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0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 204 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 205 "Tim es" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 206 29 "Polynomial Approximation - II" }}{PARA 205 "" 0 "" {TEXT 200 33 "Legandre vs Chebyshev Polynom ials" }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 42 "Worksheet by Mike May, S.J., maymk@slu.edu" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 208 8 "O verview" }}{EXCHG {PARA 0 "" 0 "" {TEXT 207 280 "In the last worksheet we looked at polynomial approximation using Legandre polynomials. Th ese polynomials form an orthogonal set with a least squares fit defini tion for the norm. In this worksheet we will look at other norms and \+ the corresponding sets of orthogonal polynomials." }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 187 "We begin by loading Ma ple's package for othogonal polynomials. We also load the plots packa ge so that we can graph our results. We repeat the technical commands from the last worksheet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(orthopoly); with(plots):\n" }{MPLTEXT 1 0 43 "assume('i',inte ger): assume('j',integer):\n" }{MPLTEXT 1 0 25 "interface(showassumed= 0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "7(I\"GG6\"I\"HGF$I\"LGF$I\"PGF$I \"TGF$I\"UGF$" }}{PARA 7 "" 1 "" {TEXT 209 49 "Warning, the name chang ecoords has been redefined" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 80 "Fo r the Legangre polynomials we saw that the norm is defined by = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typ esetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msubsup(Typesetting :-mo(\"∫\", form = \"\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = 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Verify that t hey are orthogonal under the oppropriate inner product." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 208 47 "An extended example, back to f(x) = sin(3*Pi*x)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 207 127 "Time to look at an example to see how t his works out in a particular case. Once again we will use sin(3\271x ) as our function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "func \+ := x -> sin(3*Pi*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%funcG6\"f*6#I \"xGF$F$6$I)operatorGF$I&arrowGF$F$-I$sinGF$6#,$*(\"\"$\"\"\"I#PiG%*pr otectedGF1F'F1F1F$F$F$" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 208 52 "Compu ting coefficients and approximating polynomials" }}{EXCHG {PARA 0 "" 0 "" {TEXT 207 264 "Next we start computing coefficients. We will fi nd the coefficients for both the Legangre polynomials and for Chebyshe v polynomials of the first kind. 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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:-mi(\"x\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", \+ maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", font_style_name = \"2D Comment \", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,2 55]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:- mi(\"\"), Typesetting:-mspace(height = \"0.0 ex\", width = \"0.3 em\", depth = \"0.0 ex\", linebreak = \"auto\"), Typesetting:-mo(\"&Differe ntialD;\", form = \"prefix\", fence = \"false\", separator = \"false\" , lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", 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./%&fenceGQ&falseF'/ %*separatorGFF\\qF?/FCQ3verythickmathsp aceF'FDFFFHFKFNFPFRFTFWFZFgn-F,6#Q\"xF'-F563Q\")F'/F9Q(postfixF'F[qF=F ]q/FCQ2verythinmathspaceF'F`qFFFHFKFNFPFRFTFWFZFgnF+-F563Q1&InvisibleT imes;F'FdpF:F=F?FBFDFFFHFKFNFPFRFTFWFZFgnF\\pF+F+-I'mspaceGF$6&/%'heig htGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&depthGF^s/%*linebreakGQ%autoF'-F563 Q0ⅆF'F_oF:F=F?FBFDFFFHFKFNFPFRFTFWFZFgnF\\rF+" }{TEXT 207 126 " . Similarly, if T(n,x) is the nth Chebyshev polynomial of t he first kind, the projection coefficient of f(x) onto T(n,x) is " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-msubsup(Typesetting:-mo(\"&int ;\", form = \"\", fence = \"false\", separator = \"false\", lspace = \+ \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\" , maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", font_style_name = \"2D Commen t\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255, 255]\"), Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", form = \"pre fix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rsp ace = \"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:-mn(\"1\")), Typesetting:-mn(\"1\"), superscriptshift = \"2\" , subscriptshift = \"0\"), Typesetting:-mi(\"\"), Typesetting:-mrow(Ty pesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"wtCheb\"), Ty pesetting:-mo(\"⁡\", form = \"infix\", fence = \"false\" , separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy \+ = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+ \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[ 0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetti ng:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false \", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", 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]\"), Typesetting:-mrow(Typesettin g:-mi(\"x\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"tr ue\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"ve rythinmathspace\", stretchy = \"true\", 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]\")), Typesetting:-mi(\"\")), Typesetting:-mo(\"⁢\", form = \+ \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsiz e = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", font_style_name = \"2D Comment\", siz e = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Typesetting:-mo(\"&ApplyFun ction;\", form = \"infix\", fence = \"false\", separator = \"false\", \+ lspace = \"0em\", rspace = \"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]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \" prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmath space\", rspace = \"thinmathspace\", stretchy = \"true\", 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]\"), Typesetting:-mrow(Typesetting:-mi(\"x\")), Typeset ting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"fa lse\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", str etchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsi ze = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\" )), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \" false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", st retchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", min size = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foregroun d = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Ty pesetting:-mi(\"P\"), Typesetting:-mo(\"⁡\", form = \"in fix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rsp ace = \"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:-mn(\"0\"), 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 ?/FCQ3verythickmathspaceF'FDFFFHFKFNFPFRFTFWFZFgnFcqFfqF+F+F+-I'mspace GF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&depthGF\\t/%*linebreakG Q%autoF'-F563Q0ⅆF'F_oF:F=F?FBFDFFFHFKFNFPFRFTFWFZFgnFcqF +" }{TEXT 207 14 " divided by " }{XPPEDIT 18 0 "Typesetting:-mrow(Ty pesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesett ing:-msubsup(Typesetting:-mo(\"∫\", form = \"\", fence = \"false\" , separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy \+ = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+ \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[ 0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetti ng:-mo(\"&uminus0;\", form = \"prefix\", fence = \"false\", separator \+ = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", large op = \"false\", movablelimits = \"false\", accent = \"false\", font_st yle_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", ba ckground = \"[255,255,255]\"), Typesetting:-mn(\"1\")), Typesetting:-m n(\"1\"), superscriptshift = \"2\", subscriptshift = \"0\"), Typesetti ng:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mr ow(Typesetting:-mi(\"wtCheb\"), Typesetting:-mo(\"⁡\", f orm = \"infix\", fence = \"false\", separator = \"false\", lspace = \" 0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", \+ maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", font_style_name = \"2D Comment \", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,2 55]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fe nce = \"true\", separator = \"false\", lspace = \"thinmathspace\", rsp ace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", m axsize = \"infinity\", minsize = \"1\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", font_style_name = \"2D Comment\" , size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255 ]\"), Typesetting:-mrow(Typesetting:-mi(\"x\")), Typesetting:-mo(\")\" , form = \"postfix\", fence = \"true\", separator = \"false\", lspace \+ = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true \", 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(\"\")), Typesetting :-mo(\"⁢\", form = \"infix\", fence = \"false\", separa tor = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"fals e\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", fon t_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\" , background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi( \"P\"), Typesetting:-mo(\"⁡\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", \+ minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", font_style_name = \"2D Comment\", size = \"12\", foregr ound = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow (Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", \+ stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", mi nsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", font_style_name = \"2D Comment\", size = \"12\", foregrou nd = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(T ypesetting:-mi(\"n\"), Typesetting:-mo(\",\", form = \"infix\", fence \+ = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"veryth ickmathspace\", 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]\"), T ypesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \"postfix\", fen ce = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspa ce = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\" , maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", font_style_name = \"2D Commen t\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255, 255]\")), Typesetting:-mi(\"\")), Typesetting:-mo(\"⁢\" , form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", 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:-mrow(Typesetting:-mi(\"P\"), Typesetting:-mo( \"⁡\", form = \"infix\", fence = \"false\", separator = \+ \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", s ymmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_styl e_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", back ground = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", \+ form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \+ \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", sy mmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", font_style _name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backg round = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"n\"), \+ Typesetting:-mo(\",\", form = \"infix\", fence = \"false\", separator \+ = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", stretch y = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize \+ = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", font_style_name = \"2D Comment\", size = \"12\", foreground = \+ \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separato r = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmathspac e\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[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 = \"p refix\", fence = \"false\", separator = \"false\", lspace = \"0em\", r space = \"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]\"), T ypesetting:-mi(\"x\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6*F+-I(msubsupGF$6 '-I#moGF$63Q&∫F'/%%formGF./%&fenceGQ&falseF'/%*separatorGFF\\qF?/FCQ3verythickmathspaceF'FDFFFHFKFNFPFRFTFWFZFgnFcqFfqF+F]rF`r F+F+-I'mspaceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&depthGFgs/ %*linebreakGQ%autoF'-F563Q0ⅆF'F_oF:F=F?FBFDFFFHFKFNFPFRF TFWFZFgnFcqF+" }{TEXT 207 37 " , where wtCheb(x) = (1-x^2)^(-1/2). " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "wtchev := (1-x^2)^(-1/2): \n" }{MPLTEXT 1 0 23 "for i from 0 to 15 do\n" }{MPLTEXT 1 0 46 "normP [i] := evalf(Int((P(i,x))^2, x=-1..1));\n" }{MPLTEXT 1 0 65 "coeffP[i] := evalf(Int(simplify(P(i,x)*func(x)*1.0), x=-1..1));\n" }{MPLTEXT 1 0 53 "normT[i] := evalf(Int(wtchev*(T(i,x))^2, x=-1..1));\n" }{MPLTEXT 1 0 58 "coeffT[i] := evalf(Int(wtchev*T(i,x)*func(x), x=-1..1)):\n" } {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\" \"!$\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"!$F'F '" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\"!$\"+aEfTJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\"!$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\"\"$\"+nmmmm!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"\"$\"+2f1A@!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\"\"$\"+Fjzq:!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 ">&I'coeffTG6\"6#\"\"\"$\"+se)>b&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\"#$\"+++++S!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"#$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\"#$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\"#$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\"$$\"+dG9dG!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"$$\"+Cbrj&I&normTG6\"6#\"\"$$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\"$$\"+KJ!ej#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\"%$\"+AAAAA!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"%$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\"%$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\"%$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\"&$\"+=====!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"&$!+\"ytAK\"!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\"&$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\"&$!+7n*3i%!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\"'$\"+Q:YQ:!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"'$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\"'$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\"'$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\"($\"+LLLL8!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"($!+A7%zl#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\"($\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\"($!+a,Nj#*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\")$\"+)eqk<\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\")$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\")$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\")$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"\"*$\"+z:j_5!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"\"*$\"+8=`p;!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"\"*$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"\"*$\"+?gw1z!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"#5$\"+C&4Q_*!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"#5$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"#5$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"#5$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"#6$\"+u@l&p)!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"#6$!+'[e9%[!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"#6$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"#6$!+SW]vE!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"#7$\"+++++!)!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"#7$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"#7$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"#7$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"#8$\"+2uS2u!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"#8$\"+7]C(f)!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"#8$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"#8$\"+`dS%H&!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"#9$\"+C&I'coeffPG6\"6#\"#9$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"#9$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"#9$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normPG6\"6#\"#:$\"+.Hh^k!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffPG6\"6#\"#:$!+coD`5!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&normTG6\"6#\"#:$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'coeffTG6\"6#\"#:$!+A>Piq!#7" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 207 44 "Now we define the approximating polynomials." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Lapprox := proc(m,x) \n" } {MPLTEXT 1 0 16 " local count:\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 26 "Chebapprox := proc(m,x) \n" }{MPLTEXT 1 0 16 " local count:\n" }{MPLTEXT 1 0 58 " sum(T(count,x)*coeffT[count]/normT[coun t],count=0..m):\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(LapproxG6\"f*6$I\"mGF$I\"xGF$6#I&countGF$F$F$-I$sumGF$6$*(-I\"P GF$6$F*F(\"\"\"&I'coeffPGF$F)F2&I&normPGF$F)!\"\"/F*;\"\"!F'F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I+ChebapproxG6\"f*6$I\"mGF$I\"xGF$6#I&c ountGF$F$F$-I$sumGF$6$*(-I\"TGF$6$F*F(\"\"\"&I'coeffTGF$F)F2&I&normTGF $F)!\"\"/F*;\"\"!F'F$F$F$" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 208 28 "C omparing the approximations" }}{EXCHG {PARA 0 "" 0 "" {TEXT 207 208 "S ince our function has 6 bumps and looks like it could be a 7th degreee polynomial, we want to compare the 7th degree approximations. Note fi rst that the polynomial approximations are significantly different." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "n:= 7:\n" }{MPLTEXT 1 0 15 "Lapprox(n,x);\n" }{MPLTEXT 1 0 16 "Chebapprox(n,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&$\"+qXo;O!\"*\"\"\"I\"xG6\"F'F'*&$\"+.ol1P!\")F')F (\"\"$F'!\"\"*&$\"+e\\)od)F-F')F(\"\"&F'F'*&$\"+xO&\\M&F-F')F(\"\"(F'F 0" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&$\"+\"GMs]#!\"*\"\"\"I\"xG6\"F' F'*&$\"+E)ypk#!\")F')F(\"\"$F'!\"\"*&$\"+;!>U8'F-F')F(\"\"&F'F'*&$\"+% RGUx$F-F')F(\"\"(F'F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 64 "Next co mpare the graphs of the approximations and of the errors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "lowy := -1.2: highy := 1.6: n:= 7 :\n" }{MPLTEXT 1 0 57 "pl1 := plot(func(x), x=-1..1,y=lowy..highy, col or=red):\n" }{MPLTEXT 1 0 64 "pl2 := plot(Lapprox(n,x), x=-1..1,y=lowy ..highy, color=green):\n" }{MPLTEXT 1 0 66 "pl3 := plot(Chebapprox(n,x ), x=-1..1,y=lowy..highy, color=blue):\n" }{MPLTEXT 1 0 47 "B := textp lot(\{[-0.8,highy-(highy-lowy)/20, \n" }{MPLTEXT 1 0 64 " `The function and the `||n||` term approximations`],\n" }{MPLTEXT 1 0 79 " [-0.8,highy-(highy-lowy)/9, `Chebyshev (blue) vs Legandre (gre en)`]\},\n" }{MPLTEXT 1 0 46 " align=RIGHT, font = [TIMES, BOLD, 12 ] ):\n" }{MPLTEXT 1 0 56 "display(\{pl1, pl2, pl3, B\},view=[-1..1, lo wy..highy]);" }}{PARA 13 "" 1 "" {TEXT 210 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6)-%+AXESLABELSG6'Q\"x6\"Q\"yF'-%%FONTG6$%*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++vV 5Su$*F=$!3\"yDt`6&RgbF=7$$!3vKL$3A)F=$!34VO$4AE\\%**F=7$$!3NnT5l^]o\")F=$!3 P/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$H.xF=$!3QVaV 1Y@)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$$!3*pmmT?F1n'F=$!3[KS<')*4Ft$!#?7$$!3tLL$e\\g\"fkF=$\"3/kvv*fcK%> F=7$$!3[++](y$pZiF=$\"3Wi^V*G1p%QF=7$$!3MnmT&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$$!3%RL3_&4-_]F=$\"3p[en\"R$)z)**F= 7$$!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$$!3>+++v7CtYF=$\"3>T*3q XI&H&*F=7$$!38+++]_&\\c%F=$\"3YUD+O%G6<*F=7$$!33+++]zCcVF=$\"35offFY@: #)F=7$$!30+++]1aZTF=$\"3?y<*3'RYUpF=7$$!3RLL3FW,eRF=$\"3(HB(4(*H?`bF=7 $$!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$$!3!QLL$eW%o7$F=$!3/;(z>p]Q$>F=7$$! 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This becomes even clearer when we look at the graphs of the error s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "lowy := -1.0: highy \+ := 1.4: n:= 7:\n" }{MPLTEXT 1 0 77 "pl1 := plot(func(x) - Chebapprox( n,x), x=-1..1,y=lowy..highy, color=green):\n" }{MPLTEXT 1 0 73 "pl2 := plot(func(x) - Lapprox(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 54 " `The `||n||` term Legandre error (blue)`], \n" }{MPLTEXT 1 0 70 " [-0.8,highy-(highy-lowy)/9, `and Chebychev error (green)`]\}, \n" }{MPLTEXT 1 0 55 " align=RIGHT, font = [HELVETICA, BOLD, 12] ):\n" }{MPLTEXT 1 0 27 "display(\{pl1, pl2, B\} );\n" }{MPLTEXT 1 0 1 " " }}{PARA 13 "" 1 "" {TEXT 210 0 "" } {GLPLOT2D 400 400 400 {PLOTDATA 2 "6(-%+AXESLABELSG6'Q\"x6\"Q\"yF'-%%F ONTG6$%*HELVETICAG\"#5%+HORIZONTALGF.-%'CURVESG6$7gt7$$!\"\"\"\"!$!35] ++!fxki$!#=7$$!3.MLe9r]X**F9$!3S>ddvOeTBF97$$!3%pmm\"HU,\"*)*F9$!3e*[; P;+>?\"F97$$!3')***\\PM@l$)*F9$!31$Hq!H(pE*>!#>7$$!3!RLL$e%G?y*F9$\"3h qyzm_iUnFI7$$!3t****\\(oUIn*F9$\"3bEl5)*Htk?F97$$!3ommm;p0k&*F9$\"3X!= xxuN)GIF97$$!3#HL3-)*G#p%*F9$\"31+]3C3WkNF97$$!3E++vV5Su$*F9$\"3Ajc[F, WaQF97$$!35*=p@6F97$$!3%emmmwnMa)F9$\"3OVUSF u'))p$FI7$$!3+mm;Hp6O%)F9$!3c3#))*))[()zQFI7$$!3:mmm\"4m(G$)F9$!3B@%e? $[JI6F97$$!3)****\\i&[3:\")F9$!3J;OU&zV&*[#F97$$!3\"QLL3i.9!zF9$!3)\\4 \"oHl3(f$F97$$!3'3++Dw$H.xF9$!3mb+ifugHVF97$$!3\"ommT!R=0vF9$!3'3,e\" \\R(=u%F97$$!37MLL$3,RX(F9$!3K$HE3tXWz%F97$$!3K++]i#=ES(F9$!3%4#*=(Rfn C[F97$$!3`mmmTaL^tF9$!39vZ)>y%oK[F97$$!3tKL$3i_+I(F9$!3CO*ffgy'=[F97$$ !3Cmm;zp[(>(F9$!3ObR?*Rbes%F97$$!3u****\\P8#\\4(F9$!3aX=12.W\\XF97$$!3 #RLL3FuF)oF9$!3!QJpmHp#RRF97$$!3*pmmT?F1n'F9$!3Sn5:lf*H/$F97$$!3tLL$e \\g\"fkF9$!35hvtM\"3+$>F97$$!3[++](y$pZiF9$!38y**>9=e=nFI7$$!3MnmT&Gu, .'F9$\"3e\\7HT#G,\"pFI7$$!33LLL$yaE\"eF9$\"3E-n)p\"olQ?F97$$!3%)**** \\([j5i&F9$\"3kGbrr))3[JF97$$!3hmmm\">s%HaF9$\"3I&H@(4h\">8%F97$$!3]LL $3x&y8_F9$\"3%R\"=e2x-N]F97$$!3Q+++]$*4)*\\F9$\"3j%4Zu_@Tn&F97$$!3K+++ DL\")*)[F9$\"3IcHo\"pWR)eF97$$!3E++++t_\"y%F9$\"3eo\"e?1Ja,'F97$$!3]++ ](G%QFZF9$\"3:eu=&p.70'F97$$!3>+++v7CtYF9$\"3ALte;D\"o1'F97$$!3))**** \\i#)4>YF9$\"3$*=#)o5L?igF97$$!38+++]_&\\c%F9$\"3End2+#)QPgF97$$!33+++ ]zCcVF9$\"3.\"Q0YXN\\v&F97$$!30+++]1aZTF9$\"3h\\9^s*Hy=&F97$$!3RLL3FW, eRF9$\"31KQxf6O[WF97$$!3umm;/#)[oPF9$\"3jr$HI-CW_$F97$$!3=L$e*)p0el$F9 $\"3cQfWN3)=!HF97$$!3<++v$>BJa$F9$\"3!*\\9dpz([B#F97$$!3`K$*G&H#F9$!3YZWAY*eRO&F97 $$!3EMLLL7i)4#F9$!3=i(HYCmX7'F97$$!3#pm;aVXH)=F9$!3)yk&Q(Qnqp'F97$$!3c ****\\P'psm\"F9$!3a3Szz(o)opF97$$!3u**\\(o/Efh\"F9$!3am-lWAY()pF97$$!3 k***\\iX#ek:F9$!3*e\\%\\Mr6))pF97$$!3a**\\il)QK^\"F9$!3SSRKQ\">3(pF97$ $!3r*****\\F&*=Y\"F9$!3=z8#GM,c$pF97$$!3z***\\P43#f8F9$!3L8qT\\Zx6oF97 $$!3')****\\74_c7F9$!3i/YD'yTyh'F97$$!3ZmmT5VBU5F9$!3!eeH#f0s'*fF97$$! 3)3LLL3x%z#)FI$!3cl6Q=332^F97$$!3gKL$e9d;J'FI$!3kB?C8rQ(3%F97$$!3KMLL3 s$QM%FI$!310fh\"[hI\"HF97$$!3'ym;aQdDG$FI$!3%==2=tY4B#F97$$!3T,+]ivF@A FI$!3!=%HLou@C:F97$$!3'\\L$eRx**f6FI$!3ugWO4'*p1!)FI7$$!3\\^omm;zr)*!# @$!3o7%R-CY(Go!#?7$$\"3&yK$3_Nl.5FI$\"3%>&HM_FRJpFI7$$\"3@CL$3-Dg5#FI$ \"3Y#)[z$R9jW\"F97$$\"3e?Le*['R3KFI$\"34:!G(\\x'F97$$\"3y lm;HCjV9F9$\"3O[q`^vw=pF97$$\"3;**\\(oCor\\\"F9$\"3>?fCL(><'pF97$$\"3a KLekSq]:F9$\"3M(*4K)R6_)pF97$$\"3#fm\"H#))RUg\"F9$\"3$>PcCR*=*)pF97$$ \"3H*******pvxl\"F9$\"3Jw%pfbcO(pF97$$\"3g)***\\7JFn=F9$\"3#p(3-Sq>FnF 97$$\"3#z****\\_qn2#F9$\"3eui'zHnb>'F97$$\"3=)**\\P/q%zAF9$\"3g0E4Kp]K aF97$$\"3U)***\\i&p@[#F9$\"3$\\kmEZwuX%F97$$\"3Q)*\\i!>,Zf#F9$\"3%yG \\)=0^RQF97$$\"3L)**\\(=GB2FF9$\"3pVvF$\\no<$F97$$\"3F)*\\(oWk(>GF9$\" 3V-(*\\]c@yCF97$$\"3A)****\\2'HKHF9$\"3\\!\\UJW2Fv\"F97$$\"3uJL$3UDX8$ F9$\"3j4([Uia&=TFI7$$\"3ElmmmZvOLF9$!3)4ay.*Hgz#*FI7$$\"3VKLLexn_NF9$! 3YG&Q-MTHH#F97$$\"3h******\\2goPF9$!3SvL,S2-DNF97$$\"3,mm\"H2fU'RF9$!3 pv:CE0%eZ%F97$$\"3UKL$eR<*fTF9$!3CH`c%[z*G_F97$$\"3.mm\"HiBQP%F9$!36DX NGb(**y&F97$$\"3l******\\)Hxe%F9$!3XT34t\\G]gF97$$\"30LeRAr/QYF9$!3OOw /-o6mgF97$$\"3*em\"z%Rk$)o%F9$!3&>?3<`'[kgF97$$\"3s)\\(=n;oQZF9$!3j
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"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 1 3" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" " Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curv e 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" "C urve 39" "Curve 40" "Curve 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" "Cu rve 58" "Curve 59" "Curve 60" }}{TEXT 210 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT 200 10 "Exercis e: " }{TEXT 211 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 89 "4) Expl ain why, with our example, the error only changes with odd degree appr oximations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 206 "" 0 "" {TEXT 205 0 "" }{TEXT 200 17 "More exercises: " } }{EXCHG {PARA 0 "" 0 "" {TEXT 207 267 "5) Find the 10th degree Legand re and Chebyshev polynomial approximations to sin(Pi*x)+cos(2*Pi*x). \+ Approximately what is the maximum error for each of these polynomials? Approximately what is the error at the endpoints of the domain for e ach of the approximations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 113 "6) Give a function that has a norm under either the Legandre or Chebyshev inner product but n ot under the other." }}}{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 }