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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 174 "Let's  now try to fit a cubic po
lynomial to some data.  Again, maple's built-in package can do this fo
r us, but we like to work harder than we need to, so we understand mor
e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Fir
st, let's get some data, using the provided function" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "read(`/home/mat331/lsq_data`);" }}{PARA 6 "" 1 "" 
{TEXT -1 66 "defined line_pts(), quadratic_pts(), cubic_pts(), and cir
cle_pts()" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "cubdata := cub
ic_pts():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(cubdata,s
tyle=point,axes=boxed);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$77
7$$!1+++?emj?!#:$\"1+++Z7mZ:F*7$$!1+++g=Sm\")!#;$\"1+++\"*G)\\g%F07$$
\"1+++g%yAt\"F0$\"1+++ZOqZIF*7$$!1,++!p#R&y'F0$\"1+++x(e)H=F07$$\"1+++
+yTc>!#<$\"1++++r=w=F*7$$\"1+++x&G<?\"F*$\"1+++g`cK\\F*7$$!1+++tFI&)=F
*$\"1+++7<+8#)F07$$\"1+++(e$\\P;F*$\"1+++X)*4zjF*7$$\"1+++^9&H-#F*$\"1
+++.*p\\_&F*7$$\"1+++a9@(y\"F*$\"1+++3:l#='F*7$$\"1+++a$4qE\"F*$\"1+++
uw-N\\F*7$$\"1+++5nk*y\"F*$\"1+++Jj8\\hF*7$$\"1+++R%=cj\"F*$\"1+++*Qa3
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+AL[M?F*$\"1+++FUe!Q\"F*7$$!1+++DBbL>F*$\"1+++L#HZ:\"F*7$$\"1+++W'3dr#
F*$\"1+++XwY/UF*7$$\"1+++h'oc#HF*$\"1+++@^#**H%F*7$$!1+++jkcT@F*$\"1++
+)>/Y)=F*7$$!1+++1#=+W\"F*$\"1+++t')30KF0-%'COLOURG6&%$RGBG$\"#5!\"\"
\"\"!Fdr-%&STYLEG6#%&POINTG-%*AXESSTYLEG6#%$BOXG" 2 409 409 409 5 0 1 
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{PARA 0 "" 0 "" {TEXT -1 57 "As in the linear case, we will do the wor
k \"by hand\".    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 135 "To simplify typing, let's define the type of function we
 want to fit.  We know (somehow) that our goal is to fit a cubic polyn
omial, so" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "cub := (x,a,b,c,d) ->  a*x^3 + b*x^2 + c*x + d;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cubG:6'%\"xG%\"aG%\"bG%\"cG%\"dG6\"
6$%)operatorG%&arrowGF,,**&9%\"\"\"9$\"\"$F3*&9&F3F4\"\"#F3*&9'F3F4F3F
39(F3F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 203 " Now we define an \"error functional\" which gives us the dist
ance from our cubic to our data.  Note that we use maple's \"nops\"  c
ommand so that we don't need to explicitly say how many points there a
re.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 " G := (a,b,c,d) ->
 sum((cub(cubdata[i][1],a,b,c,d) - cubdata[i][2] )^2,                 \+
       i=1..nops(cubdata));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG:
6&%\"aG%\"bG%\"cG%\"dG6\"6$%)operatorG%&arrowGF+-%$sumG6$*$,&-%$cubG6'
&&%(cubdataG6#%\"iG6#\"\"\"9$9%9&9'F=&F86#\"\"#!\"\"FD/F;;F=-%%nopsG6#
F9F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
85 "Now we just ask maple to find the (unique) critical point of G, wh
ich is our minimum." }}{PARA 0 "" 0 "" {TEXT -1 68 "Since we used assi
gn to set the coefficients, we won't see a result." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 184 "assign(solve( \{diff( G(a,b,c,d), a)=0,\n  \+
             diff( G(a,b,c,d), b)=0,\n               diff( G(a,b,c,d),
 c)=0,\n               diff( G(a,b,c,d), d)=0\},\n              \{a,b,
c,d\} ));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 42 "but we can readily see what the answer is:" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 15 "cub(x,a,b,c,d);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,**$%\"xG\"\"$$!+-D.)o$!#5*$F%\"\"#$\"+\"\\/4#RF)F%$\"+tTFcE!\"*
$\"+&[%3**>F0\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 58 "Finally, lets mak
e a picture, just to see how well it fits" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 124 "with(plots):\ncplot:= plot(cub(x,a,b,c,d), x=-3..3, \+
axes=boxed):\npplot:= plot(cubdata, style=point):\ndisplay(\{cplot, pp
lot\});" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7in7$$!\"$\"\"!$\"
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\"F-7$$!1+++DBbL>F-$\"1+++L#HZ:\"F-7$$\"1+++W'3dr#F-$\"1+++XwY/UF-7$$
\"1+++h'oc#HF-$\"1+++@^#**H%F-7$$!1+++jkcT@F-$\"1+++)>/Y)=F-7$$!1+++1#
=+W\"F-$\"1+++t')30KFcpFg^l-%&STYLEG6#%&POINTG-%+AXESLABELSG6$%\"xG%!G
-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F(Fc^l%(DEFAULTG" 2 409 409 409 2 0 1 
0 2 9 0 2 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "just for comparison, we can let ma
ple do this for us using the built-in least-squares package:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 119 "fit[leastsquare[[x,y], y=A*x^3+B*x^2+C*x+D]]([[cub
data[i][1]$i=1..nops(cubdata)],[cubdata[i][2] $i=1..nops(cubdata)]]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,**$%\"xG\"\"$$!+3D.)o$!#5*$
F'\"\"#$\"+0X!4#RF+F'$\"+xTFcE!\"*$\"+![%3**>F2\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "29 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 }