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0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 204 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 205 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 202 "" 0 "" {TEXT 206 27 "Polynomial Interpolation II" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "restart;\n" }{MPLTEXT 1 0 14 "with(plots):\n" }{MPLTEXT 1 0 15 "with(linalg):\n" }{MPLTEXT 1 0 13 "Digits := 15:" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 204 16 "Illustrati on of " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Types etting:-mrow(Typesetting:-mi(\"O\"), Typesetting:-mo(\"⁡ \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", 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 = 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Typesetting:-mrow(Typesetting:-mi(\"n\"), Typese tting:-mo(\"+\", form = \"infix\", fence = \"false\", separator = \"fa lse\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", str etchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", mins ize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"1\" )), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separ ator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmaths pace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\")), superscrip tshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thin mathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", 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(\"\")), Typesetting:-mi(\" \"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF $6#Q!F'-F#6&-F,6#Q\"OF'-I#moGF$63Q0⁡F'/%%formGQ&infixF'/ %&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ$0emF'/%'rspaceGFB/%)stretc hyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeop GF=/%.movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF'/%% sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F# 6%-F563Q\"(F'/F9Q'prefixF'/F/FAQ.thinmathspaceF'/FDFeo/FFFc oFGFIFLFOFQFSFUFXFenFhn-F#6%F+-I%msupGF$6%-F,6#Q\"hF'-F#6%F]o-F#6%-F,6 #Q\"nF'-F563Q\"+F'F8F;F>/FAQ0mediummathspaceF'/FDF[qFEFGFIFLFOFQFSFUFX FenFhn-I#mnGF$6#FN-F563Q\")F'/F9Q(postfixF'FboF>Fdo/FDQ2verythinmathsp aceF'FgoFGFIFLFOFQFSFUFXFenFhn/%1superscriptshiftGQ\"0F'F+F`qF+F+" } {TEXT 205 14 " Error Formula" }{TEXT 207 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 48 "Definition of test function, and graph on [0,1]." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f := x -> 3*x^2*exp(5*x^2-7)+sin(3* x);\n" }{MPLTEXT 1 0 53 "pf := plot(f(x), x=0..1, color=green, thickne ss=1):\n" }{MPLTEXT 1 0 12 "display(pf);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 41 "In order to \+ illustrate the error formula " }}{PARA 0 "" 0 "" {TEXT 208 13 " \+ " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Types etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Typesetting:-mo(\"&Ap plyFunction;\", form = \"infix\", fence = \"false\", separator = \"fal se\", lspace = 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Typesetting:-mo(\"=\", form = \"infix\", fence = \+ \"false\", separator = \"false\", lspace = \"thickmathspace\", rspace \+ = \"thickmathspace\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi(\"\"), Types etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mo( \"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lsp ace = \"thinmathspace\", rspace = \"thinmathspace\", 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:-mrow(Typesetting:-mi(\" \"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxs ize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", font_style_name = \"2D Comment\", s ize = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\" ), Typesetting:-msup(Typesetting:-mi(\"D\"), Typesetting:-mrow(Typeset ting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"fal se\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", 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:-mi(\"n\"), Typesetting:-mo(\"+\", form = \"infix\", fence = \"fals e\", 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\"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:-mo(\"⁡\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\" , rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsi ze = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", si ze = \"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(\"f\")), 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 _name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backg round = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mo( \")\", form = \"postfix\", fence = \"true\", separator = \"false\", ls pace = \"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:-mo(\"&ApplyFunct ion;\", form = \"infix\", 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maxsize = \"inf inity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mi(\"\"), Typesetting:-munderover(Typesetting: -mo(\"∏\", form = \"prefix\", fence = \"false\", separator = \+ \"false\", lspace = \"0em\", rspace = \"thinmathspace\", stretchy = \" true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\" , largeop = \"true\", movablelimits = \"true\", accent = \"false\", fo nt_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi (\"i\"), Typesetting:-mo(\"=\", form = \"infix\", fence = \"false\", s eparator = \"false\", lspace = \"thickmathspace\", rspace = \"thickmat hspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"inf inity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mn(\"0\")), Typesetting:-mi(\"n\"), accent = \"false\", accentun der = \"false\"), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \+ \"0.0 ex\", width = \"5.0 \", depth = \"0.0 ex\", linebreak = \"auto\" ), Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mo(\"(\", for m = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"th inmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmet ric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", font_style_nam e = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgroun d = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"x\"), Type setting:-mo(\"−\", form = \"infix\", fence = \"false\", separato r = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspac e\", stretchy = \"false\", 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:- msub(Typesetting:-mi(\"x\"), Typesetting:-mrow(Typesetting:-mi(\"i\")) , subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\" \")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", sep arator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmat hspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typese tting:-mi(\"\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \+ \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \+ \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \")), Typesetting:-mi(\"\")), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesettin g:-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(\"n\"), Typesetting:-mo(\"+\", form = \"infix\", fence = \"false \", separator = \"false\", lspace = \"mediummathspace\", rspace = \"me diummathspace\", 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:-mn(\"1\")), 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:-mo(\"!\", form = \"infix\", fence = \"false\", \+ separator = \"false\", lspace = \"verythinmathspace\", rspace = \"very thinmathspace\", 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(\"\")), linethickness = \"1\", denomalign = \"center\" , numalign = \"center\", bevelled = \"false\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mi(\"\")), Typesetti ng:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF' 6%-I#miGF$6#Q!F'-F#6'F+-F#6'F+-F#6&-F,6#Q\"fF'-I#moGF$63Q0&ApplyFuncti on;F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGFA/%'lspaceGQ$0em F'/%'rspaceGFF/%)stretchyGFA/%*symmetricGFA/%(maxsizeGQ)infinityF'/%(m insizeGQ\"1F'/%(largeopGFA/%.movablelimitsGFA/%'accentGFA/%0font_style _nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgrou ndGQ.[255,255,255]F'-F#6%-F963Q\"(F'/F=Q'prefixF'/F@Q%trueF'FB/FEQ.thi nmathspaceF'/FHFio/FJFgoFKFMFPFSFUFWFYFfnFinF\\o-F#6#-F,6#Q\"xF'-F963Q \")F'/F=Q(postfixF'FfoFBFho/FHQ2verythinmathspaceF'F[pFKFMFPFSFUFWFYFf nFinF\\oF+-F963Q(−F'F " 0 "" {MPLTEXT 1 0 9 "n : = 4:\n" }{MPLTEXT 1 0 26 "data_x := array(1..n+1):\n" }{MPLTEXT 1 0 26 "data_y := array(1..n+1):\n" }{MPLTEXT 1 0 11 "h := 1/n:\n" } {MPLTEXT 1 0 59 "printf(\"Interpolation with degree %d polynomial\\n\" ,n);\n" }{MPLTEXT 1 0 67 "printf(\" h Max. error error-ratio \+ h-ratio^(n+1)\\n\");\n" }{MPLTEXT 1 0 10 "K := 8: \n" }{MPLTEXT 1 0 22 "for k from 1 to K do\n" }{MPLTEXT 1 0 25 " for i from 0 to n do \n" }{MPLTEXT 1 0 27 " data_x[i+1] := i*h;\n" }{MPLTEXT 1 0 8 " \+ od:\n" }{MPLTEXT 1 0 27 " for i from 1 to n+1 do\n" }{MPLTEXT 1 0 34 " data_y[i] := f(data_x[i]);\n" }{MPLTEXT 1 0 8 " od:\n" } {MPLTEXT 1 0 41 " p := x -> interp(data_x, data_y, x):\n" }{MPLTEXT 1 0 28 " eval_x := array(1..11):\n" }{MPLTEXT 1 0 23 " err := vect or(11):\n" }{MPLTEXT 1 0 26 " for i from 0 to 10 do\n" }{MPLTEXT 1 0 32 " eval_x[i+1] := i/10*n*h;\n" }{MPLTEXT 1 0 8 " od:\n" } {MPLTEXT 1 0 26 " for i from 1 to 11 do\n" }{MPLTEXT 1 0 56 " e rr[i] := evalf(abs(p(eval_x[i])-f(eval_x[i])));\n" }{MPLTEXT 1 0 8 " \+ od:\n" }{MPLTEXT 1 0 36 " max_err := norm(err, infinity);\n" } {MPLTEXT 1 0 20 " if (k > 1) then\n" }{MPLTEXT 1 0 88 " printf( \"%6f %e %8.4f %6f\\n\", h, max_err, max_err_old/max_err, 2^(n+1 ));\n" }{MPLTEXT 1 0 10 " else \n" }{MPLTEXT 1 0 45 " printf(\" %6f %e \\n\", h, max_err);\n" }{MPLTEXT 1 0 8 " fi;\n" }{MPLTEXT 1 0 122 " points||k := plot([seq([data_x[i],data_y[i]], i=1..n+1)], \+ style=point, symbol=diamond, symbolsize=20, color=blue):\n" } {MPLTEXT 1 0 59 " pp||k := plot(p(x), x=0..n*h, color=red, thickness =2):\n" }{MPLTEXT 1 0 111 " frame||k := display([points||k, pp||k, p f], title=cat(\"Interpolation with h=\",convert(h,string),\".\")):\n" }{MPLTEXT 1 0 14 " h := h/2;\n" }{MPLTEXT 1 0 28 " max_err_old := \+ max_err;\n" }{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(seq(frame||k,k=1..K), insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 45 "Now we look at the right end of the inte rval:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "n := 4:\n" }{MPLTEXT 1 0 26 "data_x := array(1..n+1):\n" }{MPLTEXT 1 0 26 "data_y := array(1..n +1):\n" }{MPLTEXT 1 0 11 "h := 1/n:\n" }{MPLTEXT 1 0 59 "printf(\"Inte rpolation with degree %d polynomial\\n\",n);\n" }{MPLTEXT 1 0 67 "prin tf(\" h Max. error error-ratio h-ratio^(n+1)\\n\");\n" } {MPLTEXT 1 0 10 "K := 8: \n" }{MPLTEXT 1 0 22 "for k from 1 to K do\n" }{MPLTEXT 1 0 25 " for i from 0 to n do\n" }{MPLTEXT 1 0 29 " \+ data_x[i+1] := 1-i*h;\n" }{MPLTEXT 1 0 8 " od:\n" }{MPLTEXT 1 0 27 " for i from 1 to n+1 do\n" }{MPLTEXT 1 0 34 " data_y[i] := f(da ta_x[i]);\n" }{MPLTEXT 1 0 8 " od:\n" }{MPLTEXT 1 0 41 " p := x -> interp(data_x, data_y, x):\n" }{MPLTEXT 1 0 28 " eval_x := array(1. .11):\n" }{MPLTEXT 1 0 23 " err := vector(11):\n" }{MPLTEXT 1 0 26 " for i from 0 to 10 do\n" }{MPLTEXT 1 0 34 " eval_x[i+1] := 1-i /10*n*h;\n" }{MPLTEXT 1 0 8 " od:\n" }{MPLTEXT 1 0 26 " for i from 1 to 11 do\n" }{MPLTEXT 1 0 56 " err[i] := evalf(abs(p(eval_x[i] )-f(eval_x[i])));\n" }{MPLTEXT 1 0 8 " od:\n" }{MPLTEXT 1 0 36 " m ax_err := norm(err, infinity);\n" }{MPLTEXT 1 0 20 " if (k > 1) then \n" }{MPLTEXT 1 0 88 " printf(\"%6f %e %8.4f %6f\\n\", h, m ax_err, max_err_old/max_err, 2^(n+1));\n" }{MPLTEXT 1 0 10 " else \n " }{MPLTEXT 1 0 45 " printf(\"%6f %e \\n\", h, max_err);\n" } {MPLTEXT 1 0 8 " fi;\n" }{MPLTEXT 1 0 122 " points||k := plot([seq ([data_x[i],data_y[i]], i=1..n+1)], style=point, symbol=diamond, sy mbolsize=20, color=blue):\n" }{MPLTEXT 1 0 61 " pp||k := plot(p(x), \+ x=1-n*h..1, color=red, thickness=2):\n" }{MPLTEXT 1 0 111 " frame||k := display([points||k, pp||k, pf], title=cat(\"Interpolation with h= \",convert(h,string),\".\")):\n" }{MPLTEXT 1 0 14 " h := h/2;\n" } {MPLTEXT 1 0 28 " max_err_old := max_err;\n" }{MPLTEXT 1 0 3 "od:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display (seq(frame||k,k=1..K), insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }