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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "#This \+
program calculates the coefficients with which we sum Pj3 and Pj2 to c
reate Pj4.  b's go with Pj2, c's go with Pj3.  All formulas used herei
n were taken from the \"Calculus on the Sierpinski Gasket\" by Prof. S
trichartz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "#Hard coding the necessary startin
g points. I calculate the first b1 and c1 through hard coding since th
ere aren't formulas for some of the necessary elements.  The matrix fo
rmula is derived from identities found in the above paper and the nece
ssary matching conditions at junction points." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "al
pha[0]:=1; alpha[1]:=1/6; beta[0]:=-1/2; n[0]:=0; beta[1]:=-2/45; n[1]
:=1/2; b[0]:=-4; c[0]:=-2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "M:=Ma
trix(2,2,[[(1/2)*(2/5),(1/2)*(1+3/5)],[(1/2)*(1+5/5)]]);" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 18 "M_inv:=inverse(M);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&alphaG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%&alphaG6#\"\"\"#F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&
%%betaG6#\"\"!#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#
\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"\"\"#!\"#\"#X
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"\"\"#F'\"\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"!!\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"cG6#\"\"!!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"MG-%'RTABLEG6%\"*_[5^\"-%'MATRIXG6#7$7$#\"\"\"\"\"&#\"\"%F07$F/\"\"
!%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7
$\"\"!\"\"\"7$#\"\"&\"\"%#!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "M_inv := matrix([[0, 1], [5/4, -1/4]]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$\"\"!\"\"\"7$#\"\"&\"\"
%#!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "A:=Matrix(2,1,
[[((2/5)*b[0]*beta[1]/5)-(3*(1+3/5)*c[0]*(1/180)/5)],[-2*b[0]*alpha[1]
/5]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x_1:=evalm (M_inv &* A);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*?&)4^\"-%'MAT
RIXG6#7$7##\"#G\"%D67##\"\"%\"#:%'MatrixG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$x_1G-%'matrixG6#7$7##\"\"%\"#:7##!\")\"$D#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 
0 15 "b[1]:=x_1[1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"
\"#\"\"%\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "c[1]:=x_1[2
,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"\"#!\")\"$D#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "alpha[2]:=1/180;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"\"##\"\"\"\"$!=" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "n[2]:=1/36;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"nG6#\"\"##\"\"\"\"#O" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "#And n
ow we can calculate up to as high a level as we want, just by altering
 which j value we go to.  Beware, I use rational arithmetic to preserv
e accuracy, so once we reach j=20 or so, the values take up most of th
e page.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "for j from 2 to 22 do " }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 85 "M:=Matrix(2,2,[[(1/2)*(1-3/5^j),(1/2)*(1+3/5^j)],[(1/2)*(1+5/5
^j),(3/2)*(1-5/5^j)]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M_inv:=i
nverse(M);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "alpha[j+1]:= (4/((5^(
j+1))-5))*sum( 'alpha[j+1-k]*alpha[k]' ,'k'=1..j);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 136 "beta[j]:=(1/((5^j)-1))*sum('(2/5)*(5^(j-k))*alpha[
j-k]*beta[k] - (2/3)*alpha[j-k]*(5^k)*beta[k] + (4/5)*alpha[j-k]*beta[
k]','k'=0..j-1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "n[j+1]:=(((5^(j
+1))+1)/2)*alpha[j+1]+2*sum('n[k]*beta[j+1-k]','k'=0..j);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 239 "A:=Matrix(2,1,[[(1-3/5^j)*sum('b[i]*beta[j-
i]/5^(j-i)', 'i'=0..j-1)-3*(1+3/5^j)*sum('c[i]*alpha[j-i+1]/5^(j-i)','
i'=0..j-1)],[-(1+5^(1-j))*sum('b[i]*alpha[j-i]/5^(j-i)','i'=0..j-1) -3
*(1-5^(1-j))*sum('c[i]*n[j-i+1]/5^(j-i)','i'=0..j-1)]]);" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "x_j:=evalm (M_inv &* A);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "b[j]:=x_j[1,1]; c[j]:=x_j[2,1];" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'
RTABLEG6%\"*_Z0^\"-%'MATRIXG6#7$7$#\"#6\"#D#\"#9F07$#\"\"$\"\"&#\"\"'F
6%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7
$#\"#D\"\"%#!#N\"#77$#!#D\"\")#\"#b\"#C" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%&alphaG6#\"\"$#\"\"\"\"&+i\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%%betaG6#\"\"##!#\\\"&+'[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
\"nG6#\"\"$#\"\"\"\"%IC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RT
ABLEG6%\"*W)H7:-%'MATRIXG6#7$7##!%9o\"(voz$7##!$e%\"&D1&%'MatrixG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_jG-%'matrixG6#7$7##\"$c#\"&vo\"7#
#!%(H#\"'v=:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"##\"$c#\"
&vo\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"##!%(H#\"'v=:" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*%G>7:-%'MATRIXG6
#7$7$#\"#h\"$D\"#\"#kF07$#\"#8\"#D#\"#OF6%'MatrixG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$#\"%D6\"$T$#!$+%F,7$#!%D;\"%
k8#\"%D:F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"\"%#\"\"\"
\"(+K,$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"\"$#!#H\"(])*Q
$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"\"%#\"#P\")+%fN\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*'R)>^\"-%'MATRIXG
6#7$7##!)<#3>\"\"-Dc,;C87##!&<-%\"*DJq<\"%'MatrixG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$x_jG-%'matrixG6#7$7##\"&x\\\"\"*v$fQ97##!('*H?$\",
v$4El6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"$#\"&x\\\"\"*v$
fQ9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"$#!('*H?$\",v$4El6
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*?M@^\"-%'MATR
IXG6#7$7$#\"$6$\"$D'#\"$9$F07$#\"#j\"$D\"#\"$'=F6%'MatrixG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$#\"&v$>\"%Wj#!&D'>
\"&K!>7$#!&DJ\"\"&)o7#\"&v)Q\"&k!Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%&alphaG6#\"\"&#\"\"\"\"*++Hz*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&
%%betaG6#\"\"%#!%p\"*\"-++ZnzB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
\"nG6#\"\"&#\"#Z\"++%\\$fZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%
'RTABLEG6%\"*gU>^\"-%'MATRIXG6#7$7##!,**ztXJ\"\"2]iS;=\"HcD7##!)^df&*
\"/]i!R+u\\&%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_jG-%'mat
rixG6#7$7##\"'fAW\".+vV3s)>7##!,\"=k\\T7\"1+]7t2wz**" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>&%\"bG6#\"\"%#\"'fAW\".+vV3s)>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"cG6#\"\"%#!,\"=k\\T7\"1+]7t2wz**" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*3d%R:-%'MATRIXG6#7$7$#\"%h:\"%D
J#\"%k:F07$#\"$8$\"$D'#\"$O*F6%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&M_invG-%'matrixG6#7$7$#\"']7t\"'\"*GC#!'vVCF,7$#!'D\"y*\"'k:(
*#\"'Dc(*F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"\"'#\"$H)
\"0++gkd+8%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"\"&#!&y9&
\"0v=#3U5F_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"\"'#\"&$[8\"
0++K`[C3'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*+.\"
>;-%'MATRIXG6#7$7##!1>r.\")3jR6\":+vVBSi9Y#4_X77##!.$e\"Rg)y7\"6+v=nj_
>\">eU%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_jG-%'matrixG6#
7$7##\"+2d/#\\$\"5+DcwM5\"[zI\"7##!0rD)*3E#)H\"\"8]iS\"pB)377x>'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"&#\"+2d/#\\$\"5+DcwM5\"[z
I\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"&#!0rD)*3E#)H\"\"8
]iS\"pB)377x>'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"
*GOej\"-%'MATRIXG6#7$7$#\"%6y\"&Dc\"#\"%9yF07$#\"%j:\"%DJ#\"%'o%F6%'Ma
trixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$#\")
DJ?7\"(W[1%#!)v$4A\"\")KX>77$#!(D19)\"()oH\")#\")v$4W#\")k!*QC" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"\"(#\"%ra\"4++]fck+l,#" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"\"'#!)r8hS\"6++]'Rp'QT+
5#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"\"(#\"&f-\"\"3++5>3NP
fI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*!)ppi\"-%'M
ATRIXG6#7$7##!6RtdS+!faCz!)\"B+DJX9)481ZgC_wr5&)7##!3>N^&*Q_0oY\"?+DcE
s!\\Sa-h$oD)f\"%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_jG-%'
matrixG6#7$7##\"-doTVJK\":](=Ud)y#p%[;eM%7##!5<k)GZT<h>z%\"A+]PMF!=t;6
dmC*HHC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"'#\"-doTVJK\":
](=Ud)y#p%[;eM%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"'#!5<k
)GZT<h>z%\"A+]PMF!=t;6dmC*HHC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"M
G-%'RTABLEG6%\"*3Ski\"-%'MATRIXG6#7$7$#\"&h!R\"&D\"y#\"&k!RF07$#\"%8y
\"&Dc\"#\"&OM#F6%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M_invG
-%'matrixG6#7$7$#\"*vVtd%\"*Tmb_\"#!*]Pf_\"F,7$#!*D1R5'\"*klA5'#\"*D\"
G.hF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"\")#\")BYb7\":+
++SL3XK*G6EZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"\"(#!,joD
t)\\\"<++DO!HtV&4c0+Y'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"
\")#\"*Z+h>(\";++];y\">b(=Sp()>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"AG-%'RTABLEG6%\"*kJEj\"-%'MATRIXG6#7$7##!=29ArSs=+ui+&z+$\"M++DJ?$y_
.LD[ilg['ejg/o;7##!9r7\\[CU%z8\"eup\"H++DcE(>xym#[Le)[,jarz%'MatrixG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_jG-%'matrixG6#7$7##\"6>3zHL!GdH
J\"*\"F++vV[r204+]1j`c,Kt#7##!;;>%)f:)fQ)4x4Ts\"Kvoz;Z3V_3k(o7M#R@C'GB
/\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"(#\"6>3zHL!GdHJ\"*
\"F++vV[r204+]1j`c,Kt#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"
\"(#!;;>%)f:)fQ)4x4Ts\"Kvoz;Z3V_3k(o7M#R@C'GB/\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*?Y_N\"-%'MATRIXG6#7$7$#\"'6`>\"'D1
R#\"'9`>F07$#\"&j!R\"&D\"y#\"''=<\"F6%'MatrixG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$#\"+voHHw\"+Wt-VD#!+DJXHw\"+K?
3Hw7$#!+D\"Gj3&\"+)oag3&#\",v=ne_\"\",kS;e_\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&alphaG6#\"\"*#\"*d\\;y'\"?+++!G))RRcq_g*\\hM" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"\")#!0LcA;4B&Q\"B+++Skezs
&Qd-ITTWK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"\"*#\"/>&[pmDl
\"\"B+++qiFR-D/XvsCxn&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTA
BLEG6%\"*w7Bj\"-%'MATRIXG6#7$7##!F^BRL*z!\\J'RRP:[5J@e5\"W+]i:&yznjQ*=
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\\@7%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_jG-%'matrixG6#7$
7##!<(4`N9U]%GAC3L+*\"N++]7`Wcyw]AuZG5C)QF6P3$*7##!FHOGUbN0`AN4o3>iZ3+
\"\"W+++D1*G'>lo0W<!\\hMw34v'e?&4o:(" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>&%\"bG6#\"\")#!<(4`N9U]%GAC3L+*\"N++]7`Wcyw]AuZG5C)QF6P3$*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\")#!FHOGUbN0`AN4o3>iZ3+\"
\"W+++D1*G'>lo0W<!\\hMw34v'e?&4o:(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"MG-%'RTABLEG6%\"*w1Kj\"-%'MATRIXG6#7$7$#\"'hl(*\"(DJ&>#\"'kl(*F07$
#\"'8`>\"'D1R#\"'OfeF6%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
M_invG-%'matrixG6#7$7$#\"-+Dc,hG\",\"R]mO&*#!,D\"yvO&*F,7$#!-DJqq9Q\"-
k:gm9Q#\"-D19p9QF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"#5
#\".rNxV3e\"\"F++++!*)y#\\FfofnR2([39" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%%betaG6#\"\"*#\"68]0[1V5EcD#\"J+++v`3Wj&pu.^B<:Yd')=J" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"#5#\"0\\m7oFU5'\"G+++gP+A>G)o\\3x
`L?*)Q$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*SXUj\"
-%'MATRIXG6#7$7##\"NRQgol:.?'))RqC`ICQx\"3K%p_\"]o+++Dc,TX:_Xsx:6\"eI&
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rixG6#7$7##\"D^=\"31%eS#\\YyWCq,u@>\"X+++D19>rR8&y^yT.WZ_Vx$pM-pdX&7##
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gR\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"*#\"D^=\"31%eS#\\
YyWCq,u@>\"X+++D19>rR8&y^yT.WZ_Vx$pM-pdX&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"cG6#\"\"*#\"NJ1%ecl<k*f!e$4.#\\loVb@M%4&\"]o++]7`W
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>%\"MG-%'RTABLEG6%\"*oSRj\"-%'MATRIXG6#7$7$#\"(6G)[\"(Dcw*#\"(9G)[F07$
#\"'jl(*\"(DJ&>#\"('oHHF6%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&M_invG-%'matrixG6#7$7$#\".D19p$oZ\".W)*ea%*e\"#!.vo/t$oZ\".K&pPOoZ7
$#!.Dc,;*yJ\".)oz\"4*yJ#\".vVBSn`*\".k!RvsO&*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&alphaG6#\"#6#\"1\"\\mRAJ/5\"\"L+++](eugtGNMhBMo\"e
\\!=\\@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"#5#!<pER\"*fA%
H!4X?F'=\"R+++]incWY3l()Hk&)4E=Kc*y[EM" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"nG6#\"#6#\"7n4fM'oZj8z4\"\"Q++++b^2()R4b*>0AnFq6%)f&*fC\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*78Pj\"-%'MATRIX
G6#7$7##!en@')e;7_bumZiv$Gedsz;H()***>/2,f2^\"\\p+++]7GjtatvJvQ\"yVv#e
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{XPPMATH 20 "6#>%$x_jG-%'matrixG6#7$7##!Px#)4M%=uNgN!=7G?%*=KO#**=5EF
\"ao++]7G86kpg&He(=aG\")pEPM)GB&H-hf4@kC`k@7##!enJS)*3RZi8x(*f7mlKSrh#
QuKOw6T'=]5\"[p++++D19>@#4n0cg)[SkC6XG`\"o7%=I*z8T8nq&Q&GK)y" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#5#!Px#)4M%=uNgN!=7G?%*=KO#**=5EF
\"ao++]7G86kpg&He(=aG\")pEPM)GB&H-hf4@kC`k@" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"cG6#\"#5#!enJS)*3RZi8x(*f7mlKSrh#QuKOw6T'=]5\"[p++
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{XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*W`Oj\"-%'MATRIXG6#7$7$#\")hSTC\")D
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1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$#\"0Dc,;R\")y\"\"/TTCDYgf
#!/+vV[YgfF,7$#!0Dc^.'=%Q#\"0kl(4]=%Q##\"0DJXk&=%Q#F2" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%&alphaG6#\"#7#\"8F!pFO#fwaH^X'\"W+++++SF/2(oE@Q
>'**oJ(zFRmjq+S$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"#6#\"
B$3li%=@Sm\"[KnLm32V\"en++++]#p/vo=/&GlZeJW()p-h%3>qdb2\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"#7#\"<>vW29:gJc[2%35\"Y+++](yejY'*=)
*\\W(\\B'p>MN$*oVFKd*G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTA
BLEG6%\"*guYj\"-%'MATRIXG6#7$7##\"coZBoAxV$4\"o!)4DT=#G$p(*GJN9t5=FWmc
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{XPPMATH 20 "6#>%$x_jG-%'matrixG6#7$7##\"gnx\\bVb'>=cFS.G[d4R``k)>V3A^
%yU%=7\"`p+++]iS\">@UEH95RI4>dSmBW\"G)ekW;()44eQPxMFSV>.F7##\"cof&pCLW
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!\\SG;ZMX1(H$3Q(GCAf/4^#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#
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#6#\"cof&pCLWOFFS8SG#)=Env1lO(o!)>'fys9O[_SeP>\"\"\\q++](ozT.5f8[A9I_s
[&e?L(*>yJszP!\\SG;ZMX1(H$3Q(GCAf/4^#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"MG-%'RTABLEG6%\"*Coaj\"-%'MATRIXG6#7$7$#\"*6.2A\"\"*D19W##\"*9.
2A\"F07$#\")jSTC\")D\"G)[#\")'=UK(F6%'MatrixG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$#\"1vVtx@B!)H\"0WB:\"o5M**#!1D
1RvAB!)H\"1KqXV?B!)H7$#!1DJ?L:#o)>\"1)o/BO@o)>#\"1voH/WYgf\"1kS\"p3k/'
f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"#8#\"<p')\\r=;*oeK
\"4%y5\"hn+++++]\\,Pr;uuw?ZlOjZgY=P8RW\"*G&=" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%%betaG6#\"#7#!I@%pS3-\\U1@n$)oWZ7nuks)\"^o+++++])Gh3
(Qslrh\"yVO%[9YWfU()HMbW2T&H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"n
G6#\"#8#\"ATN-[Ou'Q;1I\")Giw:#\"\\o+++++qcb+a,%\\)o\"=q(zCAOQn,3X78xve
!>" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*w$HN;-%'MAT
RIXG6#7$7##!bpd*=A.NP6+nRd&>5l(*\\BdEjS?(H/g4s20ktt$\\w&Q(om\"z'[$\"^r
++]PMx#4X$>-v))\\^g>pPsniEG'oa\\e*fzq1jnijk:#y'o0A[K_8#fH#zy!fh/@7##!h
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KuCtB$[.wgY,3A?x\\kO%o@$*>j!3\\>^O3\\xB$%'MatrixG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$x_jG-%'matrixG6#7$7##!boH)4U'H$z0vQ0jNs\"Q!4!f0GGn0q
q-sCCh&*))[x`&\"^q+++D\"yD9'H(>?68na+&[_RnXWfM,%=T0CMuaS4mIsASuU7UZm,L
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\"f\\D6@t(*oAYarEg(eLixcH2?Q&oMqQ!3J/;(43[mI;@h@bxePE?\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#7#!boH)4U'H$z0vQ0jNs\"Q!4!f0GGn0qq-s
CCh&*))[x`&\"^q+++D\"yD9'H(>?68na+&[_RnXWfM,%=T0CMuaS4mIsASuU7UZm,LM" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"#7#!ap2PBm_QsA#*\\q70,y0N
KI6vAeT'y$4IYaLr`!y$)R(eJu%\\/#\"]r+++]7`W1\"f\\D6@t(*oAYarEg(eLixcH2?
Q&oMqQ!3J/;(43[mI;@h@bxePE?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"M
G-%'RTABLEG6%\"*_>^j\"-%'MATRIXG6#7$7$#\"*h:N5'\"+DJq?7#\"*k:N5'F07$#
\"*8.2A\"\"*D19W##\"*O4@m$F6%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%&M_invG-%'matrixG6#7$7$#\"3]P4'\\3(e<6\"2\"*yL'\\-HDP#!2v=<wI!HDP
F,7$#!3D\"G)[Dh6!\\\"\"3k:N&)*4;,\\\"#\"3DcEs:h6!\\\"F2" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"#9#\"B$yrqg[NlS+qxd(*y%=\"\"ao++++
++L'*[#**H3Zp(=rDns)=-#HM[HLF14jS;z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%%betaG6#\"#8#\"N\\@eWD!z.\\PRXKQ\\qu<#fBwI\"\"do++++DJ(*RC$o/8hO`6
E5HTOBg,qy/ptL-HiD,g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"#9#
\"K8#oi,ZQFo!*))R%*Q#R6k#omR\"\"io++++++CX_px<+^^lVCphU\\%4;`YL_`)ow\"
=h6rOa%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*!G3O;-
%'MATRIXG6#7$7##\"cq\\tIB$>&4%3\")eu.bSuBH!))p_')G?S/#*)*H\"yUc%\\UlE%
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TC*4W+l0$RPq9\"Gw))HxI(>%'MatrixG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%$x_jG-%'matrixG6#7$7##\"ep(4R(R,\\r*)y2K$*fJ@ASjBZtAD?$==*zOh9J:^P5ka
,V@#H^Y%\"cr+++++D1R]\"zt!*\\wO+OyE\\))HKB(f]c'QWh1<XDb2L#osw#QE>HkXdq
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6!Ghee)HX#\"`s++++]PMx<Knb*\\q52lDdkqTplG#o0#3'o/a8!e'>&)3,wl')Rp<c;#
\\]H2\">JL[3j%obrm:KS" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#8
#\"ep(4R(R,\\r*)y2K$*fJ@ASjBZtAD?$==*zOh9J:^P5ka,V@#H^Y%\"cr+++++D1R]
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12 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"#8#\"bq\"=i!oPEo\\&\\qX6AL2#\\:(>
@z0IpnD$y\\)y@f$RfKGngkXWn6!Ghee)HX#\"`s++++]PMx<Knb*\\q52lDdkqTplG#o0
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{XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*'zEP;-%'MATRIXG6#7$7$#\"+6yv^I\"+D
c^.h#\"+9yv^IF07$#\"*j:N5'\"+DJq?7#\"+'oa5$=F6%'MatrixG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$#\"4D\"y]qZ^ki=\"3W[r#y
9<)3i#!4vV[Y,:XE'=\"4KX9[V9XE'=7$#!4D1*y$QVj<C\"\"4)oHacHMwT7#\"4v$f3j
'H!HDP\"4k!*G'p)G!HDP" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#
\"#:#\"DP4F\"4&=#3RB!*Rjp@HPz\"ho++++++q#Go`w+h1F0!H17`TubH\"y:5:ZIFW.
Q_GC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"#9#!Z$f8,`O5$*fd7
tK1d*=L)HBGrj^#*)=3z\"\"cp++++++5W2xTk$)*)yi(HueptLi>@tG9KV0][`IDG5W\"
pC96" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"#:#\"P6xp#)H4x1+:1N
<`fdP9B,>Rv#\"bp+++++]Vg(>YvU9wpT+cNz!=o=Cc+]i,>v>(o&)))fV+U')**Q" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*)=VO;-%'MATRIXG6#
7$7##!drj^sN06wbh7-30&H.,LYSzJ;n_An2NS8dwG\\n(z\"y(32E\"G$eAO[2cLhu2\"
>z4W#z\"\"dt++++]PM-uLL#ykeL*[M*f/>QWCg=#G()z%zH[@$*=s2Dq]!=a)=!y;JxpW
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*>CDr*zJ4.^*)*[0\"R\"y-8(Q=QhIhe@l`OC9M(ycZ%[xX=%'MatrixG" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%$x_jG-%'matrixG6#7$7##!^qFM)>#z&RSRQBOgf\\4L@
.td%\\zZv!p9\"zp<1Q02:(z_3w+f0+WR*ej#\"_s++++vozm4*>27'oS1*zx.k))*pfBN
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\\Q1kWtB)z,\")H]#z1S\\qjM(4k3yd@,g#[&)z(\\]W7D%*GP.H/z/>GRtV'4')fDMCNt
&)*o\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#9#!^qFM)>#z&RSRQ
BOgf\\4L@.td%\\zZv!p9\"zp<1Q02:(z_3w+f0+WR*ej#\"_s++++vozm4*>27'oS1*zx
.k))*pfBN#QfZ3eU:^(\\!o1!\\b]x/a5!*fO'Hbt=(poH>:DeC**)eF\"" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"#9#!crR>Y:bY#zk8HhQ&fCXRXXKAs<?hN
A:\\h:\\13oYzGf1#QW=s?</5:3V,m**4*RkyO\"dt+++++voa5$oi&fI#\\Q1kWtB)z,
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*SBjj\"-%'MATRIXG6
#7$7$#\",h!*ye_\"\",D\"yv^I#\",k!*ye_\"F07$#\"+8yv^I\"+Dc^.h#\"+OMFb\"
*F6%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7
$7$#\"5voz;Z3$>\\)p\"5T;$*zJNkIGB#!5]7yvwOkIGBF,7$#!5D1k1oZdA8$*\"5kls
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aG6#\"#;#\"OHh^\"4]0@\\VqwZlXTi:.#yuR#*\"gp+++++++W%zj;R`7;l]\")4(e)Q;
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#\"#:#\"\\o()**)fC3/%>][])3uU9554w*4m`fYg#3l:)[@(\"]q+++++](GzPCUmnQQr
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"6#>&%\"nG6#\"#;#\"Uh#Q?3N$4w!)R\">Du2AV'GeQY]e#pn\"\"[q++++++5)*pr$))
>QR%)>0sI;6E+E/J7pKVQ85$3xIe'\\B!)**[%=837" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*KBrj\"-%'MATRIXG6#7$7##\"esn9iIs%>
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atrixG6#7$7##\"cr^9c*)o()\\0t+V-<g#>i))3K@2W?/DWMbr(=Z^Wqm*fU#*[(*3XG!
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#\"cr^9c*)o()\\0t+V-<g#>i))3K@2W?/DWMbr(=Z^Wqm*fU#*[(*3XG!H1&Q.$=er&\\
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\"cG6#\"#:#\"bsxL\"GW2[N(\\$3Nj%*3\\FC\\W;HL3qX#\\,GmwC[IheJXjB'Q#\\H!
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"" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*;j&Q;-%'MATRIXG6#7$7$#\",6
`%RHw\"-D1*ye_\"#\",9`%RHwF07$#\",j!*ye_\"\",D\"yv^I#\",'=njxXF6%'Matr
ixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M_invG-%'matrixG6#7$7$#\"7v=
<,7B=K:k6\"6WtFfdrs50)Q#!7D\"GjB#H=K:k6\"7K?$yFZ\"=K:k67$#!6D\"y+LQb9-
hx\"6)oa&=:VX@5w(#\"7v=#*>H\\OkIGB\"7kSmbXHOkIGB" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>&%&alphaG6#\"#<#\"U\\o4^$*>)3/Uf#[BG95M<WFeT0opv\"`q++
+++++q>'QOKFD['RgW(3hcHzMns:;@W\\;Bd4Ri2%pJV_?b4Xeg(" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>&%%betaG6#\"#;#!foLSn5&G+&*o,&f]=td;:T<r)e_n#[t(**R
dS.*o?#=lM\"iq+++++++I5!QA8P;VJz/A/gZl]^Xn>>s#H09\\B#yGpW6:B$=en$4D$f*
\\F'R" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#\"#<#\"hn^K(\\RLnhQ/
3STd!>P$>U)>p?DjI%e]V#f#\"hq+++++++'3#H$*3:-6++U*R\\X4S5`d*\\,w\"H%Hg6
3!)y1UkQ=@yEk,G]^5(*4\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RT
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{XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*;NHk\"-%'MATRIXG6#7$7##\"^yR<10S#[
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1 "" {XPPMATH 20 "6#>&%\"cG6#\"#>#\"jxZB%p.#Q[+z#ytTbS\"f$*=uTa8/*yh>h
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{XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*))G8k\"-%'MATRIXG6#7$7$#\"/6.#er$o
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20 "6#>&%\"cG6#\"#@#\"c\\lHboF1Z]b,7Yo\"z:=NdWn@>ACA\\m=fkrq&o5@RgC9`S
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{XPPMATH 20 "6#>&%%betaG6#\"#A#!bt\\gOzRg$yG[^0T(Q/+?tN-8E17$*=c$yc#*f
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{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*)Q!Gk\"-%'MATRIXG
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priate ratio, and then print them out in a nice format." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "for j from 1 to 21 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" 
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{XPPMATH 20 "6#>%&ratioG$!#:\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&ratioG$\"++D\"yv\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$
\"+dw;d9!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$\"+/m#zn%!\"
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{XPPMATH 20 "6#>%&ratioG$\"+`BoDA!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%&ratioG$!+;%zSX$!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG
$!+M<$fu#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$!+%\\&z'z#!
\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$!+KD:%z#!\"(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$!+dQG%z#!\"(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&ratioG$!+#p(G%z#!\"(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&ratioG$!+'>!H%z#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&ratioG$!+))4H%z#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ra
tioG$!+[7H%z#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$!+K8H%z
#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$!+f8H%z#!\"(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$!+o8H%z#!\"(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&ratioG$!+r8H%z#!\"(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&ratioG$!+s8H%z#!\"(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "for j from 1 to 21 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
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od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$\"++++Dc!\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$\"+oC*3N#!\"*" }}{PARA 11 "
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