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{SECT 0 {PARA 202 "" 0 "" {TEXT 203 15 "Runge's Example" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "restart;\n" }{MPLTEXT 1 0 12 "with(
plots):" }}}{SECT 1 {PARA 206 "" 0 "" {TEXT 200 16 "Runge's Function" 
}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 52 "Definition of Runge's function,
 and graph on [-1,1]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := x -> \+
1/(1+(5*x)^2);\n" }{MPLTEXT 1 0 54 "pf := plot(f(x), x=-1..1, color=gr
een, thickness=2):\n" }{MPLTEXT 1 0 12 "display(pf);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 207 "" 0 "" {TEXT 200 
46 "Polynomial Interpolation at Equidistant Points" }}{EXCHG {PARA 0 "
" 0 "" {TEXT 204 7 "Define " }{XPPEDIT 18 0 "Typesetting:-mrow(Typeset
ting:-mi(\"n\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslib
GF'6#-I#miGF$6#Q\"nF'" }{TEXT 204 49 ", the degree of the polynomial t
o be used (i.e., " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(
\"\"), Typesetting:-mrow(Typesetting:-mi(\"n\"), Typesetting:-mo(\"+\"
, form = \"infix\", fence = \"false\", separator = \"false\", lspace =
 \"mediummathspace\", rspace = \"mediummathspace\", 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:-mn(\"1\")), Typesetting
:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%
-I#miGF$6#Q!F'-F#6%-F,6#Q\"nF'-I#moGF$63Q\"+F'/%%formGQ&infixF'/%&fenc
eGQ&falseF'/%*separatorGF=/%'lspaceGQ0mediummathspaceF'/%'rspaceGFB/%)
stretchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(l
argeopGF=/%.movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~Commen
tF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255
]F'-I#mnGF$6#FNF+" }{TEXT 204 35 " distinct points are interpolated)."
 }}{PARA 0 "" 0 "" {TEXT 204 34 "Also define 2 arrays needed below." }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "n := 15;\n" }{MPLTEXT 1 0 26 "data
_x := array(1..n+1):\n" }{MPLTEXT 1 0 24 "data_y := array(1..n+1):" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 204 55 "Compute interpolation points (eq
ually spaced in [-1,1])" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i fr
om 0 to n do\n" }{MPLTEXT 1 0 31 "   data_x[i+1] := -1 + i*2/n;\n" }
{MPLTEXT 1 0 3 "od:" }{TEXT 204 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
204 52 "Generate data, i.e., sample Runge's function at the " }
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row(Typesetting:-mi(\"n\"), Typesetting:-mo(\"+\", form = \"infix\", f
ence = \"false\", separator = \"false\", lspace = \"mediummathspace\",
 rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \"fal
se\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mo
vablelimits = \"false\", accent = \"false\", font_style_name = \"2D Co
mment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,
255,255]\"), Typesetting:-mn(\"1\")), Typesetting:-mi(\"\"));" "-I%mro
wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F
,6#Q\"nF'-I#moGF$63Q\"+F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separat
orGF=/%'lspaceGQ0mediummathspaceF'/%'rspaceGFB/%)stretchyGF=/%*symmetr
icGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF=/%.movableli
mitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+f
oregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-I#mnGF$6#FNF+" }
{TEXT 204 18 " points of data_x." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 
"for i from 1 to n+1 do\n" }{MPLTEXT 1 0 31 "   data_y[i] := f(data_x[
i]);\n" }{MPLTEXT 1 0 4 "od: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 12 
"Use Maple's " }{HYPERLNK 17 "interp" 2 "interp" "" }{TEXT 204 47 " co
mmand to compute the polynomial interpolant." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "p := x -> interp(data_x, data_y, x):\n" }{MPLTEXT 1 
0 12 "'p(x)'=p(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 55 "Plot inter
polant together with original Runge function " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 126 "points := plot([seq([data_x[i],data_y[i]], i=1..n+1)
], style=point, symbol=diamond, symbolsize=20, color=blue, thickness=2
):\n" }{MPLTEXT 1 0 52 "pp := plot(p(x), x=-1..1, color=red, thickness
=2):\n" }{MPLTEXT 1 0 24 "display(points, pp, pf);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 208 "" 0 "" {TEXT 200 
44 "Polynomial Interpolation at Chebyshev Points" }{TEXT 205 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 204 24 "Compute Chebyshev points" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i from 0 to n do\n" }{MPLTEXT 
1 0 51 "   data_x[i+1] := evalf(cos((2*i+1)/(2*n+2)*Pi));\n" }{MPLTEXT
 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 62 "Generate data, i.
e., sample Runge function at Chebyshev points" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "for i from 1 to n+1 do\n" }{MPLTEXT 1 0 31 "   data_y
[i] := f(data_x[i]);\n" }{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 204 30 "Compute polynomial interpolant" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "p := x -> interp(data_x, data_y, x):\n" }{MPLTEXT 1 
0 14 "'p(x)' = p(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 55 "Plot int
erpolant together with original Runge function " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 126 "points := plot([seq([data_x[i],data_y[i]], i=1..n+1)
], style=point, symbol=diamond, symbolsize=20, color=blue, thickness=2
):\n" }{MPLTEXT 1 0 52 "pp := plot(p(x), x=-1..1, color=red, thickness
=2):\n" }{MPLTEXT 1 0 24 "display(points, pp, pf);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}}
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