Permalink
Switch branches/tags
Nothing to show
Find file Copy path
Fetching contributors…
Cannot retrieve contributors at this time
718 lines (663 sloc) 27.2 KB
(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 10.0' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 158, 7]
NotebookDataLength[ 27609, 708]
NotebookOptionsPosition[ 25599, 634]
NotebookOutlinePosition[ 25976, 651]
CellTagsIndexPosition[ 25933, 648]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell[CellGroupData[{
Cell["Euclid in the Round", "Title",
CellChangeTimes->{{3.696858114206793*^9, 3.69685812821731*^9}}],
Cell["\<\
This notebook illustrates and tests some ideas connected with Euclidean \
distance as a measure of inaccuracy. Specifically, it provides:\
\>", "Text",
CellChangeTimes->{{3.696858167910364*^9, 3.696858254780047*^9}, {
3.696858485526375*^9, 3.6968584917952003`*^9}}],
Cell[CellGroupData[{
Cell[TextData[{
"A function for computing expected inaccuracy in ",
StyleBox["N",
FontSlant->"Italic"],
" dimensions given a measure of inaccuracy."
}], "Item",
CellChangeTimes->{{3.696858257599676*^9, 3.6968582850591087`*^9},
3.696859649523774*^9}],
Cell["\<\
A counterexample to the false Conjecture 1 discussed in the accompanying blog \
post.\
\>", "Item",
CellChangeTimes->{{3.696858257599676*^9, 3.696858334652628*^9}, {
3.6968583771730223`*^9, 3.6968583809528017`*^9}, {3.69685842433598*^9,
3.696858427336335*^9}, 3.696859656470029*^9}],
Cell["\<\
Code for randomly generating examples that test the remaining Conjectures 2 \
and 3.\
\>", "Item",
CellChangeTimes->{{3.696858257599676*^9, 3.6968583424453297`*^9}, {
3.696858391813748*^9, 3.696858474709495*^9}}]
}, Open ]],
Cell[CellGroupData[{
Cell["Calculating Expected Inaccuracy", "Section",
CellChangeTimes->{{3.696858136319654*^9, 3.696858141432808*^9}}],
Cell[TextData[{
"First we define a function for computing the expected inaccuracy of \
probability assignment ",
StyleBox["q",
FontSlant->"Italic"],
", given probability assignment ",
StyleBox["p ",
FontSlant->"Italic"],
"(and a distance function, Euclidean distance in this case):"
}], "Text",
CellChangeTimes->{{3.6968585058069487`*^9, 3.696858606808199*^9}, {
3.696858646619969*^9, 3.6968586625961742`*^9}, {3.696858840643194*^9,
3.696858866234243*^9}}],
Cell[BoxData[{
RowBox[{
RowBox[{
RowBox[{"distance", "[",
RowBox[{"v1_", ",", "v2_"}], "]"}], ":=",
RowBox[{"Sqrt", "[",
RowBox[{"Total", "[",
RowBox[{
RowBox[{"(",
RowBox[{"v1", "-", "v2"}], ")"}], "^", "2"}], "]"}], "]"}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"vertices", "[", "p_", "]"}], ":=",
RowBox[{"IdentityMatrix", "[",
RowBox[{"Length", "[", "p", "]"}], "]"}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"EI", "[",
RowBox[{"p_", ",", "q_"}], "]"}], ":=",
RowBox[{"Total", "[",
RowBox[{"p", "*",
RowBox[{"distance", "[",
RowBox[{"q", ",",
RowBox[{"vertices", "[", "p", "]"}]}], "]"}]}], "]"}]}]}], "Input",
CellChangeTimes->{{3.694298229956419*^9, 3.6942982802855787`*^9}, {
3.694298340674238*^9, 3.6942984121841908`*^9}, {3.694298722821652*^9,
3.694298722958897*^9}, {3.694299909475893*^9, 3.694299918048452*^9}, {
3.694304011344225*^9, 3.6943040124402323`*^9}, {3.6943040466966267`*^9,
3.694304049586896*^9}}]
}, Open ]],
Cell[CellGroupData[{
Cell["A Counterexample", "Section",
CellChangeTimes->{{3.696858685623151*^9, 3.696858688428361*^9}}],
Cell["\<\
Next we\[CloseCurlyQuote]ll use the above function to verify a counterexample \
to Conjecture 1:\
\>", "Text",
CellChangeTimes->{{3.6968587304462147`*^9, 3.6968587606724*^9}, {
3.696858836531399*^9, 3.696858836861198*^9}, {3.696858886995626*^9,
3.696858888059914*^9}}],
Cell[BoxData[{
RowBox[{
RowBox[{"p", "=",
RowBox[{"{",
RowBox[{
RowBox[{"5", "/", "12"}], ",",
RowBox[{"4", "/", "12"}], ",",
RowBox[{"3", "/", "12"}]}], "}"}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"p", "'"}], "=",
RowBox[{"{",
RowBox[{
RowBox[{"6", "/", "12"}], ",",
RowBox[{"4", "/", "12"}], ",",
RowBox[{"2", "/", "12"}]}], "}"}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"u1", "=",
RowBox[{"{",
RowBox[{"1", ",", "0", ",", "0"}], "}"}]}], ";"}]}], "Input",
CellChangeTimes->{{3.694298826968535*^9, 3.69429882722346*^9}, {
3.694300321418326*^9, 3.69430032338974*^9}, {3.69430391852102*^9,
3.694303922347763*^9}, {3.696858704824485*^9, 3.696858706611039*^9},
3.696858813566669*^9, {3.6969430323141317`*^9, 3.6969430329282417`*^9}, {
3.6978093951084747`*^9, 3.697809414188393*^9}, {3.697809601717268*^9,
3.697809621530901*^9}, {3.697809661962631*^9, 3.697809706826128*^9}, {
3.697809982648423*^9, 3.697809998593519*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{"EI", "[",
RowBox[{"p", ",", "p"}], "]"}], "//", "N"}]], "Input",
CellChangeTimes->{{3.694298810498633*^9, 3.694298815118668*^9}}],
Cell[BoxData["0.8037895078695942`"], "Output",
CellChangeTimes->{
3.6968587898730707`*^9, 3.6968589330108767`*^9, {3.696859553404614*^9,
3.696859593500682*^9}, {3.696859923393777*^9, 3.696859967352396*^9},
3.69694098816434*^9, 3.696941354873302*^9, 3.6969416601779423`*^9, {
3.6969416933227453`*^9, 3.696941703301413*^9}, 3.696941934598238*^9, {
3.6969419721113453`*^9, 3.6969420202819653`*^9}, {3.696942052214995*^9,
3.6969420781394043`*^9}, 3.6969421712962008`*^9, 3.696942713682496*^9,
3.696942971193574*^9, 3.696943004111251*^9, 3.696943041510277*^9,
3.6969431299148397`*^9, 3.697809429053733*^9, 3.6978096241206636`*^9,
3.697809711896532*^9}]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{"EI", "[",
RowBox[{"p", ",",
RowBox[{"p", "'"}]}], "]"}], "//", "N"}]], "Input",
CellChangeTimes->{{3.694298795563883*^9, 3.694298807142515*^9}}],
Cell[BoxData["0.7999667638492747`"], "Output",
CellChangeTimes->{
3.696858789808394*^9, 3.696858933065504*^9, {3.6968595534600143`*^9,
3.6968595935552197`*^9}, {3.6968599234340763`*^9, 3.696859967389613*^9},
3.6969409882173977`*^9, 3.696941354909553*^9, 3.696941660245173*^9, {
3.696941693372797*^9, 3.696941703358409*^9}, 3.6969419346488667`*^9, {
3.6969419721608257`*^9, 3.696942020341758*^9}, {3.696942052269833*^9,
3.696942078195318*^9}, 3.696942171357782*^9, 3.6969427137431*^9,
3.69694297125605*^9, 3.696943004166643*^9, 3.696943041571138*^9,
3.696943129973826*^9, 3.697809429088571*^9, 3.697809624181242*^9,
3.697809711960217*^9}]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{"EI", "[",
RowBox[{"p", ",", "u1"}], "]"}], "//", "N"}]], "Input",
CellChangeTimes->{{3.694298820792963*^9, 3.694298832692048*^9}, {
3.696858715134721*^9, 3.696858715234432*^9}, 3.69685881670212*^9, {
3.696943034792595*^9, 3.696943035897895*^9}}],
Cell[BoxData["0.8249579113843054`"], "Output",
CellChangeTimes->{
3.696858789918792*^9, 3.696858933132029*^9, {3.696859553510372*^9,
3.696859593605788*^9}, {3.69685992350047*^9, 3.6968599674405117`*^9},
3.6969409882670403`*^9, 3.696941354966402*^9, 3.6969416603117657`*^9, {
3.6969416934224157`*^9, 3.6969417034093313`*^9}, 3.696941934697933*^9, {
3.6969419722125273`*^9, 3.696942020391197*^9}, {3.696942052320911*^9,
3.696942078244392*^9}, 3.696942171407784*^9, 3.696942713791821*^9,
3.6969429713240147`*^9, 3.696943004217387*^9, 3.696943041632897*^9,
3.6969431300241947`*^9, 3.697809429133067*^9, 3.69780962424469*^9,
3.69780971202194*^9}]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{"EI", "[",
RowBox[{"p", ",",
RowBox[{"p", "'"}]}], "]"}], "<",
RowBox[{"EI", "[",
RowBox[{"p", ",", "p"}], "]"}], "<",
RowBox[{"EI", "[",
RowBox[{"p", ",", "u1"}], "]"}]}]], "Input",
CellChangeTimes->{{3.696858910542163*^9, 3.696858922533474*^9}, {
3.6969430376417427`*^9, 3.696943037938157*^9}}],
Cell[BoxData["True"], "Output",
CellChangeTimes->{
3.69685893318218*^9, {3.6968595535599823`*^9, 3.6968595936558723`*^9}, {
3.69685992356022*^9, 3.696859967497919*^9}, 3.696940988316412*^9,
3.696941355014462*^9, 3.696941660378914*^9, {3.696941693472727*^9,
3.696941703458199*^9}, 3.6969419347478247`*^9, {3.6969419722611017`*^9,
3.696942020441607*^9}, {3.696942052369787*^9, 3.6969420782936077`*^9},
3.696942171457491*^9, 3.696942713842805*^9, 3.6969429713892937`*^9,
3.696943004266097*^9, 3.6969430416993113`*^9, 3.696943130090734*^9,
3.6978094291831713`*^9, 3.697809624296307*^9, 3.697809712072179*^9}]
}, Open ]],
Cell[TextData[{
"If the extreme assignment ",
Cell[BoxData[
FormBox[
SubscriptBox["u", "1"], TraditionalForm]]],
" isn\[CloseCurlyQuote]t the minimum point here, what is? Asking for an \
exact answer seems to take forever, so we\[CloseCurlyQuote]ll settle for a \
numerical approximation:"
}], "Text",
CellChangeTimes->{{3.696858944513562*^9, 3.696859032173798*^9}, {
3.696943146609991*^9, 3.6969431468789263`*^9}}],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{
RowBox[{"q", "=",
RowBox[{"Array", "[",
RowBox[{"x", ",",
RowBox[{"Length", "[", "p", "]"}]}], "]"}]}],
";"}], "\[IndentingNewLine]",
RowBox[{"NMinimize", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"EI", "[",
RowBox[{"p", ",", "q"}], "]"}], ",",
RowBox[{"AllTrue", "[",
RowBox[{"q", ",",
RowBox[{
RowBox[{"#", ">", "0"}], "&"}]}], "]"}], ",",
RowBox[{
RowBox[{"Total", "[", "q", "]"}], "\[Equal]", "1"}]}], "}"}], ",",
"\[IndentingNewLine]", "q"}], "\[IndentingNewLine]", "]"}]}], "Input",
CellChangeTimes->{{3.696688209093149*^9, 3.696688246384305*^9}, {
3.696940963284861*^9, 3.696940981193267*^9}, {3.696941262104432*^9,
3.696941271582412*^9}, {3.6969413432734118`*^9, 3.6969413805812283`*^9}, {
3.696941470541794*^9, 3.696941482629691*^9}, {3.696941600178646*^9,
3.696941628107565*^9}, {3.696941678751668*^9, 3.6969417006073627`*^9}, {
3.696941877350581*^9, 3.696941877512319*^9}, {3.6969419185414762`*^9,
3.696941930554678*^9}, {3.696942061559135*^9, 3.696942086586919*^9},
3.6969421671241627`*^9}],
Cell[BoxData[
RowBox[{"{",
RowBox[{"0.7974317254894209`", ",",
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{"x", "[", "1", "]"}], "\[Rule]", "0.5756608235869071`"}], ",",
RowBox[{
RowBox[{"x", "[", "2", "]"}], "\[Rule]", "0.25039203513645675`"}], ",",
RowBox[{
RowBox[{"x", "[", "3", "]"}], "\[Rule]", "0.17394714127663613`"}]}],
"}"}]}], "}"}]], "Output",
CellChangeTimes->{{3.696688227738921*^9, 3.696688246843678*^9},
3.696858790005232*^9, 3.6968590430943413`*^9, {3.696859553643774*^9,
3.696859593739819*^9}, {3.696859923640399*^9, 3.696859967578806*^9},
3.6969409884580917`*^9, 3.6969413550951767`*^9, {3.696941621464016*^9,
3.6969416288302717`*^9}, 3.6969416604432077`*^9, {3.69694169363501*^9,
3.696941703547145*^9}, 3.696941934867651*^9, {3.6969419723659563`*^9,
3.69694202058843*^9}, {3.696942052470125*^9, 3.696942078387291*^9},
3.696942171543784*^9, 3.6969427139293213`*^9, 3.696942971496037*^9,
3.696943004354645*^9, 3.6969430417907248`*^9, 3.696943130178589*^9,
3.697809435248906*^9, 3.697809624670332*^9, 3.6978097124504642`*^9}]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["Two Conjectures", "Section",
CellChangeTimes->{{3.6968590709074593`*^9, 3.696859073771244*^9}}],
Cell["Our conjectures are:", "Text",
CellChangeTimes->{{3.696859116553968*^9, 3.696859122554944*^9}}],
Cell[CellGroupData[{
Cell["\<\
Repeatedly minimizing expected inaccuracy will, eventually, lead to a \
probability assignment greater than 1/2 in the/a possibility that was most \
probable initially.\
\>", "ItemNumbered",
CellChangeTimes->{{3.696859142175696*^9, 3.696859180364551*^9}, {
3.696859686531415*^9, 3.696859687640531*^9}}],
Cell["\<\
When one of the possibilities has probability greater than 1/2, minimizing \
expected inaccuracy then leads in one step to probability 1 in that \
possibility.\
\>", "ItemNumbered",
CellChangeTimes->{{3.696859142175696*^9, 3.696859227575876*^9}}]
}, Open ]],
Cell[TextData[{
"The following code creates random tests of these conjectures. First it \
generates a random probability assignment of the specified length (",
StyleBox["dimensions",
FontWeight->"Bold"],
"). Then it repeatedly minimizes expected inaccuracy the specified number of \
times (",
StyleBox["iterations",
FontWeight->"Bold"],
").\n\nIf all goes well, the final result should be an extreme probability \
assignment (",
StyleBox["x[1], x[2], x[3],...",
FontWeight->"Bold"],
")",
StyleBox[" ",
FontSlant->"Italic"],
"that assigns 1 to the possibility ",
StyleBox["p",
FontWeight->"Bold"],
" rated most probable initially."
}], "Text",
CellChangeTimes->{{3.6968592323915443`*^9, 3.696859544726676*^9}, {
3.6968596015360537`*^9, 3.6968596241872463`*^9}, {3.696859982769079*^9,
3.6968600206305532`*^9}, {3.696860051608685*^9, 3.696860226083419*^9},
3.69694233568547*^9, {3.6969424259460506`*^9, 3.696942637368532*^9},
3.6969426732975187`*^9, {3.69694291569566*^9, 3.696942967540164*^9}}],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{
RowBox[{"dimensions", "=", "4"}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"iterations", " ", "=", " ", "5"}], ";"}], "\[IndentingNewLine]",
"\[IndentingNewLine]",
RowBox[{"(*", " ",
RowBox[{
"Function", " ", "to", " ", "randomly", " ", "generate", " ",
"probability", " ", "assignment", " ", "of", " ", "length", " ", "n"}],
" ", "*)"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"randomP", "[", "n_", "]"}], ":=",
RowBox[{"Module", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{"{", "l", "}"}], ",", "\[IndentingNewLine]",
RowBox[{
RowBox[{"l", "=",
RowBox[{"RandomReal", "[",
RowBox[{
RowBox[{"{",
RowBox[{"0", ",", "1"}], "}"}], ",",
RowBox[{"{",
RowBox[{"1", ",", "n"}], "}"}]}], "]"}]}], ";",
"\[IndentingNewLine]",
RowBox[{
RowBox[{"Flatten", "[", "l", "]"}], "/",
RowBox[{"Total", "[",
RowBox[{"Flatten", "[", "l", "]"}], "]"}]}]}]}],
"\[IndentingNewLine]", "]"}]}], "\[IndentingNewLine]",
"\[IndentingNewLine]",
RowBox[{"(*", " ",
RowBox[{
"Generate", " ", "a", " ", "random", " ", "assignment", " ", "of", " ",
"length", " ", "\"\<dimensions\>\""}], " ",
"*)"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"p", "=",
RowBox[{"randomP", "[", "dimensions", "]"}]}], " ", "\[IndentingNewLine]",
"\[IndentingNewLine]",
RowBox[{"(*", " ",
RowBox[{
"Repeatedly", " ", "minimize", " ", "EI", " ", "\"\<iterations\>\"", " ",
"times"}], " ", "*)"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"q", "=",
RowBox[{"Array", "[",
RowBox[{"x", ",", "dimensions"}], "]"}]}], ";"}], "\[IndentingNewLine]",
RowBox[{"Do", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"result", "=",
RowBox[{"NMinimize", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"EI", "[",
RowBox[{"p", ",", "q"}], "]"}], ",",
RowBox[{"AllTrue", "[",
RowBox[{"q", ",",
RowBox[{
RowBox[{"#", "\[GreaterEqual]", "0"}], "&"}]}], "]"}], ",",
RowBox[{
RowBox[{"Total", "[", "q", "]"}], "\[Equal]", "1"}]}], "}"}], ",",
"\[IndentingNewLine]", "q"}], "\[IndentingNewLine]", "]"}]}], ";",
"\[IndentingNewLine]",
RowBox[{"Print", "[", "result", "]"}], ";", "\[IndentingNewLine]",
RowBox[{"p", "=",
RowBox[{"result", "[",
RowBox[{"[",
RowBox[{"2", ",", "All", ",", "2"}], "]"}], "]"}]}], ";"}],
"\[IndentingNewLine]", ",",
RowBox[{"{",
RowBox[{"i", ",", "iterations"}], "}"}]}], "\[IndentingNewLine]",
"]"}]}], "Input",
CellChangeTimes->{{3.6942990160719337`*^9, 3.694299114339979*^9}, {
3.6942991988884163`*^9, 3.694299277747909*^9}, {3.694299426878606*^9,
3.6942994531904497`*^9}, {3.694299523914483*^9, 3.6942995355113773`*^9}, {
3.694299639119828*^9, 3.6942997481265087`*^9}, {3.6942998425138493`*^9,
3.694299859789942*^9}, {3.694299994172822*^9, 3.6943000642416553`*^9}, {
3.694300101907543*^9, 3.6943001893819313`*^9}, {3.694300442343471*^9,
3.6943004460556726`*^9}, 3.694301041688314*^9, {3.694301109209886*^9,
3.694301121008562*^9}, {3.694301290949439*^9, 3.6943013019193068`*^9}, {
3.694302130005352*^9, 3.69430214114187*^9}, {3.6943023726323967`*^9,
3.6943023853226347`*^9}, {3.69430274219938*^9, 3.694302769064809*^9}, {
3.694302850306731*^9, 3.694302850433161*^9}, {3.694302907082767*^9,
3.694302907264636*^9}, {3.6968595695239067`*^9, 3.6968595830692997`*^9}, {
3.696859790128314*^9, 3.69685991249089*^9}, {3.696860241776868*^9,
3.6968602465316563`*^9}, {3.696942189201396*^9, 3.696942226554307*^9}, {
3.696942296097682*^9, 3.696942362163374*^9}, {3.6969429850311747`*^9,
3.696943000183216*^9}, {3.696943084278946*^9, 3.6969431148184977`*^9}, {
3.697281406594698*^9, 3.697281407165967*^9}}],
Cell[BoxData[
RowBox[{"{",
RowBox[{
"0.07443974264631163`", ",", "0.3657641555042956`", ",",
"0.11080832191730797`", ",", "0.4489877799320848`"}], "}"}]], "Output",
CellChangeTimes->{{3.694302834958012*^9, 3.694302911188408*^9},
3.694303925570552*^9, 3.694304015221137*^9, {3.694304052657053*^9,
3.694304059642515*^9}, 3.6968590833158293`*^9, {3.69685955370751*^9,
3.6968595938050632`*^9}, 3.696859889042235*^9, {3.6968599236750402`*^9,
3.6968599676233883`*^9}, 3.696940988512453*^9, 3.696941355145116*^9,
3.69694166049482*^9, {3.696941693666257*^9, 3.696941703570323*^9},
3.696941934912311*^9, {3.696941972392273*^9, 3.696942020611039*^9}, {
3.696942052515861*^9, 3.696942078418625*^9}, 3.6969421715745068`*^9, {
3.69694224221039*^9, 3.696942269302615*^9}, {3.696942339672592*^9,
3.696942367732585*^9}, 3.6969427139654417`*^9, 3.696942971556038*^9,
3.696943004388494*^9, 3.6969430418243732`*^9, 3.696943130215262*^9,
3.697809435361435*^9, 3.6978096247252073`*^9, 3.697809712481452*^9}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"{",
RowBox[{"0.7754526259174528`", ",",
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{"x", "[", "1", "]"}], "\[Rule]", "0.026429862561089368`"}], ",",
RowBox[{
RowBox[{"x", "[", "2", "]"}], "\[Rule]", "0.14022732110311273`"}], ",",
RowBox[{
RowBox[{"x", "[", "3", "]"}], "\[Rule]", "0.039672458105307146`"}], ",",
RowBox[{
RowBox[{"x", "[", "4", "]"}], "\[Rule]", "0.7936703582304907`"}]}],
"}"}]}], "}"}]], "Print",
CellChangeTimes->{{3.694302835052476*^9, 3.6943029115720367`*^9},
3.6943039256382093`*^9, 3.694304015293867*^9, {3.6943040527752523`*^9,
3.6943040597448387`*^9}, 3.696859083417551*^9, {3.6968595538044987`*^9,
3.6968595938958807`*^9}, 3.696859889130157*^9, {3.696859923785305*^9,
3.69685996773172*^9}, 3.6969409885860443`*^9, 3.696941355230999*^9,
3.696941660807371*^9, {3.69694169398982*^9, 3.696941703673403*^9},
3.696941935007268*^9, {3.696941972502833*^9, 3.696942020707898*^9}, {
3.696942052587985*^9, 3.6969420785121737`*^9}, 3.696942171682938*^9, {
3.696942242556356*^9, 3.696942269425445*^9}, {3.6969423397853193`*^9,
3.6969423678382683`*^9}, 3.69694271409527*^9, 3.696942971661888*^9,
3.6969430047324867`*^9, 3.696943041947318*^9, 3.696943130330092*^9,
3.697809436154377*^9, 3.697809625715041*^9, 3.697809713060904*^9}],
Cell[BoxData[
RowBox[{"{",
RowBox[{"0.29179417798093876`", ",",
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{"x", "[", "1", "]"}], "\[Rule]",
RowBox[{"-", "1.559863349598345`*^-13"}]}], ",",
RowBox[{
RowBox[{"x", "[", "2", "]"}], "\[Rule]", "2.0742552502893209`*^-10"}],
",",
RowBox[{
RowBox[{"x", "[", "3", "]"}], "\[Rule]", "1.4863670451617386`*^-10"}],
",",
RowBox[{
RowBox[{"x", "[", "4", "]"}], "\[Rule]", "0.9999999996440938`"}]}],
"}"}]}], "}"}]], "Print",
CellChangeTimes->{{3.694302835052476*^9, 3.6943029115720367`*^9},
3.6943039256382093`*^9, 3.694304015293867*^9, {3.6943040527752523`*^9,
3.6943040597448387`*^9}, 3.696859083417551*^9, {3.6968595538044987`*^9,
3.6968595938958807`*^9}, 3.696859889130157*^9, {3.696859923785305*^9,
3.69685996773172*^9}, 3.6969409885860443`*^9, 3.696941355230999*^9,
3.696941660807371*^9, {3.69694169398982*^9, 3.696941703673403*^9},
3.696941935007268*^9, {3.696941972502833*^9, 3.696942020707898*^9}, {
3.696942052587985*^9, 3.6969420785121737`*^9}, 3.696942171682938*^9, {
3.696942242556356*^9, 3.696942269425445*^9}, {3.6969423397853193`*^9,
3.6969423678382683`*^9}, 3.69694271409527*^9, 3.696942971661888*^9,
3.6969430047324867`*^9, 3.696943041947318*^9, 3.696943130330092*^9,
3.697809436154377*^9, 3.697809625715041*^9, 3.697809713841054*^9}],
Cell[BoxData[
RowBox[{"{",
RowBox[{"5.045013347302312`*^-10", ",",
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{"x", "[", "1", "]"}], "\[Rule]",
RowBox[{"-", "6.238343175368755`*^-13"}]}], ",",
RowBox[{
RowBox[{"x", "[", "2", "]"}], "\[Rule]", "5.863377320745637`*^-13"}],
",",
RowBox[{
RowBox[{"x", "[", "3", "]"}], "\[Rule]", "5.863377320745637`*^-13"}],
",",
RowBox[{
RowBox[{"x", "[", "4", "]"}], "\[Rule]", "0.9999999999994511`"}]}],
"}"}]}], "}"}]], "Print",
CellChangeTimes->{{3.694302835052476*^9, 3.6943029115720367`*^9},
3.6943039256382093`*^9, 3.694304015293867*^9, {3.6943040527752523`*^9,
3.6943040597448387`*^9}, 3.696859083417551*^9, {3.6968595538044987`*^9,
3.6968595938958807`*^9}, 3.696859889130157*^9, {3.696859923785305*^9,
3.69685996773172*^9}, 3.6969409885860443`*^9, 3.696941355230999*^9,
3.696941660807371*^9, {3.69694169398982*^9, 3.696941703673403*^9},
3.696941935007268*^9, {3.696941972502833*^9, 3.696942020707898*^9}, {
3.696942052587985*^9, 3.6969420785121737`*^9}, 3.696942171682938*^9, {
3.696942242556356*^9, 3.696942269425445*^9}, {3.6969423397853193`*^9,
3.6969423678382683`*^9}, 3.69694271409527*^9, 3.696942971661888*^9,
3.6969430047324867`*^9, 3.696943041947318*^9, 3.696943130330092*^9,
3.697809436154377*^9, 3.697809625715041*^9, 3.697809714902437*^9}],
Cell[BoxData[
RowBox[{"{",
RowBox[{"1.9500772430147418`*^-12", ",",
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{"x", "[", "1", "]"}], "\[Rule]",
RowBox[{"-", "6.238343175368755`*^-13"}]}], ",",
RowBox[{
RowBox[{"x", "[", "2", "]"}], "\[Rule]", "5.863377320745637`*^-13"}],
",",
RowBox[{
RowBox[{"x", "[", "3", "]"}], "\[Rule]", "5.863377320745637`*^-13"}],
",",
RowBox[{
RowBox[{"x", "[", "4", "]"}], "\[Rule]", "0.9999999999994511`"}]}],
"}"}]}], "}"}]], "Print",
CellChangeTimes->{{3.694302835052476*^9, 3.6943029115720367`*^9},
3.6943039256382093`*^9, 3.694304015293867*^9, {3.6943040527752523`*^9,
3.6943040597448387`*^9}, 3.696859083417551*^9, {3.6968595538044987`*^9,
3.6968595938958807`*^9}, 3.696859889130157*^9, {3.696859923785305*^9,
3.69685996773172*^9}, 3.6969409885860443`*^9, 3.696941355230999*^9,
3.696941660807371*^9, {3.69694169398982*^9, 3.696941703673403*^9},
3.696941935007268*^9, {3.696941972502833*^9, 3.696942020707898*^9}, {
3.696942052587985*^9, 3.6969420785121737`*^9}, 3.696942171682938*^9, {
3.696942242556356*^9, 3.696942269425445*^9}, {3.6969423397853193`*^9,
3.6969423678382683`*^9}, 3.69694271409527*^9, 3.696942971661888*^9,
3.6969430047324867`*^9, 3.696943041947318*^9, 3.696943130330092*^9,
3.697809436154377*^9, 3.697809625715041*^9, 3.697809715591621*^9}],
Cell[BoxData[
RowBox[{"{",
RowBox[{"1.9500772430147418`*^-12", ",",
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{"x", "[", "1", "]"}], "\[Rule]",
RowBox[{"-", "6.238343175368755`*^-13"}]}], ",",
RowBox[{
RowBox[{"x", "[", "2", "]"}], "\[Rule]", "5.863377320745637`*^-13"}],
",",
RowBox[{
RowBox[{"x", "[", "3", "]"}], "\[Rule]", "5.863377320745637`*^-13"}],
",",
RowBox[{
RowBox[{"x", "[", "4", "]"}], "\[Rule]", "0.9999999999994511`"}]}],
"}"}]}], "}"}]], "Print",
CellChangeTimes->{{3.694302835052476*^9, 3.6943029115720367`*^9},
3.6943039256382093`*^9, 3.694304015293867*^9, {3.6943040527752523`*^9,
3.6943040597448387`*^9}, 3.696859083417551*^9, {3.6968595538044987`*^9,
3.6968595938958807`*^9}, 3.696859889130157*^9, {3.696859923785305*^9,
3.69685996773172*^9}, 3.6969409885860443`*^9, 3.696941355230999*^9,
3.696941660807371*^9, {3.69694169398982*^9, 3.696941703673403*^9},
3.696941935007268*^9, {3.696941972502833*^9, 3.696942020707898*^9}, {
3.696942052587985*^9, 3.6969420785121737`*^9}, 3.696942171682938*^9, {
3.696942242556356*^9, 3.696942269425445*^9}, {3.6969423397853193`*^9,
3.6969423678382683`*^9}, 3.69694271409527*^9, 3.696942971661888*^9,
3.6969430047324867`*^9, 3.696943041947318*^9, 3.696943130330092*^9,
3.697809436154377*^9, 3.697809625715041*^9, 3.6978097167021847`*^9}]
}, Open ]]
}, Open ]]
}, Open ]]
}, Open ]]
},
WindowSize->{1118, 1261},
WindowMargins->{{719, Automatic}, {62, Automatic}},
Magnification->1.5,
FrontEndVersion->"10.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (June 27, \
2014)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)
(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[580, 22, 101, 1, 137, "Title"],
Cell[684, 25, 278, 5, 76, "Text"],
Cell[CellGroupData[{
Cell[987, 34, 259, 7, 43, "Item"],
Cell[1249, 43, 300, 6, 43, "Item"],
Cell[1552, 51, 225, 5, 43, "Item"]
}, Open ]],
Cell[CellGroupData[{
Cell[1814, 61, 116, 1, 95, "Section"],
Cell[1933, 64, 471, 12, 76, "Text"],
Cell[2407, 78, 1053, 29, 94, "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[3497, 112, 101, 1, 95, "Section"],
Cell[3601, 115, 284, 6, 46, "Text"],
Cell[3888, 123, 1047, 26, 94, "Input"],
Cell[CellGroupData[{
Cell[4960, 153, 169, 4, 43, "Input"],
Cell[5132, 159, 680, 10, 43, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[5849, 174, 189, 5, 43, "Input"],
Cell[6041, 181, 670, 10, 43, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[6748, 196, 291, 6, 43, "Input"],
Cell[7042, 204, 674, 10, 43, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[7753, 219, 356, 10, 43, "Input"],
Cell[8112, 231, 634, 9, 43, "Output"]
}, Open ]],
Cell[8761, 243, 426, 10, 77, "Text"],
Cell[CellGroupData[{
Cell[9212, 257, 1168, 27, 144, "Input"],
Cell[10383, 286, 1133, 22, 43, "Output"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell[11565, 314, 102, 1, 95, "Section"],
Cell[11670, 317, 102, 1, 46, "Text"],
Cell[CellGroupData[{
Cell[11797, 322, 315, 6, 72, "ItemNumbered"],
Cell[12115, 330, 257, 5, 72, "ItemNumbered"]
}, Open ]],
Cell[12387, 338, 1029, 25, 194, "Text"],
Cell[CellGroupData[{
Cell[13441, 367, 3998, 93, 619, "Input"],
Cell[17442, 462, 1039, 16, 43, "Output"],
Cell[CellGroupData[{
Cell[18506, 482, 1374, 26, 30, "Print"],
Cell[19883, 510, 1412, 28, 39, "Print"],
Cell[21298, 540, 1413, 28, 39, "Print"],
Cell[22714, 570, 1414, 28, 39, "Print"],
Cell[24131, 600, 1416, 28, 39, "Print"]
}, Open ]]
}, Open ]]
}, Open ]]
}, Open ]]
}
]
*)
(* End of internal cache information *)