&%\"yG6#%\"iG-%\"fG6#&%\"xG 6#F'" }{TEXT -1 10 " for each " }{TEXT 276 1 "i" }{TEXT -1 95 ", we ca n apply the formula (i) on each consecutive pair of intervals to give \+ the approximation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b)" "6#-%$IntG6$-% \"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 2 " " }{TEXT 260 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "[y[0]+4*y[1]+y[2]];" "6#7#,(&%\"yG6#\"\"!\" \"\"*&\"\"%F)&F&6#F)F)F)&F&6#\"\"#F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 " h/3+[y[2]+4*y[3]+y[4]];" "6#,&*&%\"hG\"\"\"\"\"$!\"\"F&7#,(&%\"yG6#\" \"#F&*&\"\"%F&&F,6#F'F&F&&F,6#F0F&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 " h/3" "6#*&%\"hG\"\"\"\"\"$!\"\"" }{TEXT -1 4 " + " }{TEXT 274 5 ". . \+ ." }}{PARA 0 "" 0 "" {TEXT -1 92 " \+ + " }{XPPEDIT 18 0 "[y[n-2]+4*y[n-1]+y[n]];" "6#7#,(&%\"yG6#,&%\"nG\"\"\"\"\"#!\"\"F **&\"\"%F*&F&6#,&F)F*F*F,F*F*&F&6#F)F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h/3" "6#*&%\"hG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = [y[0]+4*y[1]+2*y[2]+4*y[3]+2*y[4 ]*`+ . . . +`*2*y[n-2]+4*y[n-1]+y[n]];" "6#/%!G7#,0&%\"yG6#\"\"!\"\"\" *&\"\"%F+&F(6#F+F+F+*&\"\"#F+&F(6#F1F+F+*&F-F+&F(6#\"\"$F+F+*,F1F+&F(6 #F-F+%*+~.~.~.~+GF+F1F+&F(6#,&%\"nGF+F1!\"\"F+F+*&F-F+&F(6#,&F?F+F+F@F +F+&F(6#F?F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h/3;" "6#*&%\"hG\"\"\"\" \"$!\"\"" }{TEXT -1 16 " ------- (ii), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 277 37 "_____________________________________" }{TEXT -1 14 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "h=(b-a)/n" "6#/%\"hG*&,&%\"bG\" \"\"%\"aG!\"\"F(%\"nGF*" }{TEXT -1 28 " is the spacing between the " } {TEXT 272 1 "x" }{TEXT -1 13 " coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The formula (ii) is the " }{TEXT 262 13 "compound form" }{TEXT -1 4 " of " }{TEXT 262 14 "Simpson's rule" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The formula (ii) can be rearranged in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[``(y[0]+y[n])+4*(y[1]+y[3]+` . . . `+y[n-1])+2*(y[2]+y [4]+` . . . `+y[n-2])];" "6#7#,(-%!G6#,&&%\"yG6#\"\"!\"\"\"&F*6#%\"nGF -F-*&\"\"%F-,*&F*6#F-F-&F*6#\"\"$F-%(~.~.~.~GF-&F*6#,&F0F-F-!\"\"F-F-F -*&\"\"#F-,*&F*6#F?F-&F*6#F2F-F9F-&F*6#,&F0F-F?F=F-F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h/3" "6#*&%\"hG\"\"\"\"\"$!\"\"" }{TEXT -1 16 " --- ---- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Describing the sum" }{XPPEDIT 18 0 " ``(y[0]+y[n])" "6#-%!G6#,& &%\"yG6#\"\"!\"\"\"&F(6#%\"nGF+" }{TEXT -1 5 " as \"" }{TEXT 271 4 "en ds" }{TEXT -1 10 "\", the sum" }{XPPEDIT 18 0 " ``(y[1]+y[3]+` . . . ` +y[n-1])" "6#-%!G6#,*&%\"yG6#\"\"\"F*&F(6#\"\"$F*%(~.~.~.~GF*&F(6#,&% \"nGF*F*!\"\"F*" }{TEXT -1 39 " in which the subsbscripts are odd as \+ \"" }{TEXT 271 4 "odds" }{TEXT -1 13 "\" and the sum" }{XPPEDIT 18 0 " ``(y[2]+y[4]+` . . . `+y[n-2])" "6#-%!G6#,*&%\"yG6#\"\"#\"\"\"&F(6#\" \"%F+%(~.~.~.~GF+&F(6#,&%\"nGF+F*!\"\"F+" }{TEXT -1 40 " in which the \+ subsbscripts are even as \"" }{TEXT 271 5 "evens" }{TEXT -1 35 "\", th e formula (iii) has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[\"ends\" + 4*\"odds\"+2*\"evens\"]" "6#7#,(Q%ends6\"\" \"\"*&\"\"%F'Q%oddsF&F'F'*&\"\"#F'Q&evensF&F'F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "h/3" "6#*&%\"hG\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 278 17 "_________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "\nNow " }{XPPEDIT 18 0 "x [i]=a+i*h" "6#/&%\"xG6#%\"iG,&%\"aG\"\"\"*&F'F*%\"hGF*F*" }{TEXT -1 4 ", so" }}{PARA 256 "" 0 "" {TEXT -1 4 "\n " }{XPPEDIT 18 0 "Int(f(x) ,x = a .. b)" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 2 " \+ " }{TEXT 261 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "h/3;" "6#*&%\"hG\" \"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[f(a)+f(b)+4*Sum(f(a+( 2*i-1)*h),i = 1 .. n/2)+2*Sum(f(a+2*i*h),i = 1 .. n/2-1)];" "6#7#,*-% \"fG6#%\"aG\"\"\"-F&6#%\"bGF)*&\"\"%F)-%$SumG6$-F&6#,&F(F)*&,&*&\"\"#F )%\"iGF)F)F)!\"\"F)%\"hGF)F)/F9;F)*&%\"nGF)F8F:F)F)*&F8F)-F06$-F&6#,&F (F)*(F8F)F9F)F;F)F)/F9;F),&*&F?F)F8F:F)F)F:F)F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Simpson's rule can be applied by means of the procedure " }{TEXT 0 7 "simpson" }{TEXT -1 8 " in the " }{TEXT 0 7 "student" }{TEXT -1 9 " package." }} {PARA 0 "" 0 "" {TEXT -1 66 "We can obtain essentially the formula abo ve (with \"=\" instead of \"" }{TEXT 275 1 "~" }{TEXT -1 5 "\" ). " }} {PARA 0 "" 0 "" {TEXT 302 4 "Note" }{TEXT -1 16 ": The procedure " } {TEXT 0 7 "simpson" }{TEXT -1 37 " can be accessed without loading the " }{TEXT 0 7 "student" }{TEXT -1 27 " package by using the name " } {TEXT 0 16 "student[simpson]" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "f := 'f': x := 'x': a := 'a': b := 'b': n := 'n': h := 'h':\nInt(f(x),x=a..b)=stu dent[simpson](f(x),x=a..a+n*h,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$IntG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG,$*&%\"hG\"\"\",*-F(6#F-F2-F(6#,&F -F2*&%\"nGF2F1F2F2F2-%$SumG6$-F(6#,&F-F2*&,&%\"iG\"\"#!\"\"F2F2F1F2F2/ FC;F2,$F:#F2FD\"\"%-F<6$-F(6#,&F-F2*&FCF2F1F2FD/FC;F2,&F:FIFEF2FDF2#F2 \"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The following code will generate specific instances of the formul a for the compound form of Simpson's rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "n := 6:\nstudent[ simpson](f(x),x=a..a+n*h,n);\nvalue(%);\nfor i from 0 to n do subs(f(a +i*h)=y[i],%) end do:\n%; n:= 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&#\"\"\"\"\"$F&*&%\"hGF&,*-%\"fG6#%\"aGF&-F,6#,&F.F&*&\"\"'F&F)F&F& F&*&\"\"%F&-%$SumG6$-F,6#,&F.F&*&,&*&\"\"#F&%\"iGF&F&F&!\"\"F&F)F&F&/F @;F&F'F&F&*&F?F&-F76$-F,6#,&F.F&*(F?F&F@F&F)F&F&/F@;F&F?F&F&F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&*&%\"hGF&,0-%\"fG6#% \"aGF&-F,6#,&F.F&*&\"\"'F&F)F&F&F&*&\"\"%F&-F,6#,&F.F&F)F&F&F&*&F5F&-F ,6#,&F.F&*&F'F&F)F&F&F&F&*&F5F&-F,6#,&F.F&*&\"\"&F&F)F&F&F&F&*&\"\"#F& -F,6#,&F.F&*&FEF&F)F&F&F&F&*&FEF&-F,6#,&F.F&*&F5F&F)F&F&F&F&F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&*&%\"hGF&,0&%\"yG6# \"\"!F&&F,6#\"\"'F&*&\"\"%F&&F,6#F&F&F&*&F3F&&F,6#F'F&F&*&F3F&&F,6#\" \"&F&F&*&\"\"#F&&F,6#F>F&F&*&F>F&&F,6#F3F&F&F&F&F&" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "A procedure for numerical integra tion using Simpson's rule: " }{TEXT 0 4 "simp" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "simp: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 18 "Calling Sequenc e:\n" }}{PARA 0 "" 0 "" {TEXT 267 2 " " }{TEXT -1 19 " simp( gx, rn g ) " }{TEXT 268 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 " " {TEXT 23 10 " gx - " }{TEXT -1 55 " an expression involving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 83 " \+ where gx evaluates to a real floating point numbe r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 23 12 " rng - " }{TEXT 269 61 "the range x=a..b for the def inite integral to be aproximated." }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "simp" }{TEXT -1 43 " calcu lates a numerical approximation for " }{XPPEDIT 18 0 "Int(gx,x = a .. b);" "6#-%$IntG6$%#gxG/%\"xG;%\"aG%\"bG" }{TEXT -1 11 " by using " } {TEXT 288 9 "Simpson's" }{TEXT -1 23 " rule with a specified " }{TEXT 262 4 "even" }{TEXT -1 282 " number of sub-intervals of the interval f rom a to b obtained using equally spaced x values. Simpson's rule can \+ alse be applied iteratively whereby the number of intervals is success ively doubled until the desired accuracy is achieved or the maximum nu mber of iterations is reached." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 270 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "intervals=n" }}{PARA 0 "" 0 "" {TEXT -1 53 "This option determines the number of intervals used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "iterate=true/fa lse" }}{PARA 0 "" 0 "" {TEXT -1 29 "This option controls whether " } {TEXT 289 9 "Simpson's" }{TEXT -1 80 " rule is to be applied iterative ly rather than with a fixed number of intervals." }}{PARA 0 "" 0 "" {TEXT -1 31 "The default is \"iterate=false\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "maxiterations=n\nThis opt ion controls the maximum number of iterations of the integration proce dure." }}{PARA 0 "" 0 "" {TEXT -1 186 "The default is \"maxiterations= 22\" when the computation can be performed using hardware floating poi nt arithmetic and \"maxiterations=16\" when software floating point a rithmetic is used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "info=true/false\nWhen " }{TEXT 290 9 "Simpson's" } {TEXT -1 191 " rule is applied iteratively, option \"info=true\" allow s the progress of the procedure to be monitored by printing each appro ximation to the integral immediately after it has been calculated. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 262 16 "How to activate:" }{TEXT -1 156 "\nTo ma ke the procedures active open the subsection, place the cursor anywher e after the prompt [ > and press [Enter].\nYou can then close up the \+ subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "simp: implementati on " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7323 "simp := proc(fx,eqn)\n local a,b,h,i,j,x,rs,Optio ns,n,saveDigits,f,fa,fb,dist,\n dosimp,oddsum,it,sm,oddsm,val,las tval,eps,maxit,\n prntflg,ends,odds,evens,t,sign,maxit_opt,usehf, vals,ff;\n\n if nargs<2 then\n error \"at least 2 arguments are required; the basic syntax is: 'simp(f(x),x=a..b)'.\"\n end if;\n\n if not type(fx,algebraic) then \n error \"the 1st argument, %1 , is invalid ..it should be an algebraic expression in a single variab le\",fx;\n end if; \n if not type(eqn,`=`) then \n error \"th e 2nd argument, %1, is invalid ..it should be an equation of the form \+ 'x=a..b' to give the required interval for the integral to be estimate d\",eqn;\n end if; \n x := op(1,eqn);\n if not type(x,name) th en\n error \"the 2nd argument equation left side, %1, should be t he independent variable\",x;\n end if;\n if not type(indets(fx,nam e) minus \{x\},set(realcons)) then\n error \"the 1st argument, %1 , must depend only on the variable %2\",fx,x;\n end if; \n rs := \+ op(2,eqn);\n if not type(rs,realcons..realcons) then\n error \"the 2nd argument,%1, is invalid .. the right side of the equation, \+ %2, should be a range of real values\",eqn,rs;\n end if;\n \n # \+ Get the allowed options.\n Options := [];\n n := 4;\n it := fals e;\n prntflg := false;\n maxit_opt := false;\n if nargs>=3 then \n Options:=[args[3..nargs]];\n if not type(Options,list(equ ation)) then\n error \"each optional argument must be an equat ion\"\n end if;\n if hasoption(Options,'intervals','n','Opti ons') then\n if not type(n,posint) or irem(n,2)>0 then\n \+ error \"\\\"intervals\\\" must be a positive even integer\"\n \+ end if;\n end if;\n if hasoption(Options,'iterate','it ','Options') then\n if not it=true then it := false end if;\n \+ end if;\n if hasoption(Options,'maxiterations','maxit','Opti ons') then\n if not type(maxit,posint) then\n error \"\\\"maxiterations\\\" must be a positive integer\"\n end if ;\n maxit_opt := true;\n elif hasoption(Options,'maxiter' ,'maxit','Options') then\n if not type(maxit,posint) then\n \+ error \"\\\"maxiter\\\" must be a positive integer\"\n \+ end if;\n maxit_opt := true;\n end if;\n if hasopt ion(Options,'info','prntflg','Options') then\n if prntflg<>tru e then prntflg := false end if;\n end if;\n if nops(Options) >0 then\n error \"%1 is not a valid option for %2 .. the recog nised options are \\\"intervals\\\", \\\"iterate\\\", \\\"maxiteration s\\\",(or \\\"maxiter\\\") and \\\"info\\\"\",op(1,Options),procname; \n end if;\n end if;\n \n saveDigits := Digits;\n Digits \+ := max(trunc(evalhf(Digits)),Digits+5);\n a := evalf(op(1,rs));\n \+ b := evalf(op(2,rs));\n if a=b then return 0 end if;\n sign := 1; \n if b `,val)\n \+ end if;\n if abs(val-lastval)0 the n\n return evalf(val);\n else\n \+ return evalf(-val);\n end if;\n end if;\n lastval := val;\n evens := odds+evens;\n \+ end do;\n else\n if not maxit_opt then maxit := 1 6 end if;\n for j to maxit do\n n := 2*n;\n \+ h := dist/n; \n odds := traperror(oddsum(f,a,h,n));\n \+ if odds=lasterror or not type(odds,numeric) then\n \+ error \"computation of Simpson's rule estimate failed\"\n \+ end if;\n val := (ends+4*odds+2*evens)*h/3;\n \+ if prntflg then\n print(`approximation with `||n||` in tervals ---> `,val)\n end if;\n if abs(val-las tval)0 then\n return evalf(val);\n \+ else\n return evalf(-val);\n \+ end if;\n end if;\n lastval := val;\n \+ evens := odds+evens;\n end do;\n end if;\n WARNING (\"reached max %1 iterations without convergence\",j-1);\n Digits := saveDigits;\n if sign>0 then\n return evalf(val);\n \+ else\n return evalf(-val);\n end if;\n else\n \+ dosimp := proc(f,a,b,n)\n local h,ends,odds,evens,i;\n \+ h := (b-a)/n;\n ends := f(a)+f(b);\n evens := 0.;\n \+ i := 2;\n while i0 then\n return evalf(sm); \n else\n return evalf(-sm);\n end if;\n end if; \+ \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Examples using: " }{TEXT 0 16 "student[simpson]" }{TEXT -1 4 "and " }{TEXT 0 4 "simp" }{TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exampl e 1 " }{TEXT 264 24 ".. Simpson's rule using " }{TEXT 259 18 "student[ trapezoid]" }{TEXT 280 5 " and " }{TEXT 259 4 "simp" }{TEXT 281 43 " - absolute and relative error calculations" }}{PARA 0 "" 0 "" {TEXT -1 44 "We first find a numerical approximation for " }{XPPEDIT 18 0 "Int( sin(x),x = Pi/4 .. 3*Pi/4);" "6#-%$IntG6$-%$sinG6#%\"xG/F);*&%#PiG\"\" \"\"\"%!\"\"*(\"\"$F.F-F.F/F0" }{TEXT -1 39 " using Simpson's rule wit h 4 intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 543 "f := x -> sin(x): # function\na := evalf(Pi/4): # lower limit of integral\nb := evalf(3*Pi/4): # upper limit of integ ral\nclr := grey: # color for shading\npp := plot([0,f(x)],x=a..b,adap tive=false,numpoints=20):\nu := op(1,op(1,pp)): v := op(1,op(2,pp)):\n p1 := plots[polygonplot]([seq([u[i],v[i],v[i+1],u[i+1]],i=1..19)],\n \+ color=clr,style=patchnogrid):\np2 := plo t([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\np3 := plot(f(x),x =0..Pi,thickness=2): # adjust plot range\nplots[display]([p1,p2,p3],la bels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 471 286 286 {PLOTDATA 2 "6(-%)POLYGONSG677&7$$\"+N;)R&y!#5$\"\"!F,7$F($\"+8y1rqF*7$$\"+6T'*= ()F*$\"+\"=/bl(F*7$F1F+7&F5F07$$\"+R2er%*F*$\"+>7f<\")F*7$F8F+7&F7$$\"+V$)*p6\"FA$\"+(>')y)*)F*7$ FGF+7&FKFF7$$\"+cSz,7FA$\"+en(oK*F*7$FNF+7&FRFM7$$\"+P0T!G\"FA$\"+o[L \"e*F*7$FUF+7&FYFT7$$\"+]L\"=O\"FA$\"+uTU#y*F*7$FfnF+7&FjnFen7$$\"+b-+ Y9FA$\"+82BA**F*7$F]oF+7&FaoF\\o7$$\"+sr\"*H:FA$\"+Pck\"***F*7$FdoF+7& FhoFco7$$\"+NgB;;FA$\"+Tzn*)**F*7$F[pF+7&F_pFjo7$$\"+PdE#p\"FA$\"+xlJE **F*7$FbpF+7&FfpFap7$$\"+ur&yx\"FA$\"+\\QR'y*F*7$FipF+7&F]qFhp7$$\"+e+ !Q'=FA$\"+0j!Qd*F*7$F`qF+7&FdqF_q7$$\"+^>iY>FA$\"+:!\\?I*F*7$FgqF+7&F[ rFfq7$$\"+k@$=-#FA$\"+f1'****)F*7$F^rF+7&FbrF]r7$$\"+#Qk76#FA$\"+L%zYd )F*7$FerF+7&FirFdr7$$\"+-_-(=#FA$\"+/#Q1;)F*7$F\\sF+7&F`sF[s7$$\"+YI:v AFA$\"+?EG?wF*7$FcsF+7&FgsFbs7$$\"+!\\%>cBFAF.7$FjsF+-%'COLOURG6&%$RGB G$\")=THv!\")FatFat-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6$7$7$$\"3C+++N; )R&y!#=F+7$F]u$\"3U+++8y1rqF_u-F^t6&F`tF,F,F,-Fit6$7$7$$\"3-+++!\\%>cB !#$\"3v9-&QTFC%oFdv7$$ \"3)\\$px*G*f!G\"F_u$\"3q>Km$*>5x7F_u7$$\"3+5@exGm]>F_u$\"3/`P()fcJQ>F _u7$$\"3[99!=3o^i#F_u$\"3/)Q8J`>^f#F_u7$$\"35!\\D0[nkH$F_u$\"3'yFM)o\" )3PKF_u7$$\"37\"=Za&z%)=RF_u$\"3b2`lK+J>QF_u7$$\"3edXa()oGjXF_u$\"33g- l[Vb1WF_u7$$\"3W%3**Hbm(H_F_u$\"3OWs$HJ6Y*\\F_u7$$\"37PRr4)3T*eF_u$\"3 g4tHMSrebF_u7$$\"3y\"[)yykYxlF_u$\"32tCIi;N8hF_u7$$\"3[s'ocGo$zrF_u$\" 3iG>vb;KylF_u7$$\"3UQ0;gr'p&yF_u$\"3S'e5DeyJ2(F_u7$$\"3vFMt?$[t`)F_u$ \"3Cf#G%e3SPvF_u7$$\"3u\"p30k@I>*F_u$\"3C$=wEj'y^zF_u7$$\"3#*R7\\HeV)y *F_u$\"3=)oD&\\m_)H)F_u7$$\"3#G[))*)3W'\\5F[v$\"3Y4SV'zgCn)F_u7$$\"30' [@XS@'46F[v$\"3oX3_YFIb*)F_u7$$\"3G9w$3G*Qz6F[v$\"3=c,/>?tV#*F_u7$$\"3 >%3*3tc9T7F[v$\"3#*zOG!*[bh%*F_u7$$\"3Ey0DBA!*38F[v$\"3'*))o/b+VIn(**F_u7$$\"3=hw\"ymX#p:F [v$\"274'Gxz)*****F[v7$$\"3mwM!*4(4&Q;F[v$\"39vB0ZK3x**F_u7$$\"3CDL:iU !))p\"F[v$\"3[oG=d;==**F_u7$$\"3o(R1/.CRw\"F[v$\"3Vz\\$>Q(39)*F_u7$$\" 32Id)H7*>J=F[v$\"3?VTu'[jGm*F_u7$$\"3Gfb7wY,(*=F[v$\"3Sm0JN*4EZ*F_u7$$ \"3cM5'zg%pg>F[v$\"3ADf%RFx%\\#*F_u7$$\"3**oJ8:.SJ?F[v$\"3?af@Y>%y&*)F _u7$$\"3)**p!zPD$\\4#F[v$\"3?%>'G5ccd')F_u7$$\"3k-G%)H)F_u7$$\"3wak3@YBCAF[v$\"3cJr!4)G)*RzF_u7$$\"39/bF_u7$$\"3H.6)e58)4IF[v$\"3M.OA\"o%)RJ\"F_u 7$$\"3Kt2RXCLtIF[v$\"3ls_wI6s?oFdv7$$\"3!)***\\/l#fTJF[v$\"3pawpOMzRJ! #E-F^t6&F`t$\"#5!\"\"F+F+-%*THICKNESSG6#\"\"#-%+AXESLABELSG6%%\"xG%\"y G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F+$\"+aEfTJFAFdfl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "simpson" }{TEXT -1 10 " from the " } {TEXT 0 7 "student" }{TEXT -1 62 " package can be used to apply Simpso n's rule with a specified " }{TEXT 262 4 "even" }{TEXT -1 83 " number \+ of intervals to obtain a numerical approximation for this definite int egral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 4 " Note" }{TEXT -1 52 ": The default number of intervals for the procedur e " }{TEXT 0 16 "student[simpson]" }{TEXT -1 43 " is 4 so the third ar gument can be omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "student[simpson](sin(x),x=Pi/4..3*P i/4,4);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#PiG\"\"\", (*$-%%sqrtG6#\"\"#F&F&*&\"\"%F&-%$SumG6$-%$sinG6#,&F%#F&F.*(#F&\"\")F& ,&%\"iGF,F&!\"\"F&F%F&F&/F;;F&F,F&F&*&F,F&-F06$-F36#,&F%F6*(F6F&F;F&F% F&F&/F;;F&F&F&F&F&#F&\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&simp4G $\"+%*QS99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "simp" }{TEXT -1 19 " can also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "simp4 := simp(sin(x),x=Pi/4..3*Pi/4,intervals=4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&simp4G$\"+%*QS99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "We can obtain a \+ numerical value for this integral which is accurate to about 10 digits by first evaluating the integral analytically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Int(sin(x),x =Pi/4..3*Pi/4);\nvalue(%);\narea := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$sinG6#%\"xG/F);,$%#PiG#\"\"\"\"\"%,$F-#\"\" $F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The absolute error \+ in the value for the integral obtained by using Simpson's rule with 4 \+ intervals is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "abserr := abs(simp4-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"'K.>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 39 " . . . and the relative error is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "relerr := ab serr/abs(area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"+z/&eM \"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We can obtain more accurate estimates for " }{XPPEDIT 18 0 "Int(sin(x ),x=Pi/4..3*Pi/4)" "6#-%$IntG6$-%$sinG6#%\"xG/F);*&%#PiG\"\"\"\"\"%!\" \"*(\"\"$F.F-F.F/F0" }{TEXT -1 59 " by using Simpson's rule with a lar ger number of intervals." }}{PARA 0 "" 0 "" {TEXT -1 232 "As the numbe r of intervals is progressively increased, the approximation will impr ove in accuracy until, eventually, further increases lead to no change . One way to do this is to keep doubling the number of intervals which are used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "simp(sin(x),x=Pi/4..3*Pi/4,intervals=16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$H9UT\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "simp(sin(x),x=Pi/4 ..3*Pi/4,intervals=32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3O@99! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "simp(sin(x),x=Pi/4..3*Pi/4,intervals=64);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "simp(sin(x),x=Pi/4..3 *Pi/4,intervals=128);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+jN@99!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "simp(sin(x),x=Pi/4..3*Pi/4,intervals=256);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "simp(sin(x),x=Pi/4..3 *Pi/4,intervals=512);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "simp" }{TEXT -1 136 " from above can be used to automate the proces s of successively doubling the number of intervals. This is achieved b y using the option \"" }{TEXT 271 12 "iterate=true" }{TEXT -1 3 "\". \+ " }}{PARA 0 "" 0 "" {TEXT -1 153 "After each approximate value for the integral is computed, the intervals used are all bisected, and the fu nction is evaluated at all of these new points." }}{PARA 0 "" 0 "" {TEXT -1 267 "Simpson's rule is then applied on the new collection of \+ intervals, by making use of all the function values previously compute d, together with the new values. The idea is to avoid re-calculating t he value of the function at points where it has already been evaluated ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "simp(sin(x),x=Pi/4.. 3*Pi/4,iterate=true,info=true); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~intervals~- -->~~~G$\"07VgzPuT\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproxima tion~with~4~intervals~--->~~~G$\"0*)oU*QS99!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~intervals~--->~~~G$\"0f:SHDUT\"! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~16~interval s~--->~~~G$\"0hvIH9UT\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happrox imation~with~32~intervals~--->~~~G$\"0p-!3O@99!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~64~intervals~--->~~~G$\"0LC_c8UT\" !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~128~interv als~--->~~~G$\"0G^Dc8UT\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iappr oximation~with~256~intervals~--->~~~G$\"0B%QiN@99!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 97 "This value checks with the value obtained from the analytical form for the value of the integral." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }{TEXT 286 35 ".. us ing Simpson's rule iteratively" }{TEXT -1 3 " I " }}{PARA 0 "" 0 "" {TEXT -1 44 "We first find a numerical approximation for " }{XPPEDIT 18 0 "Int(1/ln(x^2+1),x = 1 .. 2);" "6#-%$IntG6$*&\"\"\"F'-%#lnG6#,&*$ %\"xG\"\"#F'F'F'!\"\"/F-;F'F." }{TEXT -1 39 " using Simpson's rule wit h 8 intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 568 "f := x -> 1/ln(x^2+1); # function\na := 1: # lo wer limit of integral\nb := 2: # upper limit of integral\nclr := COLOR (RGB,.7,.9,.7): # color for shading \npp := plot([0,f(x)],x=a..b,adapt ive=false,numpoints=20):\nu := op(1,op(1,pp)): v := op(1,op(2,pp)):\np 1 := plots[polygonplot]([seq([u[i],v[i],v[i+1],u[i+1]],i=1..19)],\n \+ color=clr,style=patchnogrid):\np2 := plot ([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\np3 := plot(f(x),x= 0.5..2.5,y=0..2,thickness=2): # adjust plot range\nplots[display]([p1, p2,p3],labels=[`x`,`y`],tickmarks=[4,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F--%#l nG6#,&*$)9$\"\"#F-F-F-F-!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 508 264 264 {PLOTDATA 2 "6)-%)POLYGONSG677&7$$\"\"\"\"\"!$F*F*7$F($\"+T]pU 9!\"*7$$\"+&\\m]0\"F/$\"+4VcO8F/7$F1F+7&F5F07$$\"+d&zH5\"F/$\"+stMc7F/ 7$F8F+7&F7$$\"+JK567F/$\"+ q+`26F/7$FFF+7&FJFE7$$\"+ee3l7F/$\"+$ptj/\"F/7$FMF+7&FQFL7$$\"+lY8:8F/ $\"+snpe**!#57$FTF+7&FYFS7$$\"+0t&pO\"F/$\"+gmL*[*FX7$FfnF+7&FjnFen7$$ \"+`Bb?9F/$\"+>8=`!*FX7$F]oF+7&FaoF\\o7$$\"+Ab(RZ\"F/$\"+PLOh')FX7$Fdo F+7&FhoFco7$$\"+=j\"F/$\"+&*Qv-xFX7$FipF+7&F]qFhp7$$\" +o>`'o\"F/$\"+%oFgU(FX7$F`qF+7&FdqF_q7$$\"+B!e#RF/$\"+K'*=yjFX7$FcsF+7&FgsFbs7$$\"\"#F*$\"+Y$\\L@'FX7$FjsF+-%&COL ORG6&%$RGBG$\"\"(!\"\"$\"\"*FetFct-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6 $7$F'7$F($\"3!******4/&pU9!#<-%'COLOURG6&FbtF*F*F*-F]u6$7$F^t7$Fjs$\"3 3+++Z$\\L@'!#=Fdu-F]u6%7W7$$\"3++++++++]F]v$\"3=\\Xs<,U\"[%Fcu7$$\"35m mmT:(z@&F]v$\"3='3%3tku_TFcu7$$\"3JLLL$3VfV&F]v$\"33GEO)H/E'QFcu7$$\"3 t***\\i&*)fDcF]v$\"34PnrJm$pj$Fcu7$$\"3Dnm;H[D:eF]v$\"3-?)*yy\\$GV$Fcu 7$$\"3&3+]PpU&GgF]v$\"3<$3Gc?WdA$Fcu7$$\"3LLLLe0$=C'F]v$\"3thd%*GmLRIF cu7$$\"39MLLLA`ckF]v$\"3m:0+y:')pGFcu7$$\"3'QLL$3RBrmF]v$\"3`Cnh%[5jr# Fcu7$$\"3=mm;zjf)4(F]v$\"31na9wng]CFcu7$$\"3=LL$e4;[\\(F]v$\"3Vr4o@7@V AFcu7$$\"3C++]i'y]!zF]v$\"3cB*=.<`*f?Fcu7$$\"3,LL$ezs$H$)F]v$\"3!)ft+k yl(*=Fcu7$$\"3_****\\7iI_()F]v$\"3#*35YDLHe +5Fcu$\"3n(zSRb*HU9Fcu7$$\"3*******\\Z/N/\"Fcu$\"39'fNM?=vN\"Fcu7$$\"3 5+++NfC&3\"Fcu$\"3ULNn&zd[G\"Fcu7$$\"3LLLez6:B6Fcu$\"30#*\\q]&oaA\"Fcu 7$$\"3_mmm\"=C#o6Fcu$\"3'[#o'Go$)=;\"Fcu7$$\"3gmmmEpS17Fcu$\"3Nqs.;>7K!)zF ]v7$$\"3BLL$e\"*[Hi\"Fcu$\"3)Q)HPVC**\\xF]v7$$\"3#*******pvxl;Fcu$\"3j ^B1\\(G\"GvF]v7$$\"3z****\\_qn2%=AcsSDtF]v7$$\"3%)***\\i&p@[ ]pF]v7$$\"3_mm mwanL=Fcu$\"3w]\"*fl$*z)y'F]v7$$\"3'******\\2go(=Fcu$\"3lk%)eQ$>li'F]v 7$$\"3CLLeR<*f\">Fcu$\"3g!RBwg*p(['F]v7$$\"3'******\\)Hxe>Fcu$\"33hxg8 [;WjF]v7$$\"3Ymm\"H!o-**>Fcu$\"3=&Rom\"pN;iF]v7$$\"3))***\\7k.6/#Fcu$ \"35bLmQKp*3'F]v7$$\"3emmmT9C#3#Fcu$\"39Lseus>sfF]v7$$\"3\"****\\i!*3` 7#Fcu$\"3)*fyQyxSbeF]v7$$\"3;LLL$*zym@Fcu$\"3'3wbs)G][dF]v7$$\"30LL$3N 1#4AFcu$\"3Xz\"\\$f;VWcF]v7$$\"3kmm\"HYt7D#Fcu$\"3GdDM*f/ha&F]v7$$\"3% *******p(G**G#Fcu$\"3i(RGiv_(faF]v7$$\"3Umm;9@BMBFcu$\"3WR " 0 "" {MPLTEXT 1 0 49 "student[simpson](1/ln(x^2+1),x=1..2,8);\nevalf(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"#CF&*&F&F&-%#lnG6#\"\"#! \"\"F&F&*&F%F&*&F&F&-F*6#\"\"&F-F&F&*&#F&\"\"'F&-%$SumG6$*&F&F&-F*6#,& *$),&#\"\"(\"\")F&*&\"\"%F-%\"iGF&F&F,F&F&F&F&F-/FE;F&FDF&F&*&#F&\"#7F &-F76$*&F&F&-F*6#,&*$),&F&F&*&FDF-FEF&F&F,F&F&F&F&F-/FE;F&\"\"$F&F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2!*\\]!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Alternatively, the proced ure " }{TEXT 0 4 "simp" }{TEXT -1 14 " can be used. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "simp8 := s imp(1/ln(x^2+1),x=1..2,intervals=8);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&simp8G$\"+2!*\\]!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A value which is accurate to 10 digits can be o btained using Maple's numerical integration via " }{TEXT 0 5 "evalf" } {TEXT -1 5 " and " }{TEXT 0 3 "Int" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(1/ln(x ^2+1),x=1..2);\narea := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$*&\"\"\"F'-%#lnG6#,&*$)%\"xG\"\"#F'F'F'F'!\"\"/F.;F'F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+I:?]!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 14 " absolute error" }{TEXT -1 90 " in the value for the integral obtained \+ by using Simpson's rule with 8 intervals is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "abserr := ab s(simp8-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"'xuH!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 15 " . . . and the " }{TEXT 262 14 "rela tive error" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "relerr := abserr/abs(area) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"+ya'pG$!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We can ob tain more accurate estimates for " }{XPPEDIT 18 0 "Int(1/ln(x^2+1),x = 1 .. 2);" "6#-%$IntG6$*&\"\"\"F'-%#lnG6#,&*$%\"xG\"\"#F'F'F'!\"\"/F-; F'F." }{TEXT -1 60 " by using Simpson's rule with a larger number of i ntervals. " }}{PARA 0 "" 0 "" {TEXT -1 65 "It is convenient to double \+ the number of intervals successively. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simp(1/ln(x^2+1),x=1..2, intervals=16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\")3A]!*!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simp(1/ln(x^2+1),x=1..2,intervals=32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_F?]!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simp(1/ln(x^2+1),x=1..2,inte rvals=64);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1;?]!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "si mp(1/ln(x^2+1),x=1..2,intervals=128);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M:?]!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simp(1/ln(x^2+1),x=1..2,intervals=256);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I:?]!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "By using the option \"" } {TEXT 271 12 "iterate=true" }{TEXT -1 12 "\" procedure " }{TEXT 0 4 "s imp" }{TEXT -1 113 " can automate the process of successively doubling the number of intervals for the application of Simpson's rule." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "simp(1/ln(x^2+1),x=1..2,iterate=true,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~intervals~--->~~~G$\"0!QrfX@' 4*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~interv als~--->~~~G$\"0<_SBdV0*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gappro ximation~with~8~intervals~--->~~~G$\"0`.m+*\\]!*!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Happroximation~with~16~intervals~--->~~~G$\"0+54)3A] !*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~inter vals~--->~~~G$\"0g\")=v--0*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hap proximation~with~64~intervals~--->~~~G$\"0.(>1;?]!*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~128~intervals~--->~~~G$\"0I!R M:?]!*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~ intervals~--->~~~G$\"0B**)H:?]!*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%Iapproximation~with~512~intervals~--->~~~G$\"0]='H:?]!*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I:?]!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "This value checks with the value o btained from the analytical form for the value of the integral." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }{TEXT 279 116 ".. empirical demonstration that the absolute error is proportiona l to the fourth power of the width of the intervals" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 37 " First we find an approximation for " } {XPPEDIT 18 0 "Int(exp(-x^2),x = 0 .. 1);" "6#-%$IntG6$-%$expG6#,$*$% \"xG\"\"#!\"\"/F+;\"\"!\"\"\"" }{TEXT -1 40 " using Simpson's rule wit h 32 intervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 526 "f := x - > exp(-x^2); # function\na := 0: # lower limit of integral\nb := 1: # \+ upper limit of integral\nclr := wheat: # color for shading\npp := plot ([0,f(x)],x=a..b,adaptive=false,numpoints=20):\nu := op(1,op(1,pp)): v := op(1,op(2,pp)):\np1 := plots[polygonplot]([seq([u[i],v[i],v[i+1],u [i+1]],i=1..19)],\n color=clr,style=pat chnogrid):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black ):\np3 := plot(f(x),x=0..1.2,thickness=2): # adjust plot range\nplots[ display]([p1,p2,p3],labels=[`x`,`y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\" \"\"!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 380 299 299 {PLOTDATA 2 "6(-%)POLYGONSG677&7$$\"\"!F)F(7$F($\"\"\"F)7$$\"+u%\\m]&!#6$\"+SFsp** !#57$F.F(7&F4F-7$$\"+ubzH5F3$\"+LC^%*)*F37$F7F(7&F;F67$$\"+u!G'o:F3$\" +\"=Vpv*F37$F>F(7&FBF=7$$\"+0B.6@F3$\"+(QQTc*F37$FEF(7&FIFD7$$\"+%ee3l #F3$\"++jT@$*F37$FLF(7&FPFK7$$\"+ZmM^JF3$\"+45ia!*F37$FSF(7&FWFR7$$\"+ ZIdpOF3$\"+x1:S()F37$FZF(7&FhnFY7$$\"+KN_0UF3$\"+U\\$*y$)F37$F[oF(7&F_ oFjn7$$\"+;_vRZF3$\"+d^$z)zF37$FboF(7&FfoFao7$$\"+o\"y#*G&F3$\"+rkhfvF 37$FioF(7&F]pFho7$$\"+%G)HtdF3$\"+FA[lrF37$F`pF(7&FdpF_p7$$\"+&H!>=jF3 $\"+\"z'f3nF37$FgpF(7&F[qFfp7$$\"+%o>`'oF3$\"+[(R!\\(Fat-%&ST YLEG6#%,PATCHNOGRIDG-%'CURVESG6$7$F'F*-F\\t6&F^tF)F)F)-Fit6$7$Fjs7$F+$ \"3MBWr6WzyO!#=F\\u-Fit6%7SF*7$$\"3h*******\\ech#!#>$\"3AA(*\\q1;$***F du7$$\"3-+++v*G:*[F[v$\"3Q#epma,h(**Fdu7$$\"3u******\\L)4X(F[v$\"3OE#R +nOY%**Fdu7$$\"3)******\\MSF+\"Fdu$\"3q'Q-L*\\&***)*Fdu7$$\"3#)****\\F y:f7Fdu$\"3+52D6CqU)*Fdu7$$\"3')****\\d'*)o\\\"Fdu$\"3/&3\")3\"QUy(*Fd u7$$\"3w****\\(>ZIu\"Fdu$\"3Vf)Q(Gwu+(*Fdu7$$\"3u****\\xOi(*>Fdu$\"3eg Fb]r!)3'*Fdu7$$\"3#)****\\FPQ^AFdu$\"3;La/;)ed]*Fdu7$$\"3/+++IrS7DFdu$ \"3)*H6Mt-H)Q*Fdu7$$\"3p*****\\o;Bu#Fdu$\"3)4\"4(3m^bF*Fdu7$$\"3****** ***QS6+$Fdu$\"3(fcnCT'oQ\"*Fdu7$$\"3[******\\o-hKFdu$\"3u(z;CLi6**)Fdu 7$$\"3(*******4cZ6NFdu$\"3]Y^$Q[V*R))Fdu7$$\"3S****\\xq!*QPFdu$\"3go=' R9r`p)Fdu7$$\"3&********3X$4SFdu$\"3j6r'e9i]^)Fdu7$$\"3s******f:WQUFdu $\"3!31kNIxcN)Fdu7$$\"3f****\\<_$\\]%Fdu$\"3yf6KqhBj\")Fdu7$$\"3**)*** **fs#3u%Fdu$\"3W>V%fLBr)zFdu7$$\"3!)****\\<#Q'**\\Fdu$\"3;:86#e*G)y(Fd u7$$\"33++]_u3Y_Fdu$\"3orqHtL4%f(Fdu7$$\"3[*****\\PJK]&Fdu$\"3Mo=H0z0( Q(Fdu7$$\"3%*****\\n(p$RdFdu$\"3?kC-\"*e_$>(Fdu7$$\"3A*****\\#p2%*fFdu $\"3%pV$4T9s\")pFdu7$$\"3o****\\xgkeiFdu$\"3;utMtX-fnFdu7$$\"3g****\\- V&*)['Fdu$\"3WL;cSDZjlFdu7$$\"3.+++&\\$pPnFdu$\"3Q;\\\\\\([5N'Fdu7$$\" 37******>am%*pFdu$\"3P4KT_#R38'Fdu7$$\"3?*****\\JigC(Fdu$\"3e&p\">%3L_ \"fFdu7$$\"3G****\\Pr_Fdu7$$\"35***** *\\/;h#)Fdu$\"3O'z*o[Mo`]Fdu7$$\"3M)***\\P/&f\\)Fdu$\"3+WR)f'=re[Fdu7$ $\"3e******4zj_()Fdu$\"3_OVpa_G[YFdu7$$\"3u)***\\<3;%**)Fdu$\"3'oC*G&H dKX%Fdu7$$\"3]****\\Z=iY#*Fdu$\"35Y!yxCJGD%Fdu7$$\"39)*****\\'[M\\*Fdu $\"3mTO0EdR2\"F_bl$\"3)4K1== qc:$Fdu7$$\"3&*****\\o#R05\"F_bl$\"3-w^6TiVyHFdu7$$\"3()*****>`9V7\"F_ bl$\"3<@(y8\"=)\\#GFdu7$$\"3)****\\<#Rm\\6F_bl$\"3%eu)y>KumEFdu7$$\"3% ****\\A_ER<\"F_bl$\"3,a(4jHh0_#Fdu7$$\"3%**************>\"F_bl$\"3#z@@ oex#pBFdu-F\\t6&F^t$\"#5!\"\"F(F(-%*THICKNESSG6#\"\"#-%+AXESLABELSG6%% \"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#7F^elF[fl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "simpson" }{TEXT -1 8 " in the " }{TEXT 0 7 "student" }{TEXT -1 22 " package can be used. " }}{PARA 0 "" 0 "" {TEXT -1 26 "We obtain an estimate for " }{XPPEDIT 18 0 "Int (exp(-x^2),x = 0 .. 1)" "6#-%$IntG6$-%$expG6#,$*$%\"xG\"\"#!\"\"/F+;\" \"!\"\"\"" }{TEXT -1 52 " with the Simson's rule estimate given to 15 \+ digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "student[simpson](exp(-x^2),x=0..1,32);\nevalf(evalf(% ,18),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*#\"\"\"\"#'*F%*&F$F%-%$ expG6#!\"\"F%F%*&#F%\"#CF%-%$SumG6$-F)6#,$*$),&*&\"#;F+%\"iGF%F%#F%\"# KF+\"\"#F%F+/F:;F%F9F%F%*&#F%\"#[F%-F06$-F)6#,$*&\"$c#F+F:F=F+/F:;F%\" #:F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0&)pgST#ou!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The same result can be obtained with the procedure " }{TEXT 0 4 "simp" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "simp32 := evalf(simp(exp(-x^2),x=0..1,intervals=32),1 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'simp32G$\"0&)pgST#ou!#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "A better \+ estimate is obtained by doubling the number of sub-intervals used to 6 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "simp64 := evalf(simp(exp(-x^2),x=0..1,intervals=64),1 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'simp64G$\"0t'*HLT#ou!#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "We can o btain a numerical value for this integral which is accurate to about 1 0 digits by first evaluating the integral analytically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(exp (-x^2),x=0..1);\nvalue(%);\narea := evalf(evalf(%,20),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\" /F,;\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&-%$ erfG6#F&F&%#PiGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"0F C\"G8Cou!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The absolute errors in the values for the integral obtained by \+ using " }{TEXT 291 9 "Simpson's" }{TEXT -1 54 " rule with 32 and 64 in tervals respectively are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "abserr32 := evalf(abs(simp32 -area),15);\nabserr64 := evalf(abs(simp64-area),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr32G$\"(eXz(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr64G$\"'Ys[!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "The ratio of these absolute errors is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(abserr32/abserr64,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'s*f\"!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "When the number of intervals is doubled from 32 to 64, the absolu te error decreases by a factor of 16 approximately. " }}{PARA 0 "" 0 " " {TEXT -1 134 "It can be shown theoretically that in many situations \+ the absolute error in using Simpson's rule is approximately proportion al to the " }{TEXT 262 12 "fourth power" }{TEXT -1 82 " of the width o f the intervals used. This last result seems to support this idea. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Similar ly, when the number of intervals is tripled from 32 to 96, the absolut e error decreases by a factor of " }{XPPEDIT 18 0 "3^4=81" "6#/*$\"\"$ \"\"%\"#\")" }{TEXT -1 16 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "Digits := 15:\nsi mp96 := simp(exp(-x^2),x=0..1,intervals=96);\nabserr32 := abs(simp32-a rea);\nabserr96 := abs(simp96-area);\nevalf(abserr32/abserr96,6);\nDig its := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'simp96G$\"0w'3H8Cou!# :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr32G$\"(eXz(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr96G$\"&\\i*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'L)4)!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 4 " }{TEXT 282 35 ".. using Simpson's rule iter atively" }{TEXT -1 4 " II " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We start by finding an approximation for " }{XPPEDIT 18 0 "Int(exp(-cos( x)),x = 0 .. 2);" "6#-%$IntG6$-%$expG6#,$-%$cosG6#%\"xG!\"\"/F-;\"\"! \"\"#" }{TEXT -1 40 " using Simpson's rule with 16 intervals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 529 "f := x -> exp(-cos(x)); # function\na := 0: # lower limit of inte gral\nb := 2: # upper limit of integral\nclr := wheat: # color for sha ding\npp := plot([0,f(x)],x=a..b,adaptive=false,numpoints=20):\nu := o p(1,op(1,pp)): v := op(1,op(2,pp)):\np1 := plots[polygonplot]([seq([u[ i],v[i],v[i+1],u[i+1]],i=1..19)],\n col or=clr,style=patchnogrid):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b )]]],color=black):\np3 := plot(f(x),x=0..2.2,thickness=2): # adjust pl ot range\nplots[display]([p1,p2,p3],labels=[`x`,`y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6# ,$-%$cosG6#9$!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6(-%)POLYGONSG677&7$$\"\"!F)F(7$F($\"+7WzyO!#57$$\"+&*)H8 5\"F-$\"+y+:,PF-7$F/F(7&F3F.7$$\"+Z6ff?F-$\"+)4stv$F-7$F6F(7&F:F57$$\" +ZhDPJF-$\"+%34G'QF-7$F=F(7&FAF<7$$\"+6Y1AUF-$\"+6'[k,%F-7$FDF(7&FHFC7 $$\"+orr,`F-$\"+-a7?UF-7$FKF(7&FOFJ7$$\"+&H$p-jF-$\"+l:2eWF-7$FRF(7&FV FQ7$$\"+&4Y\"RtF-$\"+\"R\\*eZF-7$FYF(7&FgnFX7$$\"+jq/6%)F-$\"+&44V8&F- 7$FjnF(7&F^oFin7$$\"+K/^z%*F-$\"+3nE!e&F-7$FaoF(7&FeoF`o7$$\"+Mc&y0\"! \"*$\"+^Bs@hF-7$FhoF(7&F]pFgo7$$\"+d'fY:\"Fjo$\"+wn!\\n'F-7$F`pF(7&Fdp F_p7$$\"+f!QOE\"Fjo$\"+c]y!R(F-7$FgpF(7&F[qFfp7$$\"+PR1t8Fjo$\"+&f`k@) F-7$F^qF(7&FbqF]q7$$\"+Yg^y9Fjo$\"+I#*o>\"*F-7$FeqF(7&FiqFdq7$$\"+kmFu :Fjo$\"+%R'[.5Fjo7$F\\rF(7&F`rF[r7$$\"+,`9)o\"Fjo$\"+q\"4U7\"Fjo7$FcrF (7&FgrFbr7$$\"+tpg%y\"Fjo$\"+tpPO7Fjo7$FjrF(7&F^sFir7$$\"+G[\"o*=Fjo$ \"+z[]x8Fjo7$FasF(7&FesF`s7$$\"\"#F)$\"+t%3h^\"Fjo7$FhsF(-%'COLOURG6&% $RGBG$\")#)eq%)!\")Fat$\")h>!\\(Fct-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG 6$7$F'7$F($\"3MBWr6WzyO!#=-F^t6&F`tF)F)F)-F[u6$7$F\\t7$Fhs$\"3)=?gEZ3h ^\"!#$\"3KD\\:Re-$o$Fau7$$\"3?ML$3 7.y'*)Fav$\"3)z6*z[pg$p$Fau7$$\"3ymm;9O,m8Fau$\"3w#pK`@CKr$Fau7$$\"3!p mm\"*Hd$Q=Fau$\"3_r'HHB38u$Fau7$$\"3[LL3QFau7$$\"3;++vGle&>$Fau$\"3)pyMsPJ)pQFau7 $$\"35nmTv+JiOFau$\"3_AL+x)H5$RFau7$$\"39++vLo`FTFau$\"305QvhU3,SFau7$ $\"31MLLQ(zgg%Fau$\"3e9:a'>!)G3%Fau7$$\"3&pmm\"*e!eF]Fau$\"3iY\"G_x$Qj TFau7$$\"3\"4++]r!4-bFau$\"3A'RTeuQQE%Fau7$$\"3_+++D#\\&yfFau$\"3qk?#[ y%evVFau7$$\"3]+++&G0xV'Fau$\"3]vN#)yh,%\\%Fau7$$\"3emmTvHmaoFau$\"3xj 6Fyb26YFau7$$\"3EMLL)*fY]tFau$\"3gQ(zY>hDw%Fau7$$\"3QMLL$>w/x(Fau$\"3S A)oh_z(Fau7$$\"35+++F`N#G\"Fju$\"32N/BZ(> U_(Fau7$$\"3/++vdZWG8Fju$\"3ME_?uwQmyFau7$$\"3=+](=lQIP\"Fju$\"3z+c*Q* )\\i@)Fau7$$\"3)****\\#oDbA9Fju$\"3SL3E2p)oi)Fau7$$\"3ALLLCI/n9Fju$\"3 &pzOT\"Q:;!*Fau7$$\"36++]#3YX^\"Fju$\"3UMXbmeI`%*Fau7$$\"3ym;a84fd:Fju $\"3+n$>Wu<)o)*Fau7$$\"3F++]$G]Yg\"Fju$\"3bvntMoUM5Fju7$$\"3AL$3K[H*[; Fju$\"3Sh80)p!=\"3\"Fju7$$\"3?+]P0S@&p\"Fju$\"3]U?^2c7K6Fju7$$\"3LLL$e el/u\"Fju$\"3wPh*=2]R=\"Fju7$$\"3%***\\(ozRyy\"Fju$\"3pu<*Hu(GS7Fju7$$ \"3,nmm#zmM$=Fju$\"3+ut(3l*['H\"Fju7$$\"3%pm;f)p7!)=Fju$\"3Q^D&R#*ReN \"Fju7$$\"3cL$3#43SE>Fju$\"3t&e$pYZX;9Fju7$$\"3;+++Z;#*o>Fju$\"3K([vsK uNZ\"Fju7$$\"3gLLeD`l_@Fju$ \"3jmw%zng=t\"Fju7$$\"3;+++++++AFju$\"3j#=VPZ'G,=Fju-F^t6&F`t$\"#5!\" \"F(F(-%*THICKNESSG6#Fis-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG -%%VIEWG6$;F($\"#AFcelF_fl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 7 "simpson" }{TEXT -1 8 " in the " }{TEXT 0 7 "student" } {TEXT -1 22 " package can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "student[simpson](exp(-cos(x )),x=0..2,16);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\" \"\"#CF&-%$expG6#!\"\"F&F&*&F%F&-F)6#,$-%$cosG6#\"\"#F+F&F&*&#F&\"\"'F &-%$SumG6$-F)6#,$-F16#,&*&\"\"%F+%\"iGF&F&#F&\"\")F+F+/FB;F&FDF&F&*&#F &\"#7F&-F86$-F)6#,$-F16#,$*&FAF+FBF&F&F+/FB;F&\"\"(F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+Wv829!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Alternatively, the procedure " }{TEXT 0 4 "simp" }{TEXT -1 14 " can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "simp16 := simp(exp(-cos (x)),x=0..2,intervals=16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'simp1 6G$\"+Wv829!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A value which is accurate to 10 digits can be obtained us ing Maple's numerical integration via " }{TEXT 0 5 "evalf" }{TEXT -1 5 " and " }{TEXT 0 3 "Int" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(exp(-cos(x)),x =0..2);\nevalf(%,15):\narea := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$-%$cosG6#%\"xG!\"\"/F-;\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+)>SrS\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The absolute error in th e approximate value for the integral obtained using 16 intervals is ab out 2.65e-6" }{XPPEDIT 18 0 "`` = 2.65*`.`*10^(-6);" "6#/%!G*(-%&Float G6$\"$l#!\"#\"\"\"%\".GF+)\"#5,$\"\"'!\"\"F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "abserr16 := abs(simp16-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %)abserr16G$\"%aE!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "If we assume that the absolute error is proportional to the fourth power of the width of the intervals used, we can estimate how many intervals w ill be needed to obtain a value which is correct to 10 digits. " }} {PARA 0 "" 0 "" {TEXT -1 43 "If we aim for a tolerance no greater than " }{XPPEDIT 18 0 "epsilon = 2*`.`*10^(-10);" "6#/%(epsilonG*(\"\"#\" \"\"%\".GF')\"#5,$F*!\"\"F'" }{TEXT -1 95 ", this is unlikely to affec t the last decimal place, since a unit in the last place (1 ulp) is " }{XPPEDIT 18 0 "10^(-9);" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eps := 2e-10;\nabserr16/eps;\nevalf(log[16](%),4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"++++F8!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%CM!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "log[16](1327 0);" "6#-&%$logG6#\"#;6#\"&qK\"" }{TEXT -1 2 " " }{TEXT 284 1 "~" } {TEXT -1 24 " 3.424 means that 13270 " }{TEXT 285 1 "~" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "16^3.424;" "6#)\"#;-%&FloatG6$\"%CM!\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 " The desired tolerance is reached when the absolute error is divided by 16 about 4 times. " }}{PARA 0 "" 0 "" {TEXT -1 70 "Thus we should dou ble the number of intervals 11 times to reach about " }{XPPEDIT 18 0 " 16*`.`*2^4 = 2^8;" "6#/*(\"#;\"\"\"%\".GF&\"\"#\"\"%*$F(\"\")" } {XPPEDIT 18 0 "`` = 256;" "6#/%!G\"$c#" }{TEXT -1 12 " intervals. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 201 "The foll owing calculations give progressively more accurate values for the int egral by using Simpson's rule with a progressively larger number of in tervals - doubling the number of intervals each time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "2^5;\nsim p(exp(-cos(x)),x=0..2,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K+929!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "2^6;\ns imp(exp(-cos(x)),x=0..2,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(=SrS\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "2^ 7;\nsimp(exp(-cos(x)),x=0..2,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(>SrS\"! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "2^8;\nsimp(exp(-cos(x)),x=0..2,intervals=%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)>SrS\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT 283 1 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "This process can be automated by using the procedure " } {TEXT 0 4 "simp" }{TEXT -1 18 " with the option \"" }{TEXT 271 12 "ite rate=true" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 73 "The end re sult is a value for the integral which is correct to 10 digits." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 243 "Simpson 's rule is used with a progressively larger number of intervals until \+ the difference between successively computed estimates is small enough to be able to conclude that the integral has been determined to the r equired number of digits. " }}{PARA 0 "" 0 "" {TEXT -1 125 "Since the \+ number of intervals is doubled with each iteration, it is only necessa ry to calculate new values for the integrand " }{XPPEDIT 18 0 "exp(-co s(x));" "6#-%$expG6#,$-%$cosG6#%\"xG!\"\"" }{TEXT -1 258 " at the mid \+ points of the intervals of the previous step. Since the new \"evens\" \+ can be obtained as the sum of the previous \"odds\" and \"evens\", it \+ is also possible to economise on the number of additions needed to com pute each new Simpson rule approximation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "simp(exp(-cos(x)), x=0..2,iterate=true,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Ga pproximation~with~2~intervals~--->~~~G$\"0'\\l&yeZS\"!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~~~G$\"0'p' 4s~~~G$\"0$4n^!)429!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Happroximation~with~16~intervals~--->~~~G$\"0D9TaPrS\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~intervals~--->~~~G$\"0i !oJ+929!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~64~ intervals~--->~~~G$\"0>pt=SrS\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %Iapproximation~with~128~intervals~--->~~~G$\"0!G5(>SrS\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~intervals~--->~~~G$ \"0>6x>SrS\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~wit h~512~intervals~--->~~~G$\"0A\\x>SrS\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)>SrS\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 5" }{TEXT 287 45 " .. a tough example for numeri cal integration" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "We fir st find a numerical approximation for " }{XPPEDIT 18 0 "Int(ln(sqrt(x) +1),x = 0 .. 3);" "6#-%$IntG6$-%#lnG6#,&-%%sqrtG6#%\"xG\"\"\"F.F./F-; \"\"!\"\"$" }{TEXT -1 41 " using Simpson's rule with 32 intervals. 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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Th e procedure " }{TEXT 0 7 "simpson" }{TEXT -1 8 " in the " }{TEXT 0 7 " student" }{TEXT -1 45 " package can be used to obtain the estimate. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "student[simpson](ln(sqrt(x)+1),x=0..3,32);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"#KF&-%#lnG6#,&F&F&*$\"\"$#F&\" \"#F&F&F&*&#F&\"\")F&-%$SumG6$-F)6#,&*$,&*(F-F&\"#;!\"\"%\"iGF&F&#F-F' F=F.F&F&F&/F>;F&FF.F&F&F&/F> ;F&\"#:F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7u$)RA!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Alternati vely, the procedure " }{TEXT 0 4 "simp" }{TEXT -1 14 " can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "simp32 := simp(ln(sqrt(x)+1),x=0..3,intervals=32);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'simp32G$\"+7u$)RA!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "We can obtain a numerica l value for this integral which is accurate to about 10 digits by firs t evaluating the integral analytically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Int(ln(sqrt(x)+1),x=0.. 3);\nvalue(%);\narea := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$-%#lnG6#,&*$%\"xG#\"\"\"\"\"#F-F-F-/F+;\"\"!\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"#\"\"\"-%#lnG6#,&F&F&*$\"\"$#F&F%F&F&F& F+F&#F,F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+')e:UA! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 262 14 "absolute error" }{TEXT -1 91 " in the value for the integral obtained by using Simpson's rule with 32 intervals is . . . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "abserr := abs(simp32-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'abserrG$\"(u%=B!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 15 " . . . and th e " }{TEXT 262 14 "relative error" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "re lerr := abserr/abs(area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerr G$\"+Fy.M5!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can obtain more accurate estimates for " }{XPPEDIT 18 0 "Int(ln(sqrt(x)+1),x = 0 .. 3);" "6#-%$IntG6$-%#lnG6#,&-%%sqrtG6# %\"xG\"\"\"F.F./F-;\"\"!\"\"$" }{TEXT -1 73 " by using Simpson's rule \+ with a progressively larger number of intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "It is hard to obtain an \+ accurate value for this integral by using Simpson's rule. The problem \+ is that the gradient of the curve approaches infinity at the left-hand end of the interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "simp(ln(sqrt(x)+1),x=0..3,iterate=true,in fo=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~in tervals~--->~~~G$\"0Us$e2\"=5#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Gapproximation~with~4~intervals~--->~~~G$\"0$\\<0d=\">#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~intervals~--->~~~G$\" 0u7E9bQA#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~1 6~intervals~--->~~~G$\"0>*4)>HcB#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%Happroximation~with~32~intervals~--->~~~G$\"0&=f6u$)RA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~64~intervals~--->~~~G$ \"08\"\\ETLTA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~wi th~128~intervals~--->~~~G$\"0:5G(\\'=C#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~intervals~--->~~~G$\"0lCS'H0UA !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~512~interv als~--->~~~G$\"0I3[[>@C#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Jappro ximation~with~1024~intervals~--->~~~G$\"0)*eM,V@C#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~2048~intervals~--->~~~G$\"0xt QL^@C#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~4096 ~intervals~--->~~~G$\"0e+fFa@C#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %Japproximation~with~8192~intervals~--->~~~G$\"03?hJb@C#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~16384~intervals~--->~~~ G$\"08.Rob@C#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~wi th~32768~intervals~--->~~~G$\"0#e$R\"e:UA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~65536~intervals~--->~~~G$\"0i4*fe: UA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~131072~i ntervals~--->~~~G$\"0)Q;we:UA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%L approximation~with~262144~intervals~--->~~~G$\"0j5>)e:UA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~524288~intervals~--->~~ ~G$\"0RUR)e:UA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Mapproximation~w ith~1048576~intervals~--->~~~G$\"0tgY)e:UA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Mapproximation~with~2097152~intervals~--->~~~G$\"0z9\\ )e:UA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Mapproximation~with~41943 04~intervals~--->~~~G$\"0d/])e:UA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+&)e:UA!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Ex ample 6 " }{TEXT 292 124 ".. estimating the error by doubling the numb er of intervals used instead of calculating an accurate value by anoth er method " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT 302 4 "Note" }{TEXT -1 15 ": Suppose that " }{XPPEDIT 18 0 "I[1]" "6#&%\"IG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "I[2]" "6#&%\"IG6#\"\"#" }{TEXT -1 51 " are Simpson's rule approximations for an integral " }{XPPEDIT 18 0 "I=Int(f(x),x=a..b)" "6#/%\"IG-%$IntG6$- %\"fG6#%\"xG/F+;%\"aG%\"bG" }{TEXT -1 16 " obtained using " }{TEXT 303 1 "n" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2*n" "6#*&\"\"#\"\"\"%\" nGF%" }{TEXT -1 29 " intervals respectively. Let " }{XPPEDIT 18 0 "e[1 ]" "6#&%\"eG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e[2]" "6#&% \"eG6#\"\"#" }{TEXT -1 48 " be the corresponding absolute errors, that is, " }{XPPEDIT 18 0 "e[1]=abs(I[1]-I)" "6#/&%\"eG6#\"\"\"-%$absG6#,& &%\"IG6#F'F'F-!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e[2]=abs(I[2] -I)" "6#/&%\"eG6#\"\"#-%$absG6#,&&%\"IG6#F'\"\"\"F-!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 76 "An earlier example suggests that t he error is approximately proportional to " }{XPPEDIT 18 0 "h^4;" "6#* $%\"hG\"\"%" }{TEXT -1 18 ". This means that " }{XPPEDIT 18 0 "e[1]" " 6#&%\"eG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "16*e[2];" "6#*&\"#;\"\"\"&%\"eG6#\"\"#F%" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "I[1]" "6#&%\"IG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "I[2]" "6#&%\"IG6#\"\"#" }{TEXT -1 25 " are on the same \+ side of " }{TEXT 306 1 "I" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 " I[1]F-F6FA-F$6&7$F*7$F+F> F-F6FA-F$6%7$F'7$F($!\"\"F)-F.6&F0F)F)F)-%*LINESTYLEGF8-F$6%7$FH7$FI$! 3++++++++]F@FWFY-F$6%7$F*7$F+FUFWFY-F$6'7$7$$\"#NFV$!\"$FV7$F(Feo7%7$$ \"*+++0\"!\"*$!++++]O!#5Fgo7$Fjo$!++++]BF_p-FB6#%,PATCHNOGRIDG-F.6&F0$ \")#)eqkF3$\"))eqk\"F3Fjp-FZ6#\"\"$-F$6'7$7$$\"\"%F)Feo7$$\"#vFVFeo7%7 $$\"++++&R(F\\pFapFeq7$FjqF]pFcpFfpF\\q-F$6'7$7$$\"$v(!\"#Feo7$$\"$](F crFeo7%7$$\"++++vvF\\pF]pFdr7$FirFapFcpFfpF\\q-F$6'7$F`r7$$\"$+)FcrFeo 7%7$$\"++++DzF\\pFapF_s7$FdsF]pFcpFfpF\\q-F$6'7$7$$\"#QFV$F3FV7$F(F]t7 %7$$\"*+++9\"F\\p$!++++]')F_pF^t7$Fat$!++++]tF_pFcpFfpF\\q-F$6'7$7$$\" #UFVF]t7$$\"#!)FVF]t7%7$$\"++++')yF\\pFftF^u7$FcuFctFcpFfpF\\q-%%TEXTG 6%7$F($F,FVQ\"I6\"-%%FONTG6%%&TIMESG%&ROMANG\"#7-Fgu6%7$FfqFjuF[vF]v-F gu6%7$F+FjuF[vF]v-Fgu6%7$$\"$v$FcrFeoQ%15~eF\\v-F^v6$%*HELVETICAG\"#5- Fgu6%7$Far$!\"%FVQ\"eF\\vF_w-Fgu6%7$Fcq$!#wFcrFhwF_w-Fgu6%7$$\"#TFV$F \\pFVQ\"1F\\v-F^v6$FawF,-Fgu6%7$$F,Fcr$\"\"(FVFdxFex-Fgu6%7$$\"$e(FcrF [yQ\"2F\\vFex-Fgu6%7$$\"$&RFcrFfwFbyFex-Fgu6%7$$\"$&yFcr$!\"&FVFbyFex- %*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F\\vFfz-F^v6#%(DEFAULTG-%%VIEWG 6$;F(F+;FU$\"\"\"F)" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }{XPPEDIT 18 0 "abs(I[1]-I[2])" "6#-%$absG6#,&&%\"IG6#\"\"\"F*&F(6#\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 305 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "15*e[2 ];" "6#*&\"#:\"\"\"&%\"eG6#\"\"#F%" }{TEXT -1 21 ", which implies that " }{XPPEDIT 18 0 "e[2]" "6#&%\"eG6#\"\"#" }{TEXT -1 1 " " }{TEXT 304 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(I[1]-I[2])/15;" "6#*&-%$absG 6#,&&%\"IG6#\"\"\"F+&F)6#\"\"#!\"\"F+\"#:F/" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 27 "We find approximations for " }{XPPEDIT 18 0 "Int(exp(-x^2)/(x^2+1),x = 0 .. 1);" "6#-%$IntG6$*&-%$expG6#,$*$% \"xG\"\"#!\"\"\"\"\",&*$F,F-F/F/F/F./F,;\"\"!F/" }{TEXT -1 119 " using Simpson's rule with 16 and 32 intervals, and performing the calculati on with 13 digit floating point arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 541 "f := x -> exp(-x^ 2)/(x^2+1); # function\na := 0: # lower limit of integral\nb := 1: # u pper limit of integral\nclr := COLOR(RGB,.85,.8,1): # color for shadin g\npp := plot([0,f(x)],x=a..b,adaptive=false,numpoints=20):\nu := op(1 ,op(1,pp)): v := op(1,op(2,pp)):\np1 := plots[polygonplot]([seq([u[i], v[i],v[i+1],u[i+1]],i=1..19)],\n color=clr,sty le=patchnogrid):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color =black):\np3 := plot(f(x),x=0..1.2,thickness=2): # adjust plot range\n plots[display]([p1,p2,p3],labels=[`x`,`y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#,$* $)9$\"\"#\"\"\"!\"\"F5,&F1F5F5F5F6F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 387 295 295 {PLOTDATA 2 "6(-%)POLYGONSG677&7$$\"\"!F)F(7$F($ \"\"\"F)7$$\"+u%\\m]&!#6$\"+cFeR**!#57$F.F(7&F4F-7$$\"+ubzH5F3$\"+!H%o !z*F37$F7F(7&F;F67$$\"+u!G'o:F3$\"+<*HE_*F37$F>F(7&FBF=7$$\"+0B.6@F3$ \"+i15c\"*F37$FEF(7&FIFD7$$\"+%ee3l#F3$\"+\\@S4()F37$FLF(7&FPFK7$$\"+Z mM^JF3$\"+O8kO#)F37$FSF(7&FWFR7$$\"+ZIdpOF3$\"+rF37$F[oF(7&F_oFjn7$$\"+;_vRZF3$\"+9@hAlF37$FboF(7& FfoFao7$$\"+o\"y#*G&F3$\"+Xl.2fF37$FioF(7&F]pFho7$$\"+%G)HtdF3$\"+/o?u `F37$F`pF(7&FdpF_p7$$\"+&H!>=jF3$\"+s2h%z%F37$FgpF(7&F[qFfp7$$\"+%o>`' oF3$\"+gVDUUF37$F^qF(7&FbqF]q7$$\"+K-e#R(F3$\"+siuVPF37$FeqF(7&FiqFdq7 $$\"+@LQryF3$\"+M['GK$F37$F\\rF(7&F`rF[r7$$\"+0lsS%)F3$\"+$\"3#pu)za$Gj)**Ffu7$$\"3-+++v*G:*[F]v$\"3au)o,l)G_**Ffu7$$\"3u*** ***\\L)4X(F]v$\"3]e(*Q&pJ(*))*Ffu7$$\"3)******\\MSF+\"Ffu$\"3XisKUIS,) *Ffu7$$\"3#)****\\Fy:f7Ffu$\"35o5xsS3*o*Ffu7$$\"3')****\\d'*)o\\\"Ffu$ \"3%zVb*eD7k&*Ffu7$$\"3w****\\(>ZIu\"Ffu$\"3-?')ov'3ZT*Ffu7$$\"3u**** \\xOi(*>Ffu$\"3)H!G(*e;3S#*Ffu7$$\"3#)****\\FPQ^AFfu$\"3i*Ht1s\"=Z!*Ff u7$$\"3/+++IrS7DFfu$\"3vB!\\^2p3$))Ffu7$$\"3p*****\\o;Bu#Ffu$\"3,(zQi- \"zE')Ffu7$$\"3*********QS6+$Ffu$\"3wc8E0'*e$Q)Ffu7$$\"3[******\\o-hKF fu$\"34f'47W@p7)Ffu7$$\"3(*******4cZ6NFfu$\"39XQs;repyFfu7$$\"3S****\\ xq!*QPFfu$\"380l?``*)GwFfu7$$\"3&********3X$4SFfu$\"3'**fOjUPeL(Ffu7$$ \"3s******f:WQUFfu$\"3+z$y5A?K3(Ffu7$$\"3f****\\<_$\\]%Ffu$\"3;IeLRg/' y'Ffu7$$\"3**)*****fs#3u%Ffu$\"3$)G>k#*zS@lFfu7$$\"3!)****\\<#Q'**\\Ff u$\"3\"R1k]*>\"3B'Ffu7$$\"33++]_u3Y_Ffu$\"3Avam%*pFfu$\"3LmX6x)>n6%Ffu7$$\"3?*****\\JigC( Ffu$\"3]b^KPNqyQFfu7$$\"3G****\\P&[7'))o>j#Ffu7$$\"3u)***\\<3;% **)Ffu$\"3G[kKEAzhCFfu7$$\"3]****\\Z=iY#*Ffu$\"3cE8t27j#H#Ffu7$$\"39)* ****\\'[M\\*Ffu$\"3avSf!fVd8#Ffu7$$\"3y)***\\PM&=v*Ffu$\"3Y?8bV+L!)>Ff u7$$\"3*******fzs++\"!#<$\"3*4J&pPb**Q=Ffu7$$\"3(*****\\5Q_D5Fabl$\"3 \\9$R5F/Fq\"Ffu7$$\"3-++vxSw]5Fabl$\"3kN/=wn_v:Ffu7$$\"3'******>EdR2\" Fabl$\"3)ozb4*oWl9Ffu7$$\"3&*****\\o#R05\"Fabl$\"3;ka9Ha)pM\"Ffu7$$\"3 ()*****>`9V7\"Fabl$\"3_?bf]wtZ7Ffu7$$\"3)****\\<#Rm\\6Fabl$\"3=R]H-Kg[ 6Ffu7$$\"3%****\\A_ER<\"Fabl$\"3g,%)RgR!*f5Ffu7$$\"3%**************>\" Fabl$\"3W`#\\V/a,r*F]v-F^u6&F^t$\"#5FdtF(F(-%*THICKNESSG6#\"\"#-%+AXES LABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#7FdtF\\fl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "simp 16 := evalf[13](simp(exp(-x^2)/(x^2+1),x=0..1,intervals=16));\nsimp32 \+ := evalf[13](simp(exp(-x^2)/(x^2+1),x=0..1,intervals=32));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'simp16G$\".3%o$=#)='!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'simp32G$\".[[b>#)='!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Next we calculate the differenc e " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 45 " between these \+ two Simpson's rule estimates. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "delta := evalf[13](abs(simp1 6-simp32));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG$\"(Sk=\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Using the note above, this gives an approximate upper bound for the error i n the 32 interval estimate of " }{XPPEDIT 18 0 "1/15;" "6#*&\"\"\"F$\" #:!\"\"" }{XPPEDIT 18 0 "``(1.18644*`.`*10^(-7));" "6#-%!G6#*(-%&Float G6$\"'W'=\"!\"&\"\"\"%\".GF,)\"#5,$\"\"(!\"\"F," }{TEXT -1 1 " " } {TEXT 295 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "7.907*`.`*10^(-8);" "6 #*(-%&FloatG6$\"%2z!\"$\"\"\"%\".GF))\"#5,$\"\")!\"\"F)" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "abserr32_estimate := evalf(delta/15,4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%2abserr32_estimateG$\"%2z!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A value which is accur ate to 13 digits can be obtained using Maple's numerical integration v ia " }{TEXT 0 9 "evalf/Int" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 114 "This value can be used to calculate the actual error in the 32 interval Simpson's rule estimate for the integral. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Int(exp(- x^2)/(x^2+1),x=0..1);\nevalf[16](%):\narea := evalf[13](%);\nabserr32 \+ := evalf[13](abs(simp32-area));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$ IntG6$*&-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"F/,&F+F/F/F/F0/F-;\"\"!F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\".\"3L'>#)='!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr32G$\"&L#y!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "The absolute error in the approximate value for \+ the integral obtained using Simpson's rule with 32 intervals is about \+ " }{XPPEDIT 18 0 "7.8233*`.`*10^(-9)" "6#*(-%&FloatG6$\"&L#y!\"%\"\"\" %\".GF))\"#5,$\"\"*!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 216 "The error estimate calculated previously, without reference to the accurate value, does not differ greatly from the actual error. No te that both Simpson's rule estimates are less than the true value of \+ the integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 226 "Either the error estimate, or the actual error in the 32 interval Simpson's rule estimate could be used to predict the approxi mate number of intervals needed to obtain a Simpson's rule estimate wh ich is correct to 13 digits. " }}{PARA 0 "" 0 "" {TEXT -1 30 "If we a im for a tolerance of " }{XPPEDIT 18 0 "epsilon = 2*`.`*10^(-15);" "6 #/%(epsilonG*(\"\"#\"\"\"%\".GF')\"#5,$\"#:!\"\"F'" }{TEXT -1 95 ", th is is unlikely to affect the last decimal place, since a unit in the l ast place (1 ulp) is " }{XPPEDIT 18 0 "10^(-13);" "6#)\"#5,$\"#8!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eps := 2e-15;\nabserr32/eps;\nevalf[6](log[16 ](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++l6R!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'%[Z&!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "log[16](3911650);" "6#-&%$logG6#\"#;6#\"(];\"R" }{TEXT -1 1 " " }{TEXT 294 1 "~" }{TEXT -1 28 " 5.47484 means that 3911650 " }{TEXT 293 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "16^5.47484;" "6#)\"#; -%&FloatG6$\"'%[Z&!\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The desired tolerance is reached whe n the absolute error is divided by 16 about 5 times. " }}{PARA 0 "" 0 "" {TEXT -1 69 "Thus we should double the number of intervals 5 times \+ to reach about " }{XPPEDIT 18 0 "32*`.`*2^5 = 1024;" "6#/*(\"#K\"\"\"% \".GF&\"\"#\"\"&\"%C5" }{TEXT -1 12 " intervals. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "This process can be autom ated by using the procedure " }{TEXT 0 4 "simp" }{TEXT -1 18 " with th e option \"" }{TEXT 271 12 "iterate=true" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 73 "The end result is a value for the integral which \+ is correct to 13 digits." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "evalf[13](simp(exp(-x^2)/(x^2+1),x= 0..1,iterate=true,info=true));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gap proximation~with~2~intervals~--->~~~G$\"3Z%pBS/Po7'!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~~~G$\"359N&p SKx='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~int ervals~--->~~~G$\"3F*G-Bd)>)='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Happroximation~with~16~intervals~--->~~~G$\"3STwSo$=#)='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~intervals~--->~~~G$ \"3?$)z%[b>#)='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~ with~64~intervals~--->~~~G$\"3`&[/#G'>#)='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~128~intervals~--->~~~G$\"3gDoxK'># )='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~int ervals~--->~~~G$\"3`)RiIj>#)='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Iapproximation~with~512~intervals~--->~~~G$\"3(pC!3L'>#)='!#=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~1024~intervals~-- ->~~~G$\"3Lh83L'>#)='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproxim ation~with~2048~intervals~--->~~~G$\"3.K93L'>#)='!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".\"3L'>#)='!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 34 "(a) Find an approximate value for " }{XPPEDIT 18 0 "Int(cos(x^2),x = 0 .. 1);" "6#-%$IntG6$-%$cosG6#*$%\"xG\"\"#/F*; \"\"!\"\"\"" }{TEXT -1 53 " using the Simpson's rule with 256 and 512 \+ intervals." }}{PARA 0 "" 0 "" {TEXT -1 72 "(b) Calculate the definite \+ integral in (a) by using the Maple procedure " }{TEXT 0 14 "evalf(Int( ..))" }{TEXT -1 92 ", and use this value to calculate the absolute err or in the in each of the values from (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "(c) Find the ratio of the abso lute errors calculated in (b) to verify experimentally that Simpson's \+ rule has order 3, that is, the error is approximately proportional to \+ the fourth power, " }{XPPEDIT 18 0 "h^4;" "6#*$%\"hG\"\"%" }{TEXT -1 17 ", of the spacing " }{TEXT 263 1 "h" }{TEXT -1 13 " between the " } {TEXT 265 1 "x" }{TEXT -1 18 " values used (the " }{TEXT 262 9 "step-s ize" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 262 4 "Note" }{TEXT -1 128 ": You will need to increase the pr ecision used for the calculations from the default precision of 10 dig its to around 20 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 41 "_________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 57 "(a) Check that Simpson's rule with 2 in tervals gives the " }{TEXT 262 12 "exact answer" }{TEXT -1 19 " for th e integral " }{XPPEDIT 18 0 "Int(``(x^2-3*x+4),x = 0 .. 3);" "6#-%$In tG6$-%!G6#,(*$%\"xG\"\"#\"\"\"*&\"\"$F-F+F-!\"\"\"\"%F-/F+;\"\"!F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 151 " This is not surpri sing since Simpson's rule is based on the idea of approximating curves by interpolating functions based on degree 2 polynomials. " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 262 4 "Note" }{TEXT -1 20 ": Use th e procedure " }{TEXT 0 16 "student[simpson]" }{TEXT -1 49 " which will provide for the possibility of using " }{TEXT 262 16 "exact arithmeti c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "(b) Check that Simp son's rule with 2 intervals gives the " }{TEXT 262 12 "exact answer" } {TEXT -1 18 " for the integral " }{XPPEDIT 18 0 "Int(``(x^3-3*x+4),x = 0 .. 3);" "6#-%$IntG6$-%!G6#,(*$%\"xG\"\"$\"\"\"*&F,F-F+F-!\"\"\"\"%F -/F+;\"\"!F," }{TEXT -1 27 ". [ Use exact arithmetic. ]" }}{PARA 0 "" 0 "" {TEXT -1 108 "(c) It appears from part (b) that Simpson's rule gi ves exact values for the integral of a cubic polynomial. " }}{PARA 0 " " 0 "" {TEXT -1 121 " Check that this is the case for two more int egrals of cubic polynomials of your own choice over suitable intervals . " }}{PARA 0 "" 0 "" {TEXT -1 41 "___________________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 41 "_________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 32 "(a) Find approximate va lues for " }{XPPEDIT 18 0 "Int(exp(x*ln(x+1)),x = 0 .. 1);" "6#-%$IntG 6$-%$expG6#*&%\"xG\"\"\"-%#lnG6#,&F*F+F+F+F+/F*;\"\"!F+" }{TEXT -1 40 " using Simpson's rule with 32 intervals." }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 262 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "exp(x*ln (x+1)) = (x+1)^x" "6#/-%$expG6#*&%\"xG\"\"\"-%#lnG6#,&F(F)F)F)F)),&F(F )F)F)F(" }{TEXT -1 58 ", so you can use this alternative expression if you wish. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "(b) Calculate the definite integral in (a) by using the Maple p rocedure " }{TEXT 0 14 "evalf(Int(..))" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 86 " Use this value to calculate the absolute err or in the value found in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 34 "( c) Find an approximate value for " }{XPPEDIT 18 0 "Int(exp(x*ln(x+1)), x = 0 .. 1);" "6#-%$IntG6$-%$expG6#*&%\"xG\"\"\"-%#lnG6#,&F*F+F+F+F+/F *;\"\"!F+" }{TEXT -1 65 " which is correct to 10 digits using Simpson' s rule in two ways. " }}{PARA 15 "" 0 "" {TEXT -1 252 "(i) Calculate a number of approximations for the definite integral using Simpson's ru le, with a progressively larger number of intervals, until there is no change in the 10 digits of the result. It is convenient to double the intervals with each step. " }}{PARA 15 "" 0 "" {TEXT -1 23 "(ii) Use \+ the procedure " }{TEXT 0 4 "simp" }{TEXT -1 18 " with the option \"" } {TEXT 271 12 "iterate=true" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 96 "(d) Could you have predicted roughly how many intervals w ould be needed to obtain the value for " }{XPPEDIT 18 0 "Int(exp(x*ln( x+1)),x = 0 .. 1);" "6#-%$IntG6$-%$expG6#*&%\"xG\"\"\"-%#lnG6#,&F*F+F+ F+F+/F*;\"\"!F+" }{TEXT -1 27 " obtained in (c)? Explain. " }}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 32 "(a) Find approximate values for " } {XPPEDIT 18 0 "Int(arcsinh(x^2),x = 0 .. 2);" "6#-%$IntG6$-%(arcsinhG6 #*$%\"xG\"\"#/F*;\"\"!F+" }{TEXT -1 40 " using Simpson's rule with 32 \+ intervals." }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 262 4 "Note" } {TEXT -1 2 ": " }{XPPEDIT 18 0 "arcsinh(x) =ln(x+sqrt(x^2+1))" "6#/-%( arcsinhG6#%\"xG-%#lnG6#,&F'\"\"\"-%%sqrtG6#,&*$F'\"\"#F,F,F,F," } {TEXT -1 10 ", so that " }{XPPEDIT 18 0 "arcsinh(x^2)=ln(x^2+sqrt(1+x^ 4))" "6#/-%(arcsinhG6#*$%\"xG\"\"#-%#lnG6#,&*$F(F)\"\"\"-%%sqrtG6#,&F/ F/*$F(\"\"%F/F/" }{TEXT -1 55 ". You can use this alternative expressi on if you wish. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "(b) Calculate the definite integral in (a) by using the M aple procedure " }{TEXT 0 14 "evalf(Int(..))" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 85 " Use this value to calculate the abso lute error in the value found in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 262 4 "Note" }{TEXT -1 75 ": Maple can obtain a (c omplicated) analytical expression for the integral. " }}{PARA 0 "" 0 " " {TEXT -1 128 " Evaluating this expression as a floatin g point number yields a small imaginary part which should be removed w ith " }{TEXT 0 2 "Re" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 " (c) Find an approximate value for " }{XPPEDIT 18 0 "Int(arcsinh(x^2),x = 0 .. 2);" "6#-%$IntG6$-%(arcsinhG6#*$%\"xG\"\"#/F*;\"\"!F+" }{TEXT -1 65 " which is correct to 10 digits using Simpson's rule in two ways . " }}{PARA 15 "" 0 "" {TEXT -1 252 "(i) Calculate a number of approxi mations for the definite integral using Simpson's rule, with a progres sively larger number of intervals, until there is no change in the 10 \+ digits of the result. It is convenient to double the intervals with ea ch step. " }}{PARA 15 "" 0 "" {TEXT -1 23 "(ii) Use the procedure " } {TEXT 0 4 "simp" }{TEXT -1 18 " with the option \"" }{TEXT 271 12 "ite rate=true" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 96 "(d) Could you have predicted roughly how many intervals would be needed to obta in the value for " }{XPPEDIT 18 0 "Int(arcsinh(x^2),x = 0 .. 2);" "6#- %$IntG6$-%(arcsinhG6#*$%\"xG\"\"#/F*;\"\"!F+" }{TEXT -1 27 " obtained \+ in (c)? Explain. " }}{PARA 0 "" 0 "" {TEXT -1 41 "____________________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 41 "_______________________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 24 "Code for drawing picture" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 33 "Code for Si mpson's rule pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1023 "f := x -> -x^2/4+x/4+4:\ninterp([ -1,0,1,2],[3.5,4,4,4.3],x):\ng := unapply(%,x):\np1 := plot([g(x),f(x) ],x=-1.5..1.5,color=[red,blue]):\np2 := plot([[[1,0],[1,4]],[[-1,0],[- 1,3.5]]],color=black):\ncv := op(op(1,op(1,plot(-x^2/4+x/4+4,x=-1..1,a daptive=false,numpoints=20)))):\np3 := plots[polygonplot]([[-1,0],[-1, 3.5],cv,[1,4],[1,0]],\n style=patchnogrid,color=COLOR(RGB,.9 ,.85,1)):\np4 := plot([[[-1,3.5],[0,4],[1,4]]$3],style=point,color=bla ck,\n symbol=[circle,diamond,cross]): \nt1 := plots[textplot]([[-1. 15,3.85,`( -h,y )`],[-0.2,4.27,`( 0,y )`],\n [1.15,4.3,`( h,y )`] ,[1.8,-0.13,`x`],[-0.1,4.95,`y`],\n [1,-0.2,`h`],[-1,-0.2,`-h`]],col or=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-1.02,3.77,`0` ],[-0.1,4.19,`1`],\n [1.26,4.22,`2`]],color=black,font=[HELVETICA,8] ):\nt3 := plots[textplot]([1.6,3.7,`y = p(x)`],color=blue,font=[HELVET ICA,10]):\nt4 := plots[textplot]([-1.45,2.7,`y = f(x)`],color=red,font =[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3,t4],view=[-2.. 2,-.2..5],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1832 "q1 := x -> -.3*x^2+.7*x+2: \nq2 := x -> .2*x^2-1.2*x+3.8:\nq3 := x -> -.35*x^2+3.75*x-7.2:\ninter p([0,1,2,3,4,5,6],[2,2.4,2.2,2,2.2,2.8,2.7],x):\ng := unapply(%,x):\np 1 := plot(q1(x),x=-.3..2.4,color=blue,thickness=1):\np2 := plot(q2(x), x=1.5..4.3,color=COLOR(RGB,.7,.2,.7),thickness=1):\np3 := plot(q3(x),x =3.7..6.3,color=COLOR(RGB,0,.7,0),thickness=1):\np4 := plot([[[0,0],[0 ,2]],[[1,0],[1,2.4]],[[2,0],[2,2.2]],\n [[3,0],[3,2]],[[4,0],[4,2.2] ],[[5,0],[5,2.8]],\n [[6,0],[6,2.7]],[[-.5,0],[6.5,0]]],color=black) :\np5 := plot([[[0,2],[1,2.4],[2,2.2],[3,2],[4,2.2],[5,2.8],\n [6,2 .7]]$3],style=point,symbol=[circle,diamond,cross],\n color=black): \np6 := plot(q1(x),x=0..2,color=COLOR(RGB,.9,.85,1),\n adapt ive=false,numpoints=15,filled=true):\np7 := plot(q2(x),x=2..4,color=CO LOR(RGB,.9,.85,1),\n adaptive=false,numpoints=15,filled=true ):\np8 := plot(q3(x),x=4..6,color=COLOR(RGB,.9,.85,1),\n ada ptive=false,numpoints=15,filled=true):\np9 := plot(g(x),x=-.5..6.3,col or=red,thickness=1):\nt1 := plots[textplot]([[6.5,-0.1,`x`],\n [-.3,2 .15,`(x ,y )`],[.95,2.55,`(x ,y )`],\n [2.2,2.35,`(x ,y )`],[3, 2.15,`(x ,y )`],\n [3.72,2.35,`(x ,y )`],[4.74,2.93,`(x ,y )`], \n [6.4,2.77,`(x ,y )`],[0,-0.08,`x`],[1,-0.08,`x`],\n [2,-0.08,`x `],[3,-0.08,`x`],[4,-0.08,`x`],[5,-0.08,`x`],\n [6,-0.08,`x`]],font=[ HELVETICA,10]):\nt2 := plots[textplot]([[-.25,2.1,`0 0`],[1,2.5,`1 \+ 1`],\n [2.24,2.3,`2 2`],[3.04,2.1,`3 3`],[3.76,2.3,`4 4`] ,\n [4.78,2.88,`5 5`],[6.44,2.72,`6 6`],\n [0.08,-0.13,`0`],[1 .08,-0.13,`1`],[2.08,-0.13,`2`],\n [3.08,-0.13,`3`],[4.08,-0.13,`4`], [5.08,-0.13,`5`],\n [6.08,-0.13,`6`]],font=[HELVETICA,8]):\nt3 := plo ts[textplot]([6,3.1,`y = f(x)`],color=red,font=[HELVETICA,10]):\nplots [display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,t1,t2,t3],\n view=[-.5..6.5 ,-.2..3.2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 37 "Code for Simpson's rule error picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1137 "p1 := plot ([[0,0],[8,0]],thickness=2,color=navy):\np2 := plot([[[0,0],[0,.5]],[[ 7.5,0],[7.5,.5]],[[8,0],[8,.5]]],\n style=line, color=navy,thickness=2):\np3 := plot([[[0,0],[0,-1]],[[7.5,0],[7.5,-.5 ]],[[8,0],[8,-1]]],color=black,linestyle=2):\np4 := plottools[arrow]([ 3.5,-.3],[0,-.3],0,.13,.03,arrow,color=brown,linestyle=3):\np5 := plot tools[arrow]([4,-.3],[7.5,-.3],0,.13,.03,arrow,color=brown,linestyle=3 ):\np6 := plottools[arrow]([7.75,-.3],[7.5,-.3],0,.13,.3,arrow,color=b rown,linestyle=3):\np7 := plottools[arrow]([7.75,-.3],[8,-.3],0,.13,.3 ,arrow,color=brown,linestyle=3):\np8 := plottools[arrow]([3.8,-.8],[0, -.8],0,.13,.03,arrow,color=brown,linestyle=3):\np9 := plottools[arrow] ([4.2,-.8],[8,-.8],0,.13,.03,arrow,color=brown,linestyle=3):\nt1 := pl ots[textplot]([[0,.8,`I`],[7.5,.8,`I`],[8,.8,`I`]],font=[TIMES,ROMAN,1 2]):\nt2 := plots[textplot]([[3.75,-.3,`15 e`],[7.75,-.4,`e`],[4,-.76, `e`]],font=[HELVETICA,10]):\nt3 := plots[textplot]([[4.1,-.9,`1`],[.08 ,.7,`1`],[7.58,.7,`2`],\n [3.95,-.4,`2`],[7.85,-.5,`2` ]],font=[HELVETICA,8]):\nplots[display]([p||(1..9),t||(1..3)],axes=non e,view=[0..8,-1..1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }