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In this worksheet we use yet another norm, produing the Jacobi polynomials, paying attent ion to how this changes the definition of best fit and the polynomials that can be approximated." }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 206 187 "We begin by loading Maple's package for othogonal \+ polynomials. We also load the plots package 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(plot s):\n" }{MPLTEXT 1 0 43 "assume('i',integer): 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$" }}{PARA 7 "" 1 "" {TEXT 208 49 "Warning, the name changecoords has been redefine d" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 68 "For approximation by orthog onal polynomials, we define the norm by " }}{PARA 206 "" 0 "" {TEXT 200 15 " = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting :-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub sup(Typesetting:-mo(\"∫\", form = \"\", fence = \"false\", separat or = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false \", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", font _style_name = \"2D Comment\", size = \"12\", 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w(0, 0)\" = int(1,x=-0.6..0.6)/int(1,x=-1..1);\n" }{MPLTEXT 1 0 70 "\" w(1, 1)\" = int((1-x)*(1+x),x=-0.6..0.6)/int((1-x)*(1+x),x=-1..1);\n" }{MPLTEXT 1 0 78 "\"w(2, 3)\" = int((1-x)^2*(1+x)^3,x=-0.6..0.6)/int(( 1-x)^2*(1+x)^3,x=-1..1);\n" }{MPLTEXT 1 0 78 "\"w(6, 8)\" = int((1-x)^ 6*(1+x)^8,x=-0.6..0.6)/int((1-x)^6*(1+x)^8,x=-1..1);\n" }{MPLTEXT 1 0 70 "\"w(-.4, -.6)\" = evalf(Int((1-x)^(-0.4)*(1+x)^(-0.6),x=-0.6..0.6) )/\n" }{MPLTEXT 1 0 54 " evalf(Int((1-x)^(-0.4)*(1+x)^(-0.6),x=-1 ..1));\n" }{MPLTEXT 1 0 70 "\"w(-.9, -.9)\" = evalf(Int((1-x)^(-0.9)*( 1+x)^(-0.9),x=-0.6..0.6))/\n" }{MPLTEXT 1 0 52 " evalf(Int((1-x)^ (-0.9)*(1+x)^(-0.9),x=-1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q(w(0 ,~0)6\"$\"+++++g!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q(w(1,~1)6\"$\"+ +++?z!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q(w(2,~3)6\"$\"+++gT))!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q(w(6,~8)6\"$\"+w?c6)*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q,w(-.4,~-.6)6\"$\"+ar!z!R!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q,w(-.9,~-.9)6\"$\"+'>;e?\"!#5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 463 "As m entioned above, one reason to change the parameter values is to change the way closeness is defined. A second reason is to change the subsp ace of functions over which the inner product makes sense. In particu lar,using positive values for a and b produces an inner product under which functions with poles at the end of the domain have finite norm. Consider what happens to the function f(x) = 1/(1-x) with the Legand re norm and with the (3,3) Jacobi norm." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "\"_L\" = int(1*(1-x)^(-2), x=-1..1);\n" } {MPLTEXT 1 0 54 "\"_J(3,3)\" = int((1-x^2)^3*(1-x)^(-2), x=-1..1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q(_L6\"I)infinityG%*protected G" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q-_J(3,3)6\"#\"\")\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT 200 3 "Exe" }{TEXT 210 0 "" }{TEXT 200 7 "rcises:" }{TEXT 210 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 344 "1) Describe the rationa l functions that have a finite norm under the (a, b) Jacobi norm. (To simplify the problem, assume that the rational functions are in a nor mal form with partial fraction decomposition, i.e., each function is a sum of proper fractions with the denominator a power of an irreducibl e polynomial with integer coefficients.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 137 "2) For the (2,3) Jacobi norm, what percentage of the weight is on the interval [ 0, 1]? Repeat the question with the (2, 5) Jacobi norm." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 207 18 "Jacobi polynomials" }}{EXCHG {PARA 0 "" 0 "" {TEXT 206 123 "As we did in previous worksheets we check the help page for the orthogon al polynomials we are looking at in this worksheet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "?orthopoly,P" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 101 "Once we have loaded the orthopoly package, we obtain th e nth (a,b) Jacobi polynomial with P(n,a,b,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "\"P(3,1,1,x)\"=P(3,1,1,x);\n" }{MPLTEXT 1 0 28 "\"P(3,1,2,x)\"=P(3,1,2,x);\n" }{MPLTEXT 1 0 28 "\"P(3,2,1,x)\"=P(3,2, 1,x);\n" }{MPLTEXT 1 0 34 "\"P(3,-.4,-.6,x)\"=P(3,-.4,-.6,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "/Q+P(3,1,1,x)6\",*\"#9!\"\"*&\"#=\"\"\"I \"xGF$F*F**&\"#@F*),&F+F*F*F'\"\"#F*F**&\"\"(F*)F/\"\"$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "/Q+P(3,1,2,x)6\",*\"# " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 207 19 "An extende d example" }}{EXCHG {PARA 0 "" 0 "" {TEXT 206 127 "Time to look at an \+ example to see how this works out in a particular case. Once again we will use sin(3\271x) as our function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "func := x -> sin(3*Pi*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%funcG6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$-I$sin G6$%*protectedGI(_syslibGF$6#,$*(\"\"$\"\"\"I#PiGF.F4F'F4F4F$F$F$" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT 207 52 "Computing coefficients and appro ximating polynomials" }}{EXCHG {PARA 0 "" 0 "" {TEXT 206 280 "Next we \+ start computing coefficients and defining the approximating polynomia ls up to degree 12 for the inner products of type (0,0), (2,3), and (- .4, -.6). Recall that if P(n,a,b,x) is the nth (a,b) Jacobi polynomia l, the projection coeffient of f(x) onto P(n,a,b,x) should be " } {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\", 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\"false\", lspace = \"thinm athspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symme tric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", font_style_n ame = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgro und = \"[255,255,255]\")), Typesetting:-mi(\"b\"), superscriptshift = \+ \"0\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence \+ = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\" , stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreg round = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mro w(Typesetting:-mi(\"f\"), 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:-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(Typesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"th inmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", sy mmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", font_style 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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\", de pth = \"0.0 ex\", linebreak = \"auto\"), Typesetting:-mo(\"&Differenti alD;\", form = \"prefix\", fence = \"false\", separator = \"false\", l space = \"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:-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'/ %*separatorGFFepF?/FCQ3verythickmathspaceF'FDFFFHFKFNF PFRFTFWFZFgnF^rFitF^sFitFdqFgqF+F+F+-I'mspaceGF$6&/%'heightGQ'0.0~exF' /%&widthGQ'0.3~emF'/%&depthGFdu/%*linebreakGQ%autoF'-F563Q0&Differenti alD;F'F_oF:F=F?FBFDFFFHFKFNFPFRFTFWFZFgnFdqF+" }{TEXT 206 14 " divide d by " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Types etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msubsup(Typesetting:- mo(\"∫\", form = \"\", fence = \"false\", separator = \"false\", l space = \"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(\"&uminus0;\", f orm = \"prefix\", fence = \"false\", separator = \"false\", lspace = \+ \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\" , maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movabl elimits 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\"2D Comment\", size = \"12\", foregro und = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-msup( Typesetting:-mrow(Typesetting:-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]\"), T ypesetting:-mrow(Typesetting:-mn(\"1\"), Typesetting:-mo(\"+\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"mediu mmathspace\", rspace = \"mediummathspace\", stretchy = \"false\", symm etric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", font_style_n ame = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgro und = \"[255,255,255]\"), Typesetting:-mi(\"x\")), Typesetting:-mo(\") \", form = \"postfix\", fence = \"true\", separator = \"false\", lspac e = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"tr ue\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", fo nt_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\")), Typesetting:-mi(\"b\"), superscr iptshift = \"0\"), Typesetting:-mo(\"⁢\", form = \"infi x\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspac e = \"0em\", stretchy = \"false\", 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(\"P\"), Typesetting:-mo(\"⁡ \", form = \"infix\", fence = \"false\", separator = 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\"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \"postfix \", fence = \"true\", separator = \"false\", lspace = \"thinmathspace \", rspace = \"verythinmathspace\", 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:-mi(\"\")), T ypesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0 ex\", width = \"0.3 em\", depth = \"0.0 ex\", linebreak = \"auto\"), Typesetting:-m o(\"ⅆ\", form = \"prefix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\" , symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", font_s tyle_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", b ackground = \"[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./%&fenc eGQ&falseF'/%*separatorGFFe pF?/FCQ3verythickmathspaceF'FDFFFHFKFNFPFRFTFWFZFgnF^rF`tF^sF`tFdqFgqF +FbrFasF+F+-I'mspaceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&dep thGF[u/%*linebreakGQ%autoF'-F563Q0ⅆF'F_oF:F=F?FBFDFFFHFK FNFPFRFTFWFZFgnFdqF+" }{TEXT 206 4 " . " }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for i from 0 to 12 do\n" }{MPLTEXT 1 0 53 " normP00[i ] := evalf(Int((P(i,0,0,x))^2, x=-1..1)):\n" }{MPLTEXT 1 0 72 " coeffP 00[i] := evalf(Int(simplify(P(i,0,0,x)*func(x)*1.0), x=-1..1));\n" } {MPLTEXT 1 0 5 "od:\n" }{MPLTEXT 1 0 30 "Jacobi00approx := proc(m,x) \+ \n" }{MPLTEXT 1 0 16 " local count:\n" }{MPLTEXT 1 0 66 " sum(P(coun t,0,0,x)*coeffP00[count]/normP00[count],count=0..m):\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 23 "for i from 0 to 12 do\n" }{MPLTEXT 1 0 69 " normP23[i] := evalf(Int((1-x)^2*(1+x)^3*(P(i,2,3,x))^2, x=-1..1)) :\n" }{MPLTEXT 1 0 88 " coeffP23[i] := evalf(Int(simplify((1-x)^2*(1+x )^3*P(i,2,3,x)*func(x)*1.0), x=-1..1)):\n" }{MPLTEXT 1 0 5 "od:\n" } {MPLTEXT 1 0 30 "Jacobi23approx := proc(m,x) \n" }{MPLTEXT 1 0 16 " l ocal count:\n" }{MPLTEXT 1 0 66 " sum(P(count,2,3,x)*coeffP23[count]/ normP23[count],count=0..m):\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 23 "for i from 0 to 12 do\n" }{MPLTEXT 1 0 81 " normP46[i] := evalf(In t((1-x)^(-.4)*(1+x)^(-.6)*(P(i,-.4,-.6,x))^2, x=-1..1));\n" }{MPLTEXT 1 0 61 " coeffP46[i] := evalf(Int(simplify((1-x)^(-.4)*(1+x)^(-.6)*\n" }{MPLTEXT 1 0 48 " P(i,-.4,-.6,x)*func(x)*1.0), x=-1..1));\n" } {MPLTEXT 1 0 5 "od:\n" }{MPLTEXT 1 0 30 "Jacobi46approx := proc(m,x) \+ \n" }{MPLTEXT 1 0 16 " local count:\n" }{MPLTEXT 1 0 70 " sum(P(coun t,-.4,-.6,x)*coeffP46[count]/normP46[count],count=0..m):\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/Jacobi00approxG6\"f*6$ I\"mGF$I\"xGF$6#I&countGF$F$F$-I$sumGF$6$*(-I\"PGF$6&F*\"\"!F2F(\"\"\" &I)coeffP00GF$F)F3&I(normP00GF$F)!\"\"/F*;F2F'F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/Jacobi23approxG6\"f*6$I\"mGF$I\"xGF$6#I&countGF$F$F $-I$sumGF$6$*(-I\"PGF$6&F*\"\"#\"\"$F(\"\"\"&I)coeffP23GF$F)F4&I(normP 23GF$F)!\"\"/F*;\"\"!F'F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/Jaco bi46approxG6\"f*6$I\"mGF$I\"xGF$6#I&countGF$F$F$-I$sumGF$6$*(-I\"PGF$6 &F*,$$\"\"%!\"\"F5,$$\"\"'F5F5F(\"\"\"&I)coeffP46GF$F)F9&I(normP46GF$F )F5/F*;\"\"!F'F$F$F$" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 207 28 "Compar ing the approximations" }}{EXCHG {PARA 0 "" 0 "" {TEXT 206 186 "Our pa st experience with this function is that the approximation start to be interesting with the ninth degree. Note first that the polynomial app roximations are significantly different." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "n:= 9:\n" }{MPLTEXT 1 0 28 "sort(Jacobi00approx(n,x)); \n" }{MPLTEXT 1 0 28 "sort(Jacobi23approx(n,x));\n" }{MPLTEXT 1 0 26 " sort(Jacobi46approx(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",6*&$\"+$f &p[=!\")\"\"\"I\"xG6\"F'F'*&$\"+XEf)[(!\"'F'),&F(F'F'!\"\"\"\"%F'F'$\" +Pr9.=F&F0*&$\"+7\\Z/5!\"&F')F/\"\"'F'F'*&$\"+W&e94%!\"(F')F/\"\"#F'F' *&$\"+/dn8EF-F')F/\"\"$F'F'*&$\"+k.gY6F7F')F/\"\"&F'F'*&$\"+3H81:F=F') F/\"\"*F'F'*&$\"+<'>bN\"F-F')F/\"\")F'F'*&$\"+SJo\\]F-F')F/\"\"(F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",6*&$\"+Fu'z0\"!\"(\"\"\"I\"xG6\"F'F'$ \"+'\\Sc,\"F&!\"\"*&$\"+/vpS8!\"&F'),&F(F'F'F,\"\"%F'F'*&$\"+!R([C;F0F ')F2\"\"'F'F'*&$\"+!R3f5\"!\"'F')F2\"\"#F'F'*&$\"+]U'RK&FF0F')F2\"\"&F'F'*&$\"+RQ#\\P#F&F')F2\"\"*F'F'*&$\"+b9VP@F F')F2\"\"*F'F'*&$\"+Q>L`6F0F')F2\"\")F'F'*& $\"+$3;LH%F0F')F2\"\"(F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 64 "Ne xt compare the graphs of the approximations and of the errors." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "lowy := -1.2: highy := 1.6: n:= 7:\n" }{MPLTEXT 1 0 57 "pl1 := plot(func(x), x=-1..1,y=lowy..hig hy, color=red):\n" }{MPLTEXT 1 0 71 "pl2 := plot(Jacobi00approx(n,x), \+ x=-1..1,y=lowy..highy, color=green):\n" }{MPLTEXT 1 0 70 "pl3 := plot( Jacobi23approx(n,x), x=-1..1,y=lowy..highy, color=blue):\n" }{MPLTEXT 1 0 71 "pl4 := plot(Jacobi46approx(n,x), x=-1..1,y=lowy..highy, color= black):\n" }{MPLTEXT 1 0 47 "B := textplot(\{[-0.8,highy-(highy-lowy)/ 20, \n" }{MPLTEXT 1 0 70 " `The function and the `||n||` term \+ Jacobi approximations`],\n" }{MPLTEXT 1 0 36 " [-0.8,highy-(highy -lowy)/9, \n" }{MPLTEXT 1 0 66 " `(0,0) (green) vs (2,3) (blue), \+ vs (-.4, -.6) (black)`]\},\n" }{MPLTEXT 1 0 46 " align=RIGHT, font \+ = [TIMES, BOLD, 12] ):\n" }{MPLTEXT 1 0 61 "display(\{pl1, pl2, pl3, p l4, B\},view=[-1..1, lowy..highy]);" }}{PARA 13 "" 1 "" {TEXT 209 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6*-%+AXESLABELSG6'Q\"x6\"Q\"yF'-% %FONTG6$%*HELVETICAG\"#5%+HORIZONTALGF.-%'CURVESG6$7ir7$$!\"\"\"\"!$\" 3Uj!)**>CeI6!#<7$$!3,n;HdNvs**!#=$\"3?A/u*oyD*)*F=7$$!3.MLe9r]X**F=$\" 3E#)z`\\())Q`)F=7$$!3/,](=ng#=**F=$\"3GlG(HK3&GsF=7$$!3%pmm\"HU,\"*)*F =$\"3`dY$R05_(fF=7$$!3')***\\PM@l$)*F=$\"3vY#H9'H/?OF=7$$!3!RLL$e%G?y* F=$\"3gtDsfc$*e9F=7$$!3#om;HdNvs*F=$!3H6MCl7ws^!#>7$$!3t****\\(oUIn*F= $!3$Q`9'o[Z\"F97$$!3&fmT&Q75y!*F=$!39\"f]w4P]F\"F97$$!39***\\iId9(*)F=$!3i$eE/ !4[C8F97$$!3<#3xJK'zW*)F=$!3L9^$\\mg>L\"F97$$!3JmT5S`8=*)F=$!3@=#>Nf:w L\"F97$$!3W]7.dVZ\"*))F=$!3-_>`ku]T8F97$$!3YL$eRP8['))F=$!3\"H0x5\"ppV 8F97$$!3^*\\7yS\"\\6))F=$!3+,9'\\T0KM\"F97$$!3mmmmT%p\"e()F=$!3qkjAdPg O8F97$$!3%emmmwnMa)F=$!3nRCCVB*F=7$$!3\"QLL3i.9!zF=$!37%zwr`Y ^5(F=7$$!3'3++Dw$H.xF=$!3/4Z\\#R!*3/&F=7$$!3\"ommT!R=0vF=$!3'o+\\#fK?! 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This \+ becomes even clearer when we look at the graphs of the errors." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "lowy := -1.0: highy := 1.4: n:= 10:\n" }{MPLTEXT 1 0 81 "pl1 := plot(func(x) - Jacobi00approx(n, x), x=-1..1,y=lowy..highy, color=green):\n" }{MPLTEXT 1 0 80 "pl2 := p lot(func(x) - Jacobi23approx(n,x), x=-1..1,y=lowy..highy, color=blue): \n" }{MPLTEXT 1 0 81 "pl3 := plot(func(x) - Jacobi46approx(n,x), x=-1. .1,y=lowy..highy, color=black):\n" }{MPLTEXT 1 0 47 "B := textplot(\{[ -0.8,highy-(highy-lowy)/20, \n" }{MPLTEXT 1 0 46 " `The `||n| |` term Jacobi error `],\n" }{MPLTEXT 1 0 36 " [-0.8,highy-(highy -lowy)/9, \n" }{MPLTEXT 1 0 68 " `(0,0) (green) vs (2,3) (blue), vs (-.4, -.6) (black)`]\}, \n" }{MPLTEXT 1 0 55 " align=RIGHT , font = [HELVETICA, BOLD, 12] ):\n" }{MPLTEXT 1 0 32 "display(\{pl1, \+ pl2, pl3, B\});\n" }{MPLTEXT 1 0 1 " " }}{PARA 13 "" 1 "" {TEXT 209 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6)-%+AXESLABELSG6'Q\"x6\"Q\"yF' -%%FONTG6$%*HELVETICAG\"#5%+HORIZONTALGF.-%'CURVESG6$7jo7$$!\"\"\"\"!$ \"3gO()*pqQyO\"!#;7$$!3,n;HdNvs**!#=$\"3K\\vz9B%oG\"F97$$!3.MLe9r]X**F =$\"3'o>eqt-&47F97$$!3/,](=ng#=**F=$\"3c*)zw)=#pN6F97$$!3%pmm\"HU,\"*) *F=$\"3\\4U\"3L'Gl5F97$$!3')***\\PM@l$)*F=$\"3rja@*Q\"3U$*!#<7$$!3!RLL $e%G?y*F=$\"3o.!\\%\\sR`\")FT7$$!3#om;HdNvs*F=$\"3@Gdw2S.yqFT7$$!3t*** *\\(oUIn*F=$\"3](GxBE,x5'FT7$$!3wLL3-)\\&='*F=$\"3w%Ga\"=ZdM_FT7$$!3om mm;p0k&*F=$\"31ShGIkF^WFT7$$!3#HL3-)*G#p%*F=$\"3(p@E,8C>G$FT7$$!3E++vV 5Su$*F=$\"3ib4lY$H+L#FT7$$!3^m;H2Jdz#*F=$\"3>17#3TJ`c\"FT7$$!3vKL$3HbBQc'!#>7$$!3mmmmT%p \"e()F=$!3]\"\\&Ro+hPOF=7$$!3umm\"H-%\\/()F=$!3iX;/q6m'>%F=7$$!3qlm;/' =3l)F=$!3kMnra!=Gf%F=7$$!3wlmT&=Vrf)F=$!3'[V`C7hk%[F=7$$!3%emmmwnMa)F= $!3C!y^N9Dj(\\F=7$$!3#fm;zM#z*[)F=$!33%[2>^\"f**\\F=7$$!3+mm;Hp6O%)F=$ !39%4XN&p(>$\\F=7$$!32mmT5:W#Q)F=$!3,aLw$y=yy%F=7$$!3:mmm\"4m(G$)F=$!3 l'omByc,e%F=7$$!3)****\\i&[3:\")F=$!3j&R'GV'o5M$F=7$$!3\"QLL3i.9!zF=$! 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In a separate graph plot the error of \+ the approxiamtion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {XPPEDIT 19 1 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }