{VERSION 5 0 "IBM INTEL NT" "5.0" }
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mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }
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2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 83 "This sheet performs some computations needed to ve
rify various claims in the paper." }}{PARA 0 "" 0 "" {TEXT -1 108 "The
 non-math text in this sheet refers to objects discussed in the paper,
 e.g., r_1, ..., r_6, P, q_1^*, ..." }}{PARA 0 "" 0 "" {TEXT -1 84 "Th
e sheet is organized in sections that roughly correspond to sections o
f the paper." }}{PARA 0 "" 0 "" {TEXT -1 89 "The matrix factorizations
 claimed in the paper are verified in \"Section 4.1 -- Geometry\"." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 125 "The following function maps a (nonnegative) matrix to a \+
stochastic (i.e., column-stochastic) one, by normalizing the columns.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "makeStoch := lA -> Matr
ix([seq(Normalize(lA[1..RowDimension(lA),i],1), i=1..ColumnDimension(l
A))]):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Section 4.1 -- Geometr
y" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The columns of the following \+
matrix are the \"inner points\" r_1, ..., r_6." }}}{EXCHG }{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "InnerPoints := Transpose(Matrix([
\n[3/4,1/8,0],\n[3/4,1/2,0],\n[3/11,17/22,0],\n[2,0,1/2],\n[1/2,0,3/4]
,\n[1/6,0,7/12]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,InnerPointsG
-%'RTABLEG6%\")W=SM-%'MATRIXG6#7%7(#\"\"$\"\"%F.#F/\"#6\"\"##\"\"\"F3#
F5\"\"'7(#F5\"\")F4#\"#<\"#A\"\"!F>F>7(F>F>F>F4F.#\"\"(\"#7%'MatrixG" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The columns of the following ma
trix contain the vertices of the polyhedron P." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 98 "OuterPoints := Transpose(Matrix([\n[0,0,0],\n[
1,0,0],\n[1,1/2,0],\n[0,1,0],\n[9/4,0,1/2],\n[0,0,8/7]]));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%,OuterPointsG-%'RTABLEG6%\")!o+X$-%'MATRIX
G6#7%7(\"\"!\"\"\"F/F.#\"\"*\"\"%F.7(F.F.#F/\"\"#F/F.F.7(F.F.F.F.F4#\"
\")\"\"(%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "The columns \+
of the following matrix contain the \"intermediate points\" q_1^*, ...
, q_5^*." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "MediatePoints \+
:= Transpose(Matrix([\n[2-sqrt(2),0,0],\n[(3-sqrt(2))/7,(11+sqrt(2))/1
4,0],\n[1,(3+sqrt(2))/14,0],\n[0,0,(10+sqrt(2))/14],\n[(26+7*sqrt(2))/
17,0,(12-2*sqrt(2))/17]])); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%.MediatePointsG-%'RTABLEG6%\")%eDX$-%'MATRIXG6#7%7',&\"\"#\"\"\"*
$F/#F0F/!\"\",&#\"\"$\"\"(F0*&F7F3F/F2F3F0\"\"!,&#\"#E\"#<F0*(F7F0F=F3
F/F2F07'F9,&#\"#6\"#9F0*&FCF3F/F2F0,&#F6FCF0*&FCF3F/F2F0F9F97'F9F9F9,&
#\"\"&F7F0*&FCF3F/F2F0,&#\"#7F=F0*(F/F0F=F3F/F2F3%'MatrixG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")7)HX$-%'MATRIXG6#7%7'$\"*Qky&
e!\"*$\"+(>4aE#!#5$\"\"\"\"\"!$F4F4$\"+'*\\t6@F.7'F5$\"+,aHn))F1$\"+(o
4I:$F1F5F57'F5F5F5$\"+(o4I:)F1$\"+oG/&R&F1%'MatrixG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 153 "Hpre := Matrix([\n[h11,0  ,h13,h14,0  ,h
16],\n[0  ,h22,h23,0  ,0  ,0  ],\n[h31,h32,0  ,0  ,0  ,0  ],\n[0  ,0  \+
,0  ,0  ,h45,h46],\n[0  ,0  ,0  ,h54,h55,0  ]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%HpreG-%'RTABLEG6%\")C&\\X$-%'MATRIXG6#7'7(%$h11G\"\"
!%$h13G%$h14GF/%$h16G7(F/%$h22G%$h23GF/F/F/7(%$h31G%$h32GF/F/F/F/7(F/F
/F/F/%$h45G%$h46G7(F/F/F/%$h54G%$h55GF/%'MatrixG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 71 "The following matrix (denoted H below) is the matr
ix H' from the paper." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "(
\{seq(seq(InnerPoints[i,j] = (MediatePoints . Hpre)[i,j], i=1..3), j=1
..6)\}): \nsolve(%): \nH := subs(%, Hpre);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"HG-%'RTABLEG6%\")#z!fO-%'MATRIXG6#7'7(,&*&\"\"%!\"
\"\"\"##\"\"\"F2F4#F4F0F4\"\"!,$*&\"#6F1F2F3F4,&F5F4*&\"\")F1F2F3F1F6,
&#F4\"\"'F4*&\"#7F1F2F3F47(F6,&F3F4*&F<F1F2F3F1,&F4F4*&F9F1F2F3F1F6F6F
67(,&#\"\"$F0F4*&F0F1F2F3F1,&F3F4*&F<F1F2F3F4F6F6F6F67(F6F6F6F6,&#\"#@
\"#MF4*(\"\"(F4\"#oF1F2F3F4,&#\"\"&F?F4*&FAF1F2F3F17(F6F6F6,&FIF4*&F<F
1F2F3F4,&#\"#8FRF4*(FTF4FUF1F2F3F1F6%'MatrixG" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 37 "The following matrix is the matrix C." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "C := Matrix([\n[0, 10/11, 0],\n[0,
 0    , 4/11],\n[-1/11, -2/11, 1/22],\n[-1/11, 0, 5/22],\n[4/11, 0, 0]
,\n[-2/11, -8/11, -7/11]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-
%'RTABLEG6%\")sNgO-%'MATRIXG6#7(7%\"\"!#\"#5\"#6F.7%F.F.#\"\"%F17%#!\"
\"F1#!\"#F1#\"\"\"\"#A7%F6F.#\"\"&F<7%F3F.F.7%F8#!\")F1#!\"(F1%'Matrix
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The following vector is the \+
vector d." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "d := Vector([0
, 0, 2, 1, 0, 8]) / 11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG-%'RT
ABLEG6%\")O6jO-%'MATRIXG6#7(7#\"\"!F-7##\"\"#\"#67##\"\"\"F2F-7##\"\")
F2&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "
Vector[row]([1,1,1,1,1,1]) . C;\nVector[row]([1,1,1,1,1,1]) . d;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")ClqO-%'VECTORG6#7%\"\"!
F+F+&%'VectorG6#%$rowG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "The following matrix (denoted M b
elow) is the matrix M' from the paper. We verify that it is stochastic
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "M := C . InnerPoints +
 Matrix([d,d,d,d,d,d]); makeStoch(M) - M;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\")3bFM-%'MATRIXG6#7(7(#\"\"&\"#W#F/
\"#6#\"#&)\"$@\"\"\"!F6F67(F6F6F6#\"\"#F2#\"\"$F2#\"\"(\"#L7(#\"\"\"F2
#FAF0#F9F5FB#\"#:\"#))#\"#<FF7(FBFB#\"\")F5FB#\"#>FF#F/\"#C7(F:F:#\"#7
F5#FKF2F8#F9F>7(#FAF9#F/\"#A#\"#9F5#FAFX#F=F0#\"#V\"$K\"%'MatrixG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(#R!\\$-%'MATRIXG6#7(7(
\"\"!F,F,F,F,F,F+F+F+F+F+%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 98 "The following matrix is the matrix W. We verify that it is stoc
hastic. We verify that M' = W * H'." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "W := C . MediatePoints + Matrix([d,d,d,d,d]);  \nmake
Stoch(W)-W, simplify(M - W . H);\nevalf(W);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"WG-%'RTABLEG6%\"(sE(R-%'MATRIXG6#7(7'\"\"!,&#\"\"&
\"\"(\"\"\"*(F1F3\"#x!\"\"\"\"##F3F7F3,&#\"#:F5F3*(F1F3F5F6F7F8F3F.F.7
'F.F.F.,&#\"#?F5F3*(F7F3F5F6F7F8F3,&#\"#[\"$(=F3*(\"\")F3FEF6F7F8F67',
$*&\"#6F6F7F8F3F.,&#\"\"%F5F3*&F5F6F7F8F6,&#\"\"$\"#9F3*&\"$3$F6F7F8F3
,&#FSFEF3*(FGF3FEF6F7F8F67',&#F3FKF6*&FKF6F7F8F3,&FMF3*&F5F6F7F8F3F.,&
#\"#R\"$a\"F3*(F1F3FUF6F7F8F3,&#\"#@FEF3*(\"#7F3FEF6F7F8F67',&#FGFKF3*
(FNF3FKF6F7F8F6,&#FboF5F3*(FNF3F5F6F7F8F6#FNFKF.,&#\"$/\"FEF3*(\"#GF3F
EF6F7F8F37',&FjoF3*(F7F3FKF6F7F8F3,&#\"\"'F5F3*(F7F3F5F6F7F8F6,&#\"#IF
5F3*(FNF3F5F6F7F8F6,&#FRFKF3*&\"#AF6F7F8F6F.%'MatrixG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")W[sI-%'MATRIXG6#7(7'\"\"!F,F,F,F,F+
F+F+F+F+%'MatrixG-F$6%\")sz)3$-F(6#7(7(F,F,F,F,F,F,F4F4F4F4F4F-" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")oWwK-%'MATRIXG6#7(7'$\"
\"!F-$\"+Qw<h!)!#5$\"+VCPmGF0F,F,7'F,F,F,$\"+&zIZ'HF0$\"+qP$='>F07'$\"
+$p[cG\"F0F,$\"+0U;eL!#6$\"+oJx)=#F0$\"+@&>lV\"F=7'$\"+RydlPF=$\"+&=Y9
.(F=F,$\"+cw/iFF0$\"+y#zZ:#F=7'$\"+.D8I@F0$\"+@;&yB)F=$\"+OOOOOF0F,$\"
+\\j.zwF07'$\"+@5m2iF0$\"+8e#*=TF=$\"++vWhJF0$\"+\"Q[W3#F0F,%'MatrixG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "The columns of the following \+
matrix contains the points q_1^epsilon, ..., q_5^epsilon." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "EpsPoints := Matrix([\n[99/169, 12
1/534, 9337/9338, 1/42216, 813/385],\n[0, 133/150, 64/203, 0, 0],\n[1/
40560, 0, 0, 17209/21108, 997/1848]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%*EpsPointsG-%'RTABLEG6%\")%=Pk$-%'MATRIXG6#7%7'#\"#**\"$p\"#\"
$@\"\"$M&#\"%P$*\"%Q$*#\"\"\"\"&;A%#\"$8)\"$&Q7'\"\"!#\"$L\"\"$]\"#\"#
k\"$.#F>F>7'#F8\"&g0%F>F>#\"&4s\"\"&36##\"$(**\"%[=%'MatrixG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "Hepspre := Matrix([\n[1-h41
-h51,1-h22-h32,1-h23-h33,1-h44-h54,1-h45-h55],\n[0  ,h22,h23,0  ,0  ],
\n[0  ,h32,h33,0  ,0  ],\n[h41,0  ,0  ,h44,h45],\n[h51,0  ,0  ,h54,h55
]\n]); Vector[row]([1,1,1,1,1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 
"EpsPoints - MediatePoints . Hepspre:\nseq(seq(%[i,j], j=1..5), i=1..3
):\nsolHeps := solve(\{%\}):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(H
epspreG-%'RTABLEG6%\")?AuK-%'MATRIXG6#7'7',(\"\"\"F/%$h41G!\"\"%$h51GF
1,(F/F/%$h22GF1%$h32GF1,(F/F/%$h23GF1%$h33GF1,(F/F/%$h44GF1%$h54GF1,(F
/F/%$h45GF1%$h55GF17'\"\"!F4F7F@F@7'F@F5F8F@F@7'F0F@F@F:F=7'F2F@F@F;F>
%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")S6pN-%'VEC
TORG6#7'\"\"\"F+F+F+F+&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 78 "The following matrix is the matrix H_epsilon. We verify t
hat it is stochastic." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "He
ps := subs(solHeps, Hepspre); makeStoch(Heps) - Heps; evalf(Heps);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%HepsG-%'RTABLEG6%\"(/PU$-%'MATRIXG6
#7'7',&#\"&>/$\"&g0%\"\"\"*(\"&z'GF2\"'SA;!\"\"\"\"##F2F7F2,&#\"%GF\"&
Dn%F6*(\"%\"z&F2\"'v,9F6F7F8F2,&#\"%TF\"&\\!)*F2*(\"$U'F2\"&$oKF6F7F8F
6,&#\"$*o\"&a0\"F6*(\"&&f:F2\"'GxLF6F7F8F2,&#\"$*Q\"%[=F2*(\"%,bF2\"&g
p$F6F7F8F67'\"\"!,&#\"'=L;F?F2*(\"%xsF2\"&+B'F6F7F8F6,&#\"%efFFF2*(\"&
V0&F2\"''>#RF6F7F8F6FVFV7'FV,&#\"%P@\"&D+#F6*(\"%ZgF2\"&+,)F6F7F8F2,&#
\"&i5\"\"&2S\"F2*(\"%@$)F2\"&Gg&F6F7F8F2FVFV7',&#\"%Vu\"%S))F2*(\"&88&
F2\"&!>')F6F7F8F6FVFV,&#\"'(*)[\"\"'=%z\"F2*(\"'FE<F2\"(W`V\"F6F7F8F2,
&#\"%T<\"&!=EF6*(\"%Z=F2\"&q#RF6F7F8F27',&#\"'d\"3%\"'?&*oF6*(\"(tW:\"
F2\"(!3eFF6F7F8F2FVFV,&#\"%Rq\"&.*HF2*(\"'T&=$F2\"(#z8>F6F7F8F6,&#\"'h
W8\"'!3d\"F2*(\"%j6F2\"&C9\"F6F7F8F2%'MatrixG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\"([*fM-%'MATRIXG6#7'7'\"\"!F,F,F,F,F+F+F+F
+%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"('**fM-%'M
ATRIXG6#7'7'$\"++Wk****!#5$\"(1S2%!#6$\")@uc<F1$\"(\\@(>F1$\"'a76F.7'$
\"\"!F:$\"+4X7****F.$\"'cPWF.F9F97'F9$\"'\"4o%F.$\"+-&*z(***F.F9F97'$
\"'S$)>F.F9F9$\"+?`u****F.$\"((z19F17'$\"'gs:F.F9F9$\"&mu&F.$\"+m![(**
**F.%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "The following ma
trix is the matrix W_epsilon. We verify that it is stochastic." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Weps := C . EpsPoints + Matr
ix([d,d,d,d,d]);\nmakeStoch(Weps) - Weps;\nevalf(Weps);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%WepsG-%'RTABLEG6%\"(Kdz$-%'MATRIXG6#7(7'\"\"!
#\"$L\"\"$l\"#\"$S'\"%LAF.F.7'#\"\"\"\"'S:6F.F.#\"&4s\"\"&Z!e#\"$(**\"
%#3&7'#\"'@Z6\"'?B*)#F7\"']o9#\"#<\"$1&#\"$&Q\"%f<#\"%@H\"'!G.#7'#\"#Z
\"%[7#\"$8%\"%ue#F7\"'=F5#\"%:H\"&a0\"#\"%\"Q%FM7'#\"#O\"$p\"#\"#A\"$n
##\"&u'=\"&f8&#F7\"'%4;\"#\"%_K\"%NU7'#\"'`pF\"'ghW#\"%45\"&vW##\"&Ri
\"F_o#\"%+6\"%x_#F7\"'S;5%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6%\")g3eI-%'MATRIXG6#7(7'\"\"!F,F,F,F,F+F+F+F+F+%'MatrixG" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")?[RN-%'MATRIXG6#7(7'$
\"\"!F-$\"+hggg!)!#5$\"+=%*4mGF0F,F,7'$\"+\"e$Rl*)!#:F,F,$\"+jlmkHF0$
\"+`g#='>F07'$\"+Y'[cG\"F0$\"+J(p'4oF6$\"+%z$ofL!#6$\"+/Ou)=#F0$\"+yU$
pV\"FB7'$\"+Tc-mPFB$\"+(*R)4.(FB$\"+W?RN(*F6$\"+xf)>w#F0$\"+^a:b@FB7'$
\"+:v<I@F0$\"+v.qR#)FB$\"+@U(fj$F0$\"+!e4Ph)F6$\"+)em)ywF07'$\"+@;[2iF
0$\"+bSdATFB$\"+G1'=;$F0$\"+=x^%3#F0$\"+-ikQ)*F6%'MatrixG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 41 "We verify that W_epsilon = W * H_epsilon.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(Weps - W . Hep
s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")/)41$-%'MATRIXG6
#7(7'\"\"!F,F,F,F,F+F+F+F+F+%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Section 4.2 -- Type 1" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "In the following we perform determ
inant computations pertaining to the polygon P_0 (type-1 case)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 312 "Matrix([[u,0,1],[1,v/2,1],[
3/4,1/8,1]]); det1 := Determinant(%);\nMatrix([[1, v/2, 1], [1-w, 1/2 \+
+ w/2, 1], [3/4, 1/2, 1]]); det2 := Determinant(%);\nMatrix([[1-w, 1/2
 + w/2, 1], [u, 0, 1], [3/11, 17/22, 1]]); det3 := Determinant(%);\nel
iminate(\{det1, det2\}, \{v, w\}); f := simplify(subs([%[1]][1], det3)
); solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")W1WK-%'
MATRIXG6#7%7%%\"uG\"\"!\"\"\"7%F.,$*&\"\"#!\"\"%\"vGF.F.F.7%#\"\"$\"\"
%#F.\"\")F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%det1G,**&#
\"\"\"\"\"#F(*&%\"uGF(%\"vGF(F(F(*&#F(\"\")F(F+F(!\"\"#F(F/F(*&#\"\"$F
/F(F,F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Gg:O-%'MAT
RIXG6#7%7%\"\"\",$*&\"\"#!\"\"%\"vGF,F,F,7%,&F,F,%\"wGF0,&#F,F/F,*&F/F
0F4F,F,F,7%#\"\"$\"\"%F6F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%%det2G,**&#\"\"$\"\")\"\"\"%\"wGF*!\"\"#F*F)F**&#F*F)F*%\"vGF*F,*&#
F*\"\"#F**&F+F*F0F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6
%\")C'3D$-%'MATRIXG6#7%7%,&\"\"\"F-%\"wG!\"\",&#F-\"\"#F-*&F2F/F.F-F-F
-7%%\"uG\"\"!F-7%#\"\"$\"#6#\"#<\"#AF-%'MatrixG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%det3G,*#\"\"(\"#6!\"\"*&#\"#5F(\"\"\"%\"wGF-F-*&#\"
\"$F(F-%\"uGF-F-*&#F-\"\"#F-*&F2F-F.F-F-F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$<$/%\"vG*&,&%\"uG\"\"\"F*!\"\"F*,&*&\"\"%F*F)F*F*\"\"
$F+F+/%\"wG*&,&*&F/F*F)F*F*\"\"#F+F*,&*&\"\")F*F)F*F*\"\"&F+F+<\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,$**\"#:\"\"\"\"#A!\"\",(*&\"\"%
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11 "" 1 "" {XPPMATH 20 "6$,&\"\"#\"\"\"*$F$#F%F$F%,&F$F%F&!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(f, u=0..0.6);" }}{PARA 
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%\\o&F,$!38ycB*Q5?U(F07$$\"3#****\\(3'>$[dF,$!3QZS:q$\\SG&F07$$\"3D***
***4h(*3eF,$!3UCuu+7!on#F07$$\"3o***\\7hK'peF,$\"3w&G[;%f0bu!#?7$$\"3V
*\\P%eWA-fF,$\"3g6s3@x(*pIF07$$\"3=**\\i0j\"[$fF,$\"3M)>(fyLVpeF07$$\"
31\\(=#HA6^fF,$\"3%yY5$)*o$e\\(F07$$\"3#*)\\7G:3u'fF,$\"3j_g;)y9#3$*F0
7$$\"3!)[iSwSq$)fF,$\"3EoIn.42M6F,7$$\"3w**************fF,$\"3#*fjjjjj
j8F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"u6\"Q!Fg]l-%%VI
EWG6$;F($\"\"'Fb]l%(DEFAULTG" 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 "" {TEXT -1 
97 "In the following we perform determinant computations pertaining to
 the polygon P_1 (type-1 case)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 338 "Matrix([[u,0,1], [(9-9*v)/4, (7+9*v)/14, 1], [2, 1/2, 1]]); d
et1 := Determinant(%);\nMatrix([[(9-9*v)/4, (7+9*v)/14, 1], [0, (8-8*w
)/7, 1], [1/2, 3/4, 1]]); det2 := Determinant(%);\nMatrix([[(0,8-8*w)/
7, 1], [u, 0, 1], [1/6, 7/12, 1]]); det3 := Determinant(%);\neliminate
(\{det1, det2\}, \{v,w\}); f := simplify(subs([%[1]][1], det3)); solve
(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")#Re_$-%'MATRIXG
6#7%7%%\"uG\"\"!\"\"\"7%,&#\"\"*\"\"%F.*(F2F.F3!\"\"%\"vGF.F5,&#F.\"\"
#F.*(F2F.\"#9F5F6F.F.F.7%F9F8F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%det1G,(*&#\"\"*\"#9\"\"\"*&%\"uGF*%\"vGF*F*F*#F*\"\")F**&#\"$
N\"\"#cF*F-F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")_
W=L-%'MATRIXG6#7%7%,&#\"\"*\"\"%\"\"\"*(F.F0F/!\"\"%\"vGF0F2,&#F0\"\"#
F0*(F.F0\"#9F2F3F0F0F07%\"\"!,&#\"\")\"\"(F0*(F=F0F>F2%\"wGF0F2F07%F5#
\"\"$F/F0%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%det2G,*#\"\"*
\"#;\"\"\"*&\"\"#F)%\"wGF)!\"\"*&#F'F(F)%\"vGF)F-*&#\"#=\"\"(F)*&F,F)F
0F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")?()GL-%'MATR
IXG6#7%7%\"\"!,&#\"\")\"\"(\"\"\"*(F/F1F0!\"\"%\"wGF1F3F17%%\"uGF,F17%
#F1\"\"'#F0\"#7F1%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%det3G
,*#\"\"%\"#@\"\"\"*&#\"#Z\"#%)F)%\"uGF)!\"\"*&#\"\")\"\"(F)*&F.F)%\"wG
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(\"#;!\"\",&*&\"\"*\"\"\"%\"uGF.F.\"#KF*F.,&*&\"\"#F.F/F.F.\"\"(F*F*F.
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{XPPMATH 20 "6#>%\"fG,$**\"#5\"\"\"\"#@!\"\",(*&\"\"%F(%\"uGF(F*\"\"#F
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20 "6$,&\"\"#\"\"\"*$F$#F%F$!\"\",&F$F%F&F%" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 18 "plot(f, u=0.5..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 
615 615 615 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"3++++++++]!#=$\"3-%)p7%)
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$\"3py433!ot,\"F-7$$\"35n;z%4\\Y_&F*$\"3!p=@2:\"z-x!#?7$$\"3eL$eR-/Pi&
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$!3S$RKMe^xR\"F-7$$\"3wK$e*[z(yb'F*$!36*oZFTGkh\"F-7$$\"3Kmm;a/cqmF*$!
3+6E?,eRw=F-7$$\"3%fmmmJ<gw'F*$!3#f*3(RHql4#F-7$$\"3g**\\iSj0xoF*$!3kV
kM'*[i_BF-7$$\"3;mmm\"pW`(pF*$!3)z8#Rq:AzDF-7$$\"35+]i!f#=$3(F*$!3CC/I
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p>#y$F-7$$\"3#HLe*)>pxg(F*$!3A%3(=Aj&f.%F-7$$\"3>**\\Pf4t.xF*$!3g**o.A
tzcUF-7$$\"32LLe*Gst!yF*$!3u_Kx)3^_\\%F-7$$\"3#)*****\\#RW9zF*$!3^#G<W
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\"oIr!R:_F-7$$\"3b***\\(=S2L#)F*$!3!ff)*HB8SZ&F-7$$\"3Kmmm\"p)=M$)F*$!
3FqRXY+L1dF-7$$\"3N****\\(=]@W)F*$!3gh4z_pIafF-7$$\"35L$e*[$z*R&)F*$!3
)R1Dc$p%*yhF-7$$\"3#*****\\iC$pk)F*$!3nH^uttYCkF-7$$\"3el;H2qcZ()F*$!3
M0WAW]TbmF-7$$\"3q**\\7.\"fF&))F*$!3yprLg1v'*oF-7$$\"3Ymm;/Ogb*)F*$!3:
@p%35HE8(F-7$$\"3y**\\ilAFj!*F*$!3)R?8Pe&\\ztF-7$$\"3!HLLL)*pp;*F*$!3$
z#[v$e#=<wF-7$$\"3kKL3xe,t#*F*$!3+]5Tl]<gyF-7$$\"3em;HdO=y$*F*$!3$*flB
+y2,\")F-7$$\"3))*****\\#>#[Z*F*$!3rER7,MPA$)F-7$$\"31mmT&G!e&e*F*$!3?
Oj*)o3#fd)F-7$$\"3gKLL$)Qk%o*F*$!3@wN829i-))F-7$$\"37+]iSjE!z*F*$!3$*e
rRA@DW!*F-7$$\"35+]P40O\"*)*F*$!33&)He?fWv#*F-7$$\"\"\"\"\"!$!3GB&4Q_4
Q_*F--%'COLOURG6&%$RGBG$\"#5!\"\"$FhzFhzFb[l-%+AXESLABELSG6$Q\"u6\"Q!F
g[l-%%VIEWG6$;$\"\"&Fa[lFfz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 21 "Section 4.5 -- Type 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 72 "The following functions up and lo round up and down to n decima
l digits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "up := (x,n) ->
 evalf(ceil(x*10^n)/10^n):\nlo := (x,n) -> evalf(floor(x*10^n)/10^n):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "up(0.123456,3); lo(0.123456,3);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++S7!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"++++I7!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "In
 the following we verify computations of bounds for the type-4 case." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W12lo := lo(W[1,2],2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&W12loG$\"+++++!)!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W13lo := lo(W[1,3],3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&W13loG$\"++++gG!#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "W13up := up(W[1,3],3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&W13upG$\"++++qG!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "W24lo := lo(W[2,4],2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&W24loG$\"+++++H!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "W25lo := lo(W[2,5],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&W25lo
G$\"++++g>!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W33lo := l
o(W[3,3],4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&W33loG$\"++++]L!#6
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W34lo := lo(W[3,4],2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&W34loG$\"+++++@!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W35up := up(W[3,5],3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&W35upG$\"+++++:!#6" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "W42lo := lo(W[4,2],3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&W42loG$\"+++++q!#6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "W44lo := lo(W[4,4],2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&W44loG$\"+++++F!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "W45up := up(W[4,5],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&W45up
G$\"+++++A!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W55lo := l
o(W[5,5],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&W55loG$\"++++qw!#5
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W61lo := lo(W[6,1],2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&W61loG$\"+++++i!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W63up := up(W[6,3],2);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&W63upG$\"+++++K!#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "W64up := up(W[6,4],2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&W64upG$\"+++++@!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "ex := 5;\neps := 10.^(-ex);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#exG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$
\"+++++5!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "R24lo := W24
lo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&R24loG$\"+++++H!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "R25lo := W25lo;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&R25loG$\"++++g>!#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "L32up := up(W35up/R25lo,3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&L32upG$\"+++++x!#6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "L42up := up(W45up/R25lo,2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&L42upG$\"+++++7!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "L62up := up(eps/R25lo,ex);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&L62upG$\"+++++g!#9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "L52up := up(eps/R24lo,ex);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&L52upG$\"+++++S!#9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "L22lo := lo(1 - L32up - L42up - L52up - L62up,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L22loG$\"+++++!)!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "R21up := up(eps/L22lo,ex);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&R21upG$\"+++++?!#9" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "L63lo := lo(W61lo - L62up*R21up,2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L63loG$\"+++++h!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "L54lo := lo((W55lo - L52up)/(1-R25l
o),4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L54loG$\"++++R&*!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "L34up := 1 - L54lo;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L34upG$\"*+++h%!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "2*L32up, 2-3*L63lo, 2*L34up;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"++++S:!#5$\"*+++q\"!\"*$\"*+++A*F%
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "L35lo := lo( (2*W34lo -
 W64up - max(2*L32up, 2-3*L63lo, 2*L34up))/2, 3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&L35loG$\"+++++?!#6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "L44up := 1 - L54lo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%&L44upG$\"*+++h%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "
2*L42up, 2 - 3*L63lo, 2*L44up;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+
++++C!#5$\"*+++q\"!\"*$\"*+++A*F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "L45lo := lo((2*W44lo - W64up - max(2*L42up, 2 - 3*L63
lo, 2*L44up))/2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L45loG$\"+++
++X!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L11lo := W12lo;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L11loG$\"+++++!)!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "R12lo := W12lo;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&R12loG$\"+++++!)!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "R13lo := W13lo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&R13loG$\"++++gG!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L31u
p := up(eps/R12lo,ex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L31upG$\"
+++++?!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "R13up := up(W1
3up/L11lo,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&R13upG$\"+++++O!#5
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "R33up := up(W63up/L63lo
,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&R33upG$\"+++++`!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "R53up := up(eps/L45lo,ex);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&R53upG$\"+++++B!#8" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "R43lo := lo(1 - R13up - R33up - R53
up,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&R43loG$\"+++++5!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L44up := up(eps/R43lo,ex);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L44upG$\"+++++5!#8" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L41up := up(eps/R13lo,ex);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L41upG$\"+++++S!#9" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "R52up := up(eps/L35lo,ex);" }}
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