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This can also be thought of as finding the best appro ximation in the subspace under the given norm. " }}{PARA 0 "" 0 "" {TEXT 208 0 "" }}{PARA 0 "" 0 "" {TEXT 208 415 "Finding Fourier series of functions is an example of this process. When we find a Fourier p olynomial for a function, we are actually approximating a periodic fun ction by the best fitting trig polynomials of specified degree. Best \+ fitting is defined by a least sum of squares rule. To measure the dis tance between two functions we integrate the square of the difference \+ of the two functions. 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"" 0 "" {TEXT 209 17 "Technical \+ details" }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 41 "We start by loading th e packages we need." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with (orthopoly); with(plots):" }}{PARA 11 "" 1 "" {XPPMATH 20 "7(I\"GG6\"I \"HGF$I\"LGF$I\"PGF$I\"TGF$I\"UGF$" }}{PARA 7 "" 1 "" {TEXT 210 49 "Wa rning, the name changecoords has been redefined" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 205 "For this worksheet we will work with Legandre poly nomials. They are orthogonal in the space of continuous functions on \+ [-1, 1] with the usual integration inner product. Look at the help pa ge for details." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "?P" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 208 178 "Time for some technical commands . We want to assume that i and j are integers to simplify the integra tions we need to do. We also don't want them printed with a trailing \+ tilde." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "assume('i',intege r): assume('j',integer):\n" }{MPLTEXT 1 0 25 "interface(showassumed=0) :" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 209 66 "An extended example, appr oximating the function f(x) = sin(3*Pi*x)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 54 "We will start with the function func(x) = sin(3\271x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "func := x -> sin(3*Pi*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%funcG6\"f*6#I\"xGF$F$6$I)operator GF$I&arrowGF$F$-I$sinGF$6#,$*(\"\"$\"\"\"I#PiG%*protectedGF1F'F1F1F$F$ F$" }}}{SECT 0 {PARA 205 "" 0 "" {TEXT 211 41 "Computing Taylor and Le gandre polynomials" }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 242 "We would l ike to start by computing coefficients. We will find the coefficient s for both the Legangre polynomials and for Taylor polynomials. 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symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mi(\"x\")), Typesetting:-mi(\"\"));" "-I%mrowG 6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6*F+-I (msubsupGF$6'-I#moGF$63Q&∫F'/%%formGF./%&fenceGQ&falseF'/%*separat orGFF\\qF?/FCQ3verythickmathspaceF'FDFFFHFKFNFPFRFTFWFZFgnFcqFf qF+F+F+-I'mspaceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&depthGF gs/%*linebreakGQ%autoF'-F563Q0ⅆF'F_oF:F=F?FBFDFFFHFKFNFP FRFTFWFZFgnFcqF+" }{TEXT 208 14 " divided by " }{XPPEDIT 18 0 "Types etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-msubsup(Typesetting:-mo(\"∫\", form = \"\", fe nce = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0 em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinit y\", minsize = 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Typesetting:-mo(\"⁡\", form = \"inf ix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspa ce = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"2D Comment\", size = \+ \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typ esetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"tru e\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thi nmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \" infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", font_style_name = \"2D Comment\", size = \" 12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Types etting:-mrow(Typesetting:-mi(\"n\"), Typesetting:-mo(\",\", form = \"i nfix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rsp ace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"fals e\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", font_style_name = \"2D Com ment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,2 55,255]\"), Typesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \"p ostfix\", fence = \"true\", separator = \"false\", lspace = \"thinmath space\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background \+ = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0 ex\", width = \"0.3 em\", depth = \"0.0 ex\", linebreak = \"auto\"), Typesetting: -mo(\"ⅆ\", form = \"prefix\", fence = \"false\", separat or = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false \", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", font _style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"x\")), Typesetting :-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6% -I#miGF$6#Q!F'-F#6*F+-I(msubsupGF$6'-I#moGF$63Q&∫F'/%%formGF./%&fe nceGQ&falseF'/%*separatorGFF\\qF?/FCQ3 verythickmathspaceF'FDFFFHFKFNFPFRFTFWFZFgn-F,6#Q\"xF'-F563Q\")F'/F9Q( postfixF'F[qF=F]q/FCQ2verythinmathspaceF'F`qFFFHFKFNFPFRFTFWFZFgnF+-F5 63Q1⁢F'FdpF:F=F?FBFDFFFHFKFNFPFRFTFWFZFgnF\\pF+F+-I'msp aceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&depthGF^s/%*linebrea kGQ%autoF'-F563Q0ⅆF'F_oF:F=F?FBFDFFFHFKFNFPFRFTFWFZFgnF \\rF+" }{TEXT 208 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " func := x -> sin(3*Pi*x);\n" }{MPLTEXT 1 0 46 "normP[0] := evalf(Int(( P(0,x))^2, x=-1..1)):\n" }{MPLTEXT 1 0 55 "coeffP[0] := evalf(Int(P(0, x)*func(x)*1.0, x=-1..1)):\n" }{MPLTEXT 1 0 40 "taylco[0] := evalf(sub s(x=0,func(x))):\n" }{MPLTEXT 1 0 23 "for i from 1 to 20 do\n" } {MPLTEXT 1 0 46 "normP[i] := evalf(Int((P(i,x))^2, x=-1..1)):\n" } {MPLTEXT 1 0 65 "coeffP[i] := evalf(Int(simplify(P(i,x)*func(x)*1.0), \+ x=-1..1)):\n" }{MPLTEXT 1 0 53 "taylco[i] := evalf(subs(x=0,diff(func( x),x$i))/i!):\n" }{MPLTEXT 1 0 3 "od:" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%funcG6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$-I$sinGF$6#,$*(\" \"$\"\"\"I#PiG%*protectedGF1F'F1F1F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 89 "Now we define procedures that define the approximating p olynomials for specified degrees." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Lapprox := proc(m,x) \n" }{MPLTEXT 1 0 16 " local co unt:\n" }{MPLTEXT 1 0 58 " sum(P(count,x)*coeffP[count]/normP[count], count=0..m):\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 23 "Tapprox := \+ proc(m,x) \n" }{MPLTEXT 1 0 16 " local count:\n" }{MPLTEXT 1 0 43 " \+ sum(x^count*taylco[count], count=0..m):\n" }{MPLTEXT 1 0 4 "end;" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I(LapproxG6\"f*6$I\"mGF$I\"xGF$6#I&coun tGF$F$F$-I$sumGF$6$*(-I\"PGF$6$F*F(\"\"\"&I'coeffPGF$F)F2&I&normPGF$F) !\"\"/F*;\"\"!F'F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(TapproxG6\" f*6$I\"mGF$I\"xGF$6#I&countGF$F$F$-I$sumGF$6$*&)F(F*\"\"\"&I'taylcoGF$ F)F0/F*;\"\"!F'F$F$F$" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 204 0 "" } {TEXT 200 52 "Comparing the fit of Taylor and Legandre polynomials" }} {EXCHG {PARA 0 "" 0 "" {TEXT 208 86 "Compare the 3rd degree approximat ions and note that they are substantially different.." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Lapprox(3,x);\n" }{MPLTEXT 1 0 13 " Tapprox(3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&$\"+TwSwg!#5\"\"\"I \"xG6\"F'!\"\"*&$\"+%3^Ka\"!\"*F')F(\"\"$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&$\"+izxC%*!\"*\"\"\"I\"xG6\"F'F'*&$\"+^CG&R\"!\"(F')F (\"\"$F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 190 "It is worth re calling that Taylor polynomials are \"best fitting\" in the sense of f itting very well at the point of expansion. Compare the graph of the \+ function and the rwo approximations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=3:\n" }{MPLTEXT 1 0 51 "pl1 := plot(func(x), x=-1.. 1,y=-1..1, color=red):\n" }{MPLTEXT 1 0 58 "pl2 := plot(Lapprox(n,x), \+ x=-1..1,y=-1..1, color=green):\n" }{MPLTEXT 1 0 57 "pl3 := plot(Tappro x(n,x), x=-1..1,y=-1..1, color=blue):\n" }{MPLTEXT 1 0 29 "display(\{p l1, pl2, pl3\});\n" }{MPLTEXT 1 0 71 "print(\"func with its 3rd degre \+ Legandre and Taylor approximations.\"):" }}{PARA 13 "" 1 "" {TEXT 212 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6'-%+AXESLABELSG6'Q\"x6\"Q\"y F'-%%FONTG6$%*HELVETICAG\"#5%+HORIZONTALGF.-%'CURVESG6$7gt7$$!\"\"\"\" !$!3Sd?W(RSRn$!#L7$$!3%pmm\"HU,\"*)*!#=$!3Jn@\"*>9OD5F=7$$!3!RLL$e%G?y *F=$!3YPk+`S\"*R?F=7$$!3t****\\(oUIn*F=$!31-Cl_I'H.$F=7$$!3ommm;p0k&*F =$!3[J5]^-/%*RF=7$$!3E++vV5Su$*F=$!3\"yDt`6&RgbF=7$$!3vKL$3A)F=$!34VO$4AE \\%**F=7$$!3NnT5l^]o\")F=$!3P/1:'=z&z)*F=7$$!3)****\\i&[3:\")F=$!30q7^ HS>*y*F=7$$!3Mm;aQUC3!)F=$!3_U]c)[)GM&*F=7$$!3\"QLL3i.9!zF=$!3hzot??z# =*F=7$$!3'3++Dw$H.xF=$!3QVaV1Y@)G)F=7$$!3\"ommT!R=0vF=$!3+E)G(*\\Ib5(F =7$$!3tKL$3i_+I(F=$!3NQgn`RC@cF=7$$!3u****\\P8#\\4(F=$!3!\\ns&4]]FRF=7 $$!3#RLL3FuF)oF=$!3UY'=#3JrA?F=7$$!3*pmmT?F1n'F=$!3[KS<')*4Ft$!#?7$$!3 tLL$e\\g\"fkF=$\"3/kvv*fcK%>F=7$$!3[++](y$pZiF=$\"3Wi^V*G1p%QF=7$$!3Mn mT&Gu,.'F=$\"3#=S#3oPVXcF=7$$!33LLL$yaE\"eF=$\"33**>\\:[_2sF=7$$!3%)** **\\([j5i&F=$\"3^;P6F4DN$)F=7$$!3hmmm\">s%HaF=$\"3/T7SHy$>>*F=7$$!3]** *\\7)*G;K&F=$\"3LL&)yzP2W&*F=7$$!3]LL$3x&y8_F=$\"35,'G^/)p(z*F=7$$!30, ]ilT')f^F=$\"3***e'['o4n))*F=7$$!3]nmTgD%f5&F=$\"36l,SvF>]**F=7$$!3%RL 3_&4-_]F=$\"3p[en\"R$)z)**F=7$$!3Q+++]$*4)*\\F=$\"2!***fe&R)*****!#<7$ $!3j++]Pj&R%\\F=$\"3f_%*yHN0')**F=7$$!3K+++DL\")*)[F=$\"3P+h5%*e7Y**F= 7$$!3,++]7.nN[F=$\"3l(*))[%*\\I!))*F=7$$!3E++++t_\"y%F=$\"32U**)e=i()y *F=7$$!3>+++v7CtYF=$\"3>T*3qXI&H&*F=7$$!38+++]_&\\c%F=$\"3YUD+O%G6<*F= 7$$!33+++]zCcVF=$\"35offFY@:#)F=7$$!30+++]1aZTF=$\"3?y<*3'RYUpF=7$$!3R LL3FW,eRF=$\"3(HB(4(*H?`bF=7$$!3umm;/#)[oPF=$\"3)[7F!y'Gs)RF=7$$!3=L$e *)p0el$F=$\"3'[&R$\\]dE*HF=7$$!3<++v$>BJa$F=$\"3x3c7`]Ok>F=7$$!3Un7QR\"*!#>7$$!3hLLL$=exJ$F=$!3o:W8(eqyY\"Ff\\l7$$!3!QLL $eW%o7$F=$!3/;(z>p]Q$>F=7$$!3)RLLLtIf$HF=$!3\")3F'pmt%eOF=7$$!3CnmT&o_ Qr#F=$!3IjEy;aP7bF=7$$!3]++]PYx\"\\#F=$!3?8T=t7nDrF=7$$!3QnmTNz>&H#F=$ !3w$fz(R*fhH)F=7$$!3EMLLL7i)4#F=$!3u^Eys#*p#=*F=7$$!3g+]PMLy!*>F=$!3>i 5!ey[q`*F=7$$!3#pm;aVXH)=F=$!3UY*3@\")oHz*F=7$$!3%**\\Pf[E!H=F=$!3(4_U ]'=:$))*F=7$$!3CL$ek`2^x\"F=$!3;N4Nl!=y%**F=7$$!3am\"zpe)=@;D&Q)******Fdx7$$!3u**\\(o/Efh\"F=$ !3X(GR[]n&))**F=7$$!3k***\\iX#ek:F=$!3f9Dq\\=v`**F=7$$!3a**\\il)QK^\"F =$!3]N9lGHj&*)*F=7$$!3r*****\\F&*=Y\"F=$!3](GIq!oM9)*F=7$$!3z***\\P43# f8F=$!3.hi%y'z3$e*F=7$$!3')****\\74_c7F=$!3+v<%[rR@E*F=7$$!3ZmmT5VBU5F =$!3wS>B(=nwJ)F=7$$!3)3LLL3x%z#)Ff\\l$!3G\"y*HC`3NqF=7$$!3gKL$e9d;J'Ff \\l$!3!o>&fi`#Rg&F=7$$!3KMLL3s$QM%Ff\\l$!3B'*=vDCc!)RF=7$$!3T,+]ivF@AF f\\l$!3'QIrZ.Y#y?F=7$$!3\\^omm;zr)*!#@$!3\"[ehPA5QI*Fgt7$$\"3&yK$3_Nl. 5Ff\\l$\"3AhcYr=6X%*Ff\\l7$$\"3@CL$3-Dg5#Ff\\l$\"3q1#)Q?U(=(>F=7$$\"3e ?Le*['R3KFf\\l$\"3K3#Q-@rz(HF=7$$\"3%pJL$ezw5VFf\\l$\"3jM\"Qqr^>&RF=7$ $\"3K`mmmJ+IiFf\\l$\"3L*Q887A+a&F=7$$\"3s*)***\\PQ#\\\")Ff\\l$\"3@Pd1+ GKZpF=7$$\"3ilm\"z\\1A-\"F=$\"3kBbx6vS6#)F=7$$\"3GKLLe\"*[H7F=$\"3&fs< *)=(4j\"*F=7$$\"3.***\\PzglL\"F=$\"3(3d)))e3#*>&*F=7$$\"3ylm;HCjV9F=$ \"3^$)eg\")>))z(*F=7$$\"3;**\\(oCor\\\"F=$\"3_c^R6Kns)*F=7$$\"3aKLekSq ]:F=$\"33%3T\\_N.%**F=7$$\"3#fm\"H#))RUg\"F=$\"3&RL\\8q'p#)**F=7$$\"3H *******pvxl\"F=$\"3g%z7(=*['****F=7$$\"37**\\7`+:5].;***F= 7$$\"3&*)**\\iSCDw\"F=$\"3-*pnqu<#f**F=7$$\"3y)*\\Pf()*[\"=F=$\"3IgXR[ 0d-**F=7$$\"3g)***\\7JFn=F=$\"3I9b(H!**z@)*F=7$$\"3F)**\\(==-s>F=$\"3= mH(RhL()e*F=7$$\"3#z****\\_qn2#F=$\"3%3&[uo&)Gi#*F=7$$\"3=)**\\P/q%zAF =$\"3je7PU%3!y$)F=7$$\"3U)***\\i&p@[#F=$\"3)zz\\Si\"*))=(F=7$$\"3L)** \\(=GB2FF=$\"3nae$3VBVc&F=7$$\"3A)****\\2'HKHF=$\"3%*yfp&>L.p$F=7$$\"3 uJL$3UDX8$F=$\"3!Q#y56exi=F=7$$\"3ElmmmZvOLF=$!3O.'R!f$>YA$Fgt7$$\"3%) )***\\iirWMF=$!3/[GA*QKy/\"F=7$$\"3VKLLexn_NF=$!3<^WR(QzD0#F=7$$\"3-mm ;a#R1m$F=$!3!=SnJo$4OIF=7$$\"3h******\\2goPF=$!3;\"\\s6T,#))RF=7$$\"3, mm\"H2fU'RF=$!32N$f(3?0-cF=7$$\"3UKL$eR<*fTF=$!3\"Qp?xsXf-(F=7$$\"3.mm \"HiBQP%F=$!3&*R(*=b8`3$)F=7$$\"3l******\\)Hxe%F=$!3vL_V6Hda#*F=7$$\"3 *em\"z%Rk$)o%F=$!3]mf3_]wr&*F=7$$\"36KLeR*)**)y%F=$!3Q'*fM/n\"H!)*F=7$ $\"3]l\"z>@;$R[F=$!3)Q,S!esa&))*F=7$$\"3M)*\\P%[L'*)[F=$!3qMJh[+&f%**F =7$$\"3=J3xc2&*R\\F=$!3f`zrf#*)R)**F=7$$\"3ckm;H!o-*\\F=$!3K%fk(f$z&** **F=7$$\"31JL3xS'G/&F=$!3_++hR4%=***F=7$$\"3c(****\\7ga4&F=$!3ExrNW^bf **F=7$$\"31km\"HVSU(e*F=7$$\"3y)***\\7k.6aF=$!3#))>.=$ =(*e#*F=7$$\"3IKLe9as;cF=$!3p%p%e9'pxN)F=7$$\"3#emmmT9C#eF=$!3;Xw'fJfM 9(F=7$$\"3WKLeRF=7$$\"3%QLLL$*zym'F=$\"3g1N'[kw M9\"Fgt7$$\"3ILL$3sr*zoF=$\"3?1vp['Qo*>F=7$$\"3wKLL3N1#4(F=$\"3$fD<.q? 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3qzn(ysC)Q?Fdx7$Fg]m$!3\\h)>yuYh\"RFdx7$Fa^m$!3nB*=E0\"fBhFdx7$F[_m$!3 Ho-)4c(*)[\"*Fdx7$Fi`m$!3c(p'obqhj7Ffgn7$Fgbm$!3`&)4f5ze+ " 0 "" {MPLTEXT 1 0 69 "err1 := plot(func(x) - Lapprox(n,x), x=-1..1 ,y=-1..1, color=green):\n" }{MPLTEXT 1 0 68 "err2 := plot(func(x) - Ta pprox(n,x), x=-1..1,y=-1..1, color=blue):\n" }{MPLTEXT 1 0 26 "display (\{err1, err2\});\n" }{MPLTEXT 1 0 76 "print(\"The error in the 3rd de gre Legandre and Taylor approximations.\"):\n" }{MPLTEXT 1 0 1 " " }} {PARA 13 "" 1 "" {TEXT 212 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6& -%+AXESLABELSG6'Q\"x6\"Q\"yF'-%%FONTG6$%*HELVETICAG\"#5%+HORIZONTALGF. -%'CURVESG6$7gs7$$!\"\"\"\"!$\"3`'*****)>.hN*!#=7$$!3%pmm\"HU,\"*)*F9$ \"3?aZHXz'y*yF97$$!3!RLL$e%G?y*F9$\"3/\"zs'f*>8Y'F97$$!3t****\\(oUIn*F 9$\"3Qe`UAU.d]F97$$!3ommm;p0k&*F9$\"3;KBcM!f`p$F97$$!3#HL3-)*G#p%*F9$ \"3)e,rCrUIb#F97$$!3E++vV5Su$*F9$\"3MZj*Qp'*oX\"F97$$!3^m;H2Jdz#*F9$\" 3x)[MYT2%HT!#>7$$!3vKL$3A)F9$!3ruu/&)HZjjF97$$!3NnT5l^]o\")F9$!3SO$3+0ss%Ha F9$\"3QRwmf,%GO)F97$$!3]***\\7)*G;K&F9$\"3ya^V#QCij)F97$$!3]LL$3x&y8_F 9$\"3?!orqvFo\"))F97$$!30,]ilT')f^F9$\"3z8:ZC!*)F97$$!3%RL3_&4-_]F9$\"3!4vJ'RI23*)F97$$!3Q+++]$*4)*\\F9$ \"3Wc]y`)*z*)))F97$$!3E++++t_\"y%F9$\"3W=\\3>XRq&)F97$$!38+++]_&\\c%F9 $\"35^!)o7NMlyF97$$!33+++]zCcVF9$\"3Z(o6mgcR%oF97$$!30+++]1aZTF9$\"3HV n/-IIBbF97$$!3RLL3FW,eRF9$\"3[-0/Mo00TF97$$!3umm;/#)[oPF9$\"3ZU**R%zhK _#F97$$!3<++v$>BJa$F9$\"3nVbpe`Zy\\Fgn7$$!3hLLL$=exJ$F9$!3%3fdWW#>*f\" F97$$!3!QLL$eW%o7$F9$!3(\\\")Ged^?O$F97$$!3)RLLLtIf$HF9$!3>knaJs\">0&F 97$$!3CnmT&o_Qr#F9$!3eQYR!RlH&oF97$$!3]++]PYx\"\\#F9$!3a@S=,W,,%)F97$$ !3QnmTNz>&H#F9$!3SSnTY?A/&*F97$$!3EMLLL7i)4#F9$!3U++1#zE:.\"!#<7$$!3g+ ]PMLy!*>F9$!3zpguyo\\i5Fi\\l7$$!3#pm;aVXH)=F9$!3I!zo8l4M3\"Fi\\l7$$!3% **\\Pf[E!H=F9$!3v'yWLe6+4\"Fi\\l7$$!3CL$ek`2^x\"F9$!3]M2;$f7S4\"Fi\\l7 $$!3am\"zpe)=@@J\"4\"Fi\\l7$$\"3g)***\\7JFn=F9$\"3wN%Rwc&f&3\" Fi\\l7$$\"3F)**\\(==-s>F9$\"3#>0f6Nmo1\"Fi\\l7$$\"3#z****\\_qn2#F9$\"3 =\">vb!*)fQ5Fi\\l7$$\"3=)**\\P/q%zAF9$\"3I'yWtKB.e*F97$$\"3U)***\\i&p@ [#F9$\"3D%*fJ\\([6Y)F97$$\"3L)**\\(=GB2FF9$\"3#\\!4#=lUJ!pF97$$\"3A)** **\\2'HKHF9$\"3sV*=J)o,$3&F97$$\"3uJL$3UDX8$F9$\"35Q*4Cce@H$F97$$\"3El mmmZvOLF9$\"3MW#Rhpl>U\"F97$$\"3VKLLexn_NF9$!3$pd&f#*GCeeFgn7$$\"3h*** ***\\2goPF9$!3-CEfS,CCDF97$$\"3,mm\"H2fU'RF9$!3l9MZ#oZY:%F97$$\"3UKL$e R<*fTF9$!39(pTph]\"4cF97$$\"3.mm\"HiBQP%F9$!3c;%e#>x4UpF97$$\"3l****** \\)Hxe%F9$!3vDH(oiJq&zF97$$\"36KLeR*)**)y%F9$!323Dqk+$ze)F97$$\"3ckm;H !o-*\\F9$!3PQs\"3p5^)))F97$$\"31JL3xS'G/&F9$!3]_)RI2%p1*)F97$$\"3c(*** *\\7ga4&F9$!3x&)=#*3=-0*)F97$$\"31km\"Haj^mF97$$\"3WKLeRBzom@F97$$\"3%QLL L$*zym'F9$!36%f]([L#)>^Fgn7$$\"3ILL$3sr*zoF9$\"3LSZ!R/!p^6F97$$\"3wKLL 3N1#4(F9$\"3:M6caT=2FF97$$\"3c***\\(o!*R-tF9$\"3Kvlh?cKnSF97$$\"3Mmm;H Yt7vF9$\"3kn&>)frlw^F97$$\"3\"HL$ek6,1xF9$\"3#)3v?w'RI#fF97$$\"3Y***** **p(G**yF9$\"3'*H&fZ[N!ojF97$$\"37L$3-)omazF9$\"3hr[.WcfNkF97$$\"3ymmT gg/5!)F9$\"3>$H&*)4uPvkF97$$\"3K**\\iS_Ul!)F9$\"3Ee&fQT.p['F97$$\"3)HL L3U/37)F9$\"3b#\\0&[qwpkF97$$\"3jm;/,O=w\")F9$\"36\"[d?`LOU'F97$$\"3=* **\\7yi:B)F9$\"3*Qm_txS#[jF97$$\"3%GLe9'>%pG)F9$\"3oc!>G1+MC'F97$$\"3] mmmT6KU$)F9$\"3!*4#QO]'**3hF97$$\"3a****\\P$[/a)F9$\"3r8A!\\(e<'Q&F97$ $\"3fKLLLbdQ()F9$\"3U+dYQZC\"H%F97$$\"3`mm\"zW?)\\*)F9$\"3WK4?5!4Tt#F9 7$$\"3Z++]i`1h\"*F9$\"3=tU7H<\\(4)Fgn7$$\"3Y++DJ&f@E*F9$!3C_O_9!=OF#Fg n7$$\"3X++++PDj$*F9$!3a!QjXKc8L\"F97$$\"3X++voyMk%*F9$!3YEU\"\\VSa\\#F 97$$\"3W++]P?Wl&*F9$!3cwQ1iaN7PF97$$\"3K+]7G:3u'*F9$!3&f<.ayM-2&F97$$ \"3A++v=5s#y*F9$!33#yW#))yNqkF97$$\"35+]P40O\"*)*F9$!3\"z?(e!\\sC!zF97 $$\"\"\"F6$!3`'*****)>.hN*F9-%&COLORG6&%$RGBG$F6F5$F-F5F^hm-F06$7S7$F4 $!3&)***z8nM5I\"!#:7$FJ$!3_s^'4r(\\M6Ffhm7$Fin$!3DSn'\\Sz9+\"Ffhm7$Fco $!3(*pd\\&pg,k)!#;7$F]p$!3]*GH[i8jP(F`im7$Fer$!3TAdYpA4IiF`im7$F_s$!3Y 'Hb,PqAE&F`im7$Fis$!3BC@Y\"4bPN%F`im7$Fct$!3M@\">H;CK^$F`im7$F]u$!3])* oEx9QvFF`im7$Fgu$!3lM=))*y;.7#F`im7$Fav$!37NNMHFgH;F`im7$F_x$!3A_vSc_0 r6F`im7$Fix$!3zjP$e*zf`!)Fi\\l7$Fcy$!3f%p_A3C;N&Fi\\l7$F]z$!3W6#>'**3' o^$Fi\\l7$Fgz$!3%*pKn-pP$)>Fi\\l7$Fa[l$!3!G,[tr7)H6Fi\\l7$F[\\l$!3TY08 0f0G_F97$Fe\\l$!3'>hLY9O**H#F97$Fd^l$!3)*)QO\"yr1IvFgn7$Fb`l$!3Y*)R]#Febl7$Fjdl$\"3n'ok 2Nijo\"Fgn7$Fdel$\"3$QXKO(4%HK(Fgn7$Fbgl$\"3)z)*zg\"F`im7$F`^m$\"3SZExpN'Q8#F`im 7$Fj^m$\"3B_\"3LulTy#F`im7$Fd_m$\"3 " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 205 0 "" }{TEXT 200 43 "Animating the approximations and the errors" } }{EXCHG {PARA 0 "" 0 "" {TEXT 208 136 "We can also compare the results with an animation. To prepare for the animation we create a procedur e that will produce a single frame." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "framer := proc(n,x, lowy, highy)\n" }{MPLTEXT 1 0 31 "local pl1, pl2, pl3, B, C, D:\n" }{MPLTEXT 1 0 57 "pl1 := plot(func(x ), x=-1..1,y=lowy..highy, color=red):\n" }{MPLTEXT 1 0 64 "pl2 := plot (Lapprox(n,x), x=-1..1,y=lowy..highy, color=green):\n" }{MPLTEXT 1 0 63 "pl3 := plot(Tapprox(n,x), x=-1..1,y=lowy..highy, color=blue):\n" } {MPLTEXT 1 0 47 "B := textplot(\{[-0.8,highy-(highy-lowy)/20, \n" } {MPLTEXT 1 0 64 " `The function and the `||n||` term approxim ations`],\n" }{MPLTEXT 1 0 77 " [0-0.8,highy-(highy-lowy)/9, `Tay lor (blue) vs Legandre (green)`]\},\n" }{MPLTEXT 1 0 46 " align=RI GHT, font = [TIMES, BOLD, 12] ):\n" }{MPLTEXT 1 0 58 "display(\{pl1, p l2, pl3, B\},view=[-1..1, lowy..highy]);\n" }{MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 208 102 "Using that procedure, it is easy to produce an animation of the approximations of degree 1 through 20. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "highy := 1.6: lowy :=-1 .2:\n" }{MPLTEXT 1 0 62 "display([seq(framer(counta,x, lowy, highy), c ounta=1..20)], \n" }{MPLTEXT 1 0 27 " insequence = true);" } {TEXT 208 0 "" }}{PARA 13 "" 1 "" {TEXT 212 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6%-%+AXESLABELSG6'Q\"x6\"Q\"yF'-%%FONTG6$%*HELVETICAG \"#5%+HORIZONTALGF.-%(ANIMATEG667'-%'CURVESG6$7gt7$$!\"\"\"\"!$!3Sd?W( RSRn$!#L7$$!3%pmm\"HU,\"*)*!#=$!3Jn@\"*>9OD5FA7$$!3!RLL$e%G?y*FA$!3YPk +`S\"*R?FA7$$!3t****\\(oUIn*FA$!31-Cl_I'H.$FA7$$!3ommm;p0k&*FA$!3[J5]^ -/%*RFA7$$!3E++vV5Su$*FA$!3\"yDt`6&RgbFA7$$!3vKL$3A)FA$!34VO$4AE\\%**FA7$ $!3NnT5l^]o\")FA$!3P/1:'=z&z)*FA7$$!3)****\\i&[3:\")FA$!30q7^HS>*y*FA7 $$!3Mm;aQUC3!)FA$!3_U]c)[)GM&*FA7$$!3\"QLL3i.9!zFA$!3hzot??z#=*FA7$$!3 '3++Dw$H.xFA$!3QVaV1Y@)G)FA7$$!3\"ommT!R=0vFA$!3+E)G(*\\Ib5(FA7$$!3tKL $3i_+I(FA$!3NQgn`RC@cFA7$$!3u****\\P8#\\4(FA$!3!\\ns&4]]FRFA7$$!3#RLL3 FuF)oFA$!3UY'=#3JrA?FA7$$!3*pmmT?F1n'FA$!3[KS<')*4Ft$!#?7$$!3tLL$e\\g \"fkFA$\"3/kvv*fcK%>FA7$$!3[++](y$pZiFA$\"3Wi^V*G1p%QFA7$$!3MnmT&Gu,.' 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10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 1 5" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" " Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curv e 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" "C urve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Cu rve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve \+ 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Cur ve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 8 5" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" " Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "Curve 97" "Curv e 98" "Curve 99" "Curve 100" }}{TEXT 212 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 208 226 "If you step through the frames, it is noteworthy that the Legandre approximation becomes reasonable at degree 7, and that t he graph of the function we are approximating has 6 bumps, and thus lo oks like a 7th degree polynomial." }}{PARA 0 "" 0 "" {TEXT 208 0 "" }} {PARA 0 "" 0 "" {TEXT 208 82 "In our example the Legandre approximatio n becomes very good when the degree is 13." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 206 "Since we are looking at approximations it is worthwhile to to also look at the error between the approximations and the origi nal function. Once again we start with a procedure for producing a si ngle frame." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "framererr := proc(n,x, lowy, highy)\n" }{MPLTEXT 1 0 20 "local pl1, pl2, B:\n" } {MPLTEXT 1 0 60 "pl1 := plot(func(x) - Lapprox(n,x), x=-1..1, color=gr een):\n" }{MPLTEXT 1 0 59 "pl2 := plot(func(x) - Tapprox(n,x), x=-1..1 , color=blue):\n" }{MPLTEXT 1 0 18 "B := textplot(\{\n" }{MPLTEXT 1 0 63 " [-0.8,highy-(highy-lowy)/20, `The `||n||` term error`],\n" } {MPLTEXT 1 0 76 " [-0.8,highy-(highy-lowy)/9, `Taylor (blue) vs L egandre (green)`]\}, \n" }{MPLTEXT 1 0 52 " align=RIGHT, font = [TIMES, BOLD, 12] ):\n" }{MPLTEXT 1 0 54 "display(\{pl1, pl2, B\}, \+ view=[-1..1, lowy..highy]);\n" }{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 105 "We now look at the animation of the error ter ms. Initially we set the the window to see errors under .1." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "highy := .1: lowy := -.1:\n " }{MPLTEXT 1 0 65 "display([seq(framererr(counta,x, lowy, highy), cou nta=1..20)], \n" }{MPLTEXT 1 0 25 " insequence = true);" }}{PARA 13 "" 1 "" {TEXT 212 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6%-%+AXE SLABELSG6'Q\"x6\"Q!F'-%%FONTG6$%*HELVETICAG\"#5%+HORIZONTALGF.-%(ANIMA TEG667&-%%TEXTG6&7$$!\")!\"\"$\"+yxxxx!#6QBTaylor~(blue)~vs~Legandre~( green)F'%+ALIGNRIGHTG-F*6%I&TIMESGF'I%BOLDGF'\"#7-F46&7$F7$\"1-++++++! *!#F?-%'CURVESG6$7]t7$$F9\"\"!$\"3)f*****f))4$= $!#=7$$!3!RLL$e%G?y*FT$\"3j,WT5B!Q2\"FT7$$!3ommm;p0k&*FT$!3?MNfYQ1(\\* !#>7$$!3E++vV5Su$*FT$!3?9*=yeIkd#FT7$$!3vKL$3,:X4N(FT7$$!3iL$eRZD>A)FT$!3(*puhgg!yK(FT7$$!3NnT5l^]o\")F T$!3[&)eDQKYzsFT7$$!3)****\\i&[3:\")FT$!3[1!QSH#31sFT7$$!3Mm;aQUC3!)FT $!3^)oMz<&=&)pFT7$$!3\"QLL3i.9!zFT$!30M%\\\\8(pnmFT7$$!3'3++Dw$H.xFT$! 3%obpq/!=OeFT7$$!3\"ommT!R=0vFT$!3d*\\%ymib;ZFT7$$!3tKL$3i_+I(FT$!33zX 9O]c(H$FT7$$!3u****\\P8#\\4(FT$!3\\#3auS@\"p;FT7$$!3#RLL3FuF)oFT$\"3A^ /%H&)>9o\"Fin7$$!3*pmmT?F1n'FT$\"3**\\]/&[**f3#FT7$$!3tLL$e\\g\"fkFT$ \"3;w>%3Cr#**RFT7$$!3[++](y$pZiFT$\"33'>(p')*3c$eFT7$$!3MnmT&Gu,.'FT$ \"3R7NVdy*[c(FT7$$!33LLL$yaE\"eFT$\"3e'=KpH]x0*FT7$$!3%)****\\([j5i&FT $\"3Kg=]+\"\\C,\"FI7$$!3hmmm\">s%HaFT$\"3o#3x.D>?4\"FI7$$!3]***\\7)*G; K&FT$\"3v^WT*3+Q7\"FI7$$!3]LL$3x&y8_FT$\"38)4Z*f(Hd9\"FI7$$!30,]ilT')f ^FT$\"3*p@#3YX\"H:\"FI7$$!3]nmTgD%f5&FT$\"3D%*G(pZYv:\"FI7$$!3%RL3_&4- _]FT$\"3Z#y*fg\"4'f6FI7$$!3Q+++]$*4)*\\FT$\"3D9\"FI7$$!3E++++t_\"y%FT$\"3%H)ecfp2J6FI7$$!3>+++v7CtYF T$\"3\\o/XNpq,6FI7$$!38+++]_&\\c%FT$\"3P9D7#))>C1\"FI7$$!33+++]zCcVFT$ \"3d,#[fT^=g*FT7$$!30+++]1aZTFT$\"3\"RktB:nEE)FT7$$!3RLL3FW,eRFT$\"33% [o\"=\"yI\"oFT7$$!3umm;/#)[oPFT$\"3PhxoGdx'=&FT7$$!3<++v$>BJa$FT$\"3?F @,$>w@4$FT7$$!3hLLL$=exJ$FT$\"3s)e*)Rq\")G4*Fin7$$!3!QLL$eW%o7$FT$!3&y +z!*)=X&Q*Fin7$$!3)RLLLtIf$HFT$!3ny9$**)y$Rs#FT7$$!3CnmT&o_Qr#FT$!3=!* o'yFH&[YFT7$$!3]++]PYx\"\\#FT$!37(z$QsZ^KjFT7$$!3QnmTNz>&H#FT$!3dz'4mu vbc(FT7$$!3EMLLL7i)4#FT$!3MQJC(Q(o9&)FT7$$!3g+]PMLy!*>FT$!3ebaepFO.*)F T7$$!3#pm;aVXH)=FT$!3iYs@l'3O>*FT7$$!3%**\\Pf[E!H=FT$!36vF\"Gla4I*FT7$ $!3CL$ek`2^x\"FT$!39UJy(y$y#Q*FT7$$!3am\"zpe)=@IS&FT7$ $!3KMLL3s$QM%Fin$!3?#=$4$z$HUQFT7$$!3T,+]ivF@AFin$!3fI>+u0a2?FT7$$!3 \\^omm;zr)*!#@$!3wP$edL\"e*)*)!#?7$$\"3@CL$3-Dg5#Fin$\"3PY(e>OP[!>FT7$ $\"3%pJL$ezw5VFin$\"3V\"=*H6dt9QFT7$$\"3K`mmmJ+IiFin$\"3Wfs`h\\rT`FT7$ $\"3s*)***\\PQ#\\\")Fin$\"32ICD'[Czo'FT7$$\"3ilm\"z\\1A-\"FT$\"376_&z2 Hg)yFT7$$\"3GKLLe\"*[H7FT$\"3'pS!4N'Ql2&f.K*FT7$$\"3;**\\(oCor\\\"FT$\"3 Mb&4uv4hR*FT7$$\"3aKLekSq]:FT$\"3oEQ#44JnW*FT7$$\"3#fm\"H#))RUg\"FT$\" 3M?/I(G^?Z*FT7$$\"3H*******pvxl\"FT$\"3xCAjCD'>Z*FT7$$\"37**\\7`+:5J.([K*FT7$$\"3g)***\\7JFn=FT$\"3VuZS6%GuA*FT7$$\"3F)** \\(==-s>FT$\"3&=9dOd>5'*)FT7$$\"3#z****\\_qn2#FT$\"3$4%Roz>B,')FT7$$\" 3=)**\\P/q%zAFT$\"3(>y/EfICl(FT7$$\"3U)***\\i&p@[#FT$\"3[`xd8Dz)R'FT7$ $\"3L)**\\(=GB2FFT$\"36%3C4Y%e-ZFT7$$\"3A)****\\2'HKHFT$\"3J#[WjOapv#F T7$$\"3uJL$3UDX8$FT$\"3;SEQ@;$R[FT$!3O0$z\\%\\fU6FI7$$\"3M)*\\P%[L'*)[FT$!3BK1cs oB]6FI7$$\"3=J3xc2&*R\\FT$!3,H^>UCkb6FI7$$\"3ckm;H!o-*\\FT$!3*z!Qs+J!) e6FI7$$\"31JL3xS'G/&FT$!30*)*4HW.(f6FI7$$\"3c(****\\7ga4&FT$!3=ZjeZ! 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"Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Cur ve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 4 6" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" " Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curv e 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65 " "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "C urve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" }}{TEXT 212 0 "" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 209 10 "Exercises:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 208 "1 ) For our example function, what degree Taylor approximation do we ne ed to be within 0.1 on the domain [-0.5, 0.5]? What degree Legandre a pproximation do we need to be within 0.1 on the domain [-0.5, 0.5]?" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 101 "2) With the same function what is the maximum error of a 20th degree Taylor approximation on [-1, 1]?" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 219 "3) \+ With the same function, what degree approximation do we need to be wit hin 0.1 on the domain [-1, 1]? What degree for an accuracy of 0.01 on the same domain? What degree for an accuracy of 0.001 on the same do main?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 212 "4) Repeat the process above for the function \+ sin(2\271x) + cos(\271x). What degree Legandre approximation gives an accuracy of 0.01 on the whole domain? (Be warned that you need to re compute the coefficients.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }