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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "#This program will \+
graph the eigenvalues versus normal derivatives for the 5-series, for \+
each j, on a log-log scale.  (Note that this is natural log, not base \+
10)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "with(plots): with(C
urveFitting): read \"values_5.m\"; Digits:=20;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'DigitsG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "#To plot points, \+
we need to put our values into a sequence of ordered pairs, so that's \+
what I'm doing here." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for i from 1 to 512 do " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y[i]:=log(-dn5[i]):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "for j from 1 to 512 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x[j]:
=log(El5[j]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "data_list:= [seq([x[i],y[i]],i=1..5
12)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "#And here's the graph.  Ooh, pretty." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "pointplot(data_list,title=
\"logarithmic graph of eigenvalue vs. normal derivative for 5 series\"
, labels=[\"Eigenvalues\",\"Normal Derivatives\"]);" }}{PARA 0 "> " 0 
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jM+MF#3&=FW$\"51p.7lh@o>!*F)7$$\"5][jxI!zC3&=FW$\"5c^XjScKN!G*F)7$$\"5
s)*oF#*o]#3&=FW$\"5-xKN38;%Qs*F)-%+AXESLABELSG6$Q,Eigenvalues6\"Q3Norm
al~DerivativesFf[u-%&TITLEG6#Q]ologarithmic~graph~of~eigenvalue~vs.~no
rmal~derivative~for~5~seriesFf[u" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "#It se
ems that the maximum point in each grouping falls along something of a
 straight line.  Let's investigate that.  First we need to make a new \+
sequence of ordered pairs involving only the maximum points.  It turns
 out that those points are 2^i, which are sequences with only one + ch
osen for epsilon." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 0 to 9 do " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b[i]:=log(-dn5[2^i]):" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for j from 0 to 9 do
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a[j]:=log(El5[2^j]):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "#These are the values for those
 maximum points (on a logarithmic scale)." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 " data_list2:= [seq([a[i],b[i]],i=0..9)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "data_list2;" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#7,7$$\"5c\"\\w,T\"3JBS!#>$\"5L!)G<LwJ&pd#F'7$$\"5D
$yx'zR:h\\^F'$\"500VdZ\"4<mp#F'7$$\"5CZlKl>kZSrF'$\"5na>![)=?U#*RF'7$$
\"5e*p\\KuPh3$))F'$\"5Ie9Exo,rv[F'7$$\"5p.MX-7!*oX5!#=$\"5!*[9\\C3?F'p
&F'7$$\"54\")[^H4l'p?\"F<$\"58'[\"*p*\\:</lF'7$$\"56TvT#*>r(zO\"F<$\"5
QI.$>z!p_4tF'7$$\"5;<rL>yU$*G:F<$\"54Wsc:\"[tV6)F'7$$\"5V`d!oHu!))*o\"
F<$\"5FId))Q1z6>*)F'7$$\"5s)*oF#*o]#3&=F<$\"5-xKN38;%Qs*F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "#And here's what they look like." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 97 "pointplot(data_list2, title=\"Maximum Points
 Graph\", labels=[\"Eigenvalues\",\"Normal Derivatives\"]);" }}{PARA 
13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6%-%'POINTSG6,7$$\"5R>wt
3[2LAG!#>$\"5\"4^GM&*>Ayo\"F)7$$\"5I8XqQh6M\"[&F)$\"5Dh\\S@iS5\\GF)7$$
\"5i_SNqsz.6sF)$\"5@m,pC.6SsPF)7$$\"5ZW.0k26NX))F)$\"5]i6%H0744g%F)7$$
\"5B[:$))Qf!)f/\"!#=$\"5$3WH1uY5/T&F)7$$\"5mHH*4(**[-27F>$\"5I_3g+/s3;
iF)7$$\"5ud\\)*)4!)))zO\"F>$\"5@$zm&R`t*4-(F)7$$\"5'eS)f_9m$*G:F>$\"5L
O#)zy\\\\vDyF)7$$\"5-fg$R-@\"))*o\"F>$\"5!3;jWtp\"[I')F)7$$\"5%G3AxB;D
3&=F>$\"5!*zcVyXC?N%*F)-%+AXESLABELSG6$Q,Eigenvalues6\"Q3Normal~Deriva
tivesFhn-%&TITLEG6#Q5Maximum~Points~GraphFhn" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "#It'
s nearly .5 (and we're not using many points, so the approximation won
't be fantastic) which indicates that the normal derivatives are of th
e order of the square root of the eigenvalue, just like on the unit in
terval.  Now, we'll take points with better accuracy (those more to th
e right) and see if this idea holds." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "#Using \+
a Least Squares Method, we can see what the slope of the line is." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "LeastSquares(data_list2,v);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"5&4GOp5<Q&o>!#?\"\"\"*&$\"5yw
i6knlG')\\F&F'%\"vGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
d[0]:=log(-dn5[1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " for i from \+
1 to 8 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "d[i]:=log(-dn5[2^(i+1
)]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6#\"\"!$\"5
L!)G<LwJ&pd#!#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "c[0]:=lo
g(El5[1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for j from 1 to 8 do
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "c[j]:=log(El5[2^(j+1)]):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"!$\"5c\"\\w,T\"3
JBS!#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " data_list3:= [se
q([c[i],d[i]],i=0..8)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+data_lis
t3G7+7$$\"5c\"\\w,T\"3JBS!#>$\"5L!)G<LwJ&pd#F)7$$\"5CZlKl>kZSrF)$\"5na
>![)=?U#*RF)7$$\"5e*p\\KuPh3$))F)$\"5Ie9Exo,rv[F)7$$\"5p.MX-7!*oX5!#=$
\"5!*[9\\C3?F'p&F)7$$\"54\")[^H4l'p?\"F9$\"58'[\"*p*\\:</lF)7$$\"56TvT
#*>r(zO\"F9$\"5QI.$>z!p_4tF)7$$\"5;<rL>yU$*G:F9$\"54Wsc:\"[tV6)F)7$$\"
5V`d!oHu!))*o\"F9$\"5FId))Q1z6>*)F)7$$\"5s)*oF#*o]#3&=F9$\"5-xKN38;%Qs
*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "pointplot(data_list3
, title=\"Maximum Points Graph\", labels=[\"Eigenvalues\",\"Normal Der
ivatives\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 {PLOTDATA 2 "6
%-%'POINTSG6+7$$\"5c\"\\w,T\"3JBS!#>$\"5L!)G<LwJ&pd#F)7$$\"5CZlKl>kZSr
F)$\"5na>![)=?U#*RF)7$$\"5e*p\\KuPh3$))F)$\"5Ie9Exo,rv[F)7$$\"5p.MX-7!
*oX5!#=$\"5!*[9\\C3?F'p&F)7$$\"54\")[^H4l'p?\"F9$\"58'[\"*p*\\:</lF)7$
$\"56TvT#*>r(zO\"F9$\"5QI.$>z!p_4tF)7$$\"5;<rL>yU$*G:F9$\"54Wsc:\"[tV6
)F)7$$\"5V`d!oHu!))*o\"F9$\"5FId))Q1z6>*)F)7$$\"5s)*oF#*o]#3&=F9$\"5-x
KN38;%Qs*F)-%+AXESLABELSG6$Q,Eigenvalues6\"Q3Normal~DerivativesFY-%&TI
TLEG6#Q5Maximum~Points~GraphFY" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "LeastSquares(data_list3,v);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"5]PkhOXn^k]!#?\"\"\"*&$\"5!Q(
G$pf\"HYs\\F&F'%\"vGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"for i from 1 to 8 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f[i]:=log
(-dn5[2^(i+1)]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }{TEXT -1 
0 "" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for j fro
m 1 to 8 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e[j]:=log(El5[2^(j+1
)]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "data_list4:= [seq([e[i],f[i]],i=1..8)];" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+data_list4G7*7$$\"5CZlKl>kZSr!#>$\"
5na>![)=?U#*RF)7$$\"5e*p\\KuPh3$))F)$\"5Ie9Exo,rv[F)7$$\"5p.MX-7!*oX5!
#=$\"5!*[9\\C3?F'p&F)7$$\"54\")[^H4l'p?\"F4$\"58'[\"*p*\\:</lF)7$$\"56
TvT#*>r(zO\"F4$\"5QI.$>z!p_4tF)7$$\"5;<rL>yU$*G:F4$\"54Wsc:\"[tV6)F)7$
$\"5V`d!oHu!))*o\"F4$\"5FId))Q1z6>*)F)7$$\"5s)*oF#*o]#3&=F4$\"5-xKN38;
%Qs*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "pointplot(data_li
st4, title=\"Maximum Points Graph\", labels=[\"Eigenvalues\",\"Normal \+
Derivatives\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 {PLOTDATA 
2 "6%-%'POINTSG6*7$$\"5CZlKl>kZSr!#>$\"5na>![)=?U#*RF)7$$\"5e*p\\KuPh3
$))F)$\"5Ie9Exo,rv[F)7$$\"5p.MX-7!*oX5!#=$\"5!*[9\\C3?F'p&F)7$$\"54\")
[^H4l'p?\"F4$\"58'[\"*p*\\:</lF)7$$\"56TvT#*>r(zO\"F4$\"5QI.$>z!p_4tF)
7$$\"5;<rL>yU$*G:F4$\"54Wsc:\"[tV6)F)7$$\"5V`d!oHu!))*o\"F4$\"5FId))Q1
z6>*)F)7$$\"5s)*oF#*o]#3&=F4$\"5-xKN38;%Qs*F)-%+AXESLABELSG6$Q,Eigenva
lues6\"Q3Normal~DerivativesFT-%&TITLEG6#Q5Maximum~Points~GraphFT" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "LeastSquares(data_list4,v);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"5Rovu8^%3dD%!#?\"\"\"*&$\"59B
%f%eAFcG]F&F'%\"vGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f
or i from 4 to 9 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "h[i]:=log(-
dn5[2^(i)]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }{TEXT -1 0 "" 
}{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for j from 4 t
o 9 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g[j]:=log(El5[2^(j)]):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "data_list5:=[seq([g[i],h[i]],i=4..9)];" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%+data_list5G7(7$$\"5p.MX-7!*oX5!#=$\"5!*[9\\C3
?F'p&!#>7$$\"54\")[^H4l'p?\"F)$\"58'[\"*p*\\:</lF,7$$\"56TvT#*>r(zO\"F
)$\"5QI.$>z!p_4tF,7$$\"5;<rL>yU$*G:F)$\"54Wsc:\"[tV6)F,7$$\"5V`d!oHu!)
)*o\"F)$\"5FId))Q1z6>*)F,7$$\"5s)*oF#*o]#3&=F)$\"5-xKN38;%Qs*F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "pointplot(data_list5, title=
\"Maximum Points Graph\", labels=[\"Eigenvalues\",\"Normal Derivatives
\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 {PLOTDATA 2 "6%-%'POIN
TSG6(7$$\"5p.MX-7!*oX5!#=$\"5!*[9\\C3?F'p&!#>7$$\"54\")[^H4l'p?\"F)$\"
58'[\"*p*\\:</lF,7$$\"56TvT#*>r(zO\"F)$\"5QI.$>z!p_4tF,7$$\"5;<rL>yU$*
G:F)$\"54Wsc:\"[tV6)F,7$$\"5V`d!oHu!))*o\"F)$\"5FId))Q1z6>*)F,7$$\"5s)
*oF#*o]#3&=F)$\"5-xKN38;%Qs*F,-%+AXESLABELSG6$Q,Eigenvalues6\"Q3Normal
~DerivativesFJ-%&TITLEG6#Q5Maximum~Points~GraphFJ" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "LeastSquares(data_list5,v);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&$\"5*>!yl$RV'*em%!#?\"\"\"*&$\"5A5N2$=R()=+&F&F'%\"
vGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 117 "#As we can see, our conjecture was corre
ct.  The slope of the line gets closer to .5 as we take more accurate \+
points." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "42" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }