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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 2-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 51 "with(plots): with(C
urveFitting): read \"values_2.m\";" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "#To pl
ot 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 \+
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0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 303 "#It seems that the maximum point in each grouping falls along s
omething of a straight line.  Let's investigate that.  First we need t
o make a new sequence of ordered pairs involving only the maximum poin
ts.  It turns out that those points are 2^i, which are sequences with \+
only one + chosen 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 20 "b[i]:=log(-dn[2^i]):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{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 19 "a[j]:=log(El[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 39 "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$$
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "#And here's what they look like. " }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 97 "pointplot(data_list2, title=\"Maximum Point
s Graph\", labels=[\"Eigenvalues\",\"Normal Derivatives\"]);" }}{PARA 
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0 "" {MPLTEXT 1 0 72 "#Using a Least Squares Method, we can see what t
he slope of the line is." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"LeastSquares(data_list2,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+
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320 "#It's nearly .5 (and we're not using many points, so the approxim
ation won't be fantastic) which indicates that the normal derivatives \+
are of the order of the square root of the eigenvalue, just like on th
e unit interval.  Now, we'll take points with better accuracy (those m
ore to the right) and see if this idea holds." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fo
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1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 3 to \+
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ta_list4, title=\"Maximum Points Graph\", labels=[\"Eigenvalues\",\"No
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}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "#As we can see, our conjec
ture was correct.  The slope of the line gets closer to .5 as we take \+
more accurate points." }}}}{MARK "2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 
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