(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 21353, 578] NotebookOptionsPosition[ 20278, 535] NotebookOutlinePosition[ 20621, 550] CellTagsIndexPosition[ 20578, 547] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Cholesky", "Section", CellChangeTimes->{{3.7528526993294487`*^9, 3.7528527021568923`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Cholesky", "[", "A_", "]"}], ":=", RowBox[{"(", "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", RowBox[{"Length", "[", "A", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"L", "=", RowBox[{"Table", "[", RowBox[{"0", ",", RowBox[{"{", RowBox[{"i", ",", "n"}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "n"}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"k", "=", "1"}], ",", " ", RowBox[{"k", "\[LessEqual]", "n"}], ",", " ", RowBox[{"k", "++"}], ",", " ", RowBox[{"(*", RowBox[{"Iterate", " ", "over", " ", "the", " ", "columns"}], "*)"}], " ", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"Find", " ", "the", " ", "diagonal", " ", "element"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sum", "=", "0"}], ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"p", "=", "1"}], ",", RowBox[{"p", "\[LessEqual]", RowBox[{"k", "-", "1"}]}], ",", RowBox[{"p", "++"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"sum", "+=", RowBox[{ RowBox[{"L", "[", RowBox[{"[", RowBox[{"k", ",", "p"}], "]"}], "]"}], "^", "2"}]}], ";"}]}], "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"L", "[", RowBox[{"[", RowBox[{"k", ",", "k"}], "]"}], "]"}], "=", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{ RowBox[{"A", "[", RowBox[{"[", RowBox[{"k", ",", "k"}], "]"}], "]"}], "-", "sum"}], ")"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"j", "=", RowBox[{"k", "+", "1"}]}], ",", RowBox[{"j", "\[LessEqual]", "n"}], ",", RowBox[{"j", "++"}], ",", " ", RowBox[{"(*", RowBox[{ "Find", " ", "all", " ", "the", " ", "elements", " ", "below", " ", "the", " ", "main", " ", "diagonal"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sum", "=", "0"}], ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", RowBox[{"i", "\[LessEqual]", RowBox[{"k", "-", "1"}]}], ",", RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", RowBox[{"sum", "+=", RowBox[{ RowBox[{"L", "[", RowBox[{"[", RowBox[{"k", ",", "i"}], "]"}], "]"}], RowBox[{"L", "[", RowBox[{"[", RowBox[{"j", ",", "i"}], "]"}], "]"}]}]}]}], "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"L", "[", RowBox[{"[", RowBox[{"j", ",", "k"}], "]"}], "]"}], "=", RowBox[{ FractionBox["1", RowBox[{"L", "[", RowBox[{"[", RowBox[{"k", ",", "k"}], "]"}], "]"}]], RowBox[{"(", RowBox[{ RowBox[{"A", "[", RowBox[{"[", RowBox[{"k", ",", "j"}], "]"}], "]"}], "-", "sum"}], ")"}]}]}]}]}], "\[IndentingNewLine]", "]"}]}]}], "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", "L"}], "\[IndentingNewLine]", ")"}]}]], "Input", CellChangeTimes->{{3.6557929521573105`*^9, 3.6557929531373124`*^9}, { 3.655792992582371*^9, 3.655793106106538*^9}, {3.655793138812117*^9, 3.655793169457162*^9}, {3.6557932145632286`*^9, 3.6557935316707597`*^9}, { 3.655793580854902*^9, 3.6557936143739595`*^9}, 3.655793659345024*^9, { 3.655793741346494*^9, 3.6557937470565023`*^9}, {3.6557939568350487`*^9, 3.655794013999318*^9}, {3.655794130158848*^9, 3.655794131032898*^9}, { 3.6558002898616323`*^9, 3.6558003467928886`*^9}, {3.655801062470823*^9, 3.6558010922085238`*^9}, {3.6558012685996127`*^9, 3.655801272430832*^9}, { 3.65607152594147*^9, 3.6560716019998207`*^9}, {3.656071642224121*^9, 3.6560721417216907`*^9}, {3.6560722196281466`*^9, 3.656072224283413*^9}, 3.656072399894457*^9, {3.656073073059471*^9, 3.65607307514359*^9}, { 3.688455147842004*^9, 3.68845518106205*^9}, {3.688455223272109*^9, 3.688455233862124*^9}, {3.754031528371827*^9, 3.7540316013002625`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L", "=", RowBox[{"Cholesky", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"10", ",", "45", ",", "285"}], "}"}], ",", RowBox[{"{", RowBox[{"45", ",", "285", ",", "2025"}], "}"}], ",", RowBox[{"{", RowBox[{"285", ",", "2025", ",", "15333"}], "}"}]}], "}"}], "]"}]}]], "Input", CellChangeTimes->{{3.656072402458604*^9, 3.6560724057937946`*^9}, { 3.6884551618220234`*^9, 3.6884552104220915`*^9}, {3.75285263308751*^9, 3.7528526334794807`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SqrtBox["10"], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"9", " ", SqrtBox[ FractionBox["5", "2"]]}], ",", SqrtBox[ FractionBox["165", "2"]], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"57", " ", SqrtBox[ FractionBox["5", "2"]]}], ",", RowBox[{"9", " ", SqrtBox[ FractionBox["165", "2"]]}], ",", RowBox[{"4", " ", SqrtBox["33"]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.6884552120920935`*^9, 3.752852637565536*^9, 3.752853553941119*^9, 3.754031605329527*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L", ".", RowBox[{"Transpose", "[", "L", "]"}]}]], "Input", CellChangeTimes->{{3.7528526398753705`*^9, 3.752852644738364*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"10", ",", "45", ",", "285"}], "}"}], ",", RowBox[{"{", RowBox[{"45", ",", "285", ",", "2025"}], "}"}], ",", RowBox[{"{", RowBox[{"285", ",", "2025", ",", "15333"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.752852645207127*^9, 3.7528535552785425`*^9, 3.7540316069285965`*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Least-squares method", "Section", CellChangeTimes->{{3.7528525852434206`*^9, 3.7528525925977373`*^9}}], Cell[CellGroupData[{ Cell["LUSolve", "Subsection", CellChangeTimes->{{3.7528531021507297`*^9, 3.7528531043588247`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"LUSolve", "[", RowBox[{"L_", ",", "U_", ",", "b_"}], "]"}], ":=", RowBox[{"(", "\[IndentingNewLine]", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}], ",", " ", RowBox[{"(*", RowBox[{ "Define", " ", "x", " ", "and", " ", "y", " ", "to", " ", "be", " ", "local", " ", "variables"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", RowBox[{"Length", "[", "b", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"y", "=", RowBox[{"Table", "[", RowBox[{"0", ",", RowBox[{"{", RowBox[{"i", ",", "n"}], "}"}]}], "]"}]}], " ", ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", RowBox[{"i", "\[LessEqual]", "n"}], ",", " ", RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"y", "[", RowBox[{"[", "i", "]"}], "]"}], "=", RowBox[{ FractionBox["1", RowBox[{"L", "[", RowBox[{"[", RowBox[{"i", ",", "i"}], "]"}], "]"}]], RowBox[{"(", RowBox[{ RowBox[{"b", "[", RowBox[{"[", "i", "]"}], "]"}], "-", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"L", "[", RowBox[{"[", RowBox[{"i", ",", "j"}], "]"}], "]"}], RowBox[{"y", "[", RowBox[{"[", "j", "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"j", ",", "1", ",", RowBox[{"i", "-", "1"}]}], "}"}]}], "]"}]}], ")"}]}]}]}], "\[IndentingNewLine]", "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", RowBox[{"x", "=", RowBox[{"Table", "[", RowBox[{"0", ",", RowBox[{"{", RowBox[{"i", ",", "n"}], "}"}]}], "]"}]}], " ", ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "n"}], ",", RowBox[{"i", "\[GreaterEqual]", "1"}], ",", " ", RowBox[{"i", "--"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"x", "[", RowBox[{"[", "i", "]"}], "]"}], "=", RowBox[{ FractionBox["1", RowBox[{"U", "[", RowBox[{"[", RowBox[{"i", ",", "i"}], "]"}], "]"}]], RowBox[{"(", RowBox[{ RowBox[{"y", "[", RowBox[{"[", "i", "]"}], "]"}], "-", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"U", "[", RowBox[{"[", RowBox[{"i", ",", "j"}], "]"}], "]"}], RowBox[{"x", "[", RowBox[{"[", "j", "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"j", ",", RowBox[{"i", "+", "1"}], ",", "n"}], "}"}]}], "]"}]}], ")"}]}]}]}], "\[IndentingNewLine]", "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", "x"}]}], "\[IndentingNewLine]", "]"}], ")"}]}]], "Input", CellChangeTimes->{{3.688454461859042*^9, 3.6884545743691998`*^9}, { 3.6884547177694006`*^9, 3.688454753759451*^9}, {3.752853524940637*^9, 3.7528535715929356`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Least-squares fit for the falling body problem", "Subsection", CellChangeTimes->{{3.75285311332985*^9, 3.752853121789238*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"t", "=", RowBox[{"Range", "[", RowBox[{"0", ",", "9"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"X", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"1", ",", RowBox[{"t", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{ RowBox[{"t", "[", RowBox[{"[", "i", "]"}], "]"}], "^", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "10"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"y", "=", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{ "450", ",", "445", ",", "431", ",", "408", ",", "375", ",", "332", ",", "279", ",", "216", ",", "143", ",", "61"}], "}"}], "}"}], "//", "Transpose"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"A", "=", RowBox[{ RowBox[{ RowBox[{"Transpose", "[", "X", "]"}], ".", "X"}], "//", "N"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"b", "=", RowBox[{ RowBox[{ RowBox[{"Transpose", "[", "X", "]"}], ".", "y"}], "//", "N"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"L", "=", RowBox[{"Cholesky", "[", "A", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"coefs", "=", RowBox[{"LUSolve", "[", RowBox[{"L", ",", RowBox[{"Transpose", "[", "L", "]"}], ",", "b"}], "]"}]}]}], "Input", CellChangeTimes->{{3.7528529741518607`*^9, 3.752853091011501*^9}, { 3.752853179816147*^9, 3.7528532152464633`*^9}, {3.7528532512212744`*^9, 3.752853303513504*^9}, {3.7528533813115473`*^9, 3.752853389040886*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", "449.36363636363615`", "}"}], ",", RowBox[{"{", "0.9621212121213638`", "}"}], ",", RowBox[{"{", RowBox[{"-", "4.901515151515168`"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.7528531500776515`*^9, {3.752853196006855*^9, 3.752853215946557*^9}, 3.7528532551856723`*^9, {3.7528532957961254`*^9, 3.752853304662425*^9}, { 3.752853398077732*^9, 3.7528534192751074`*^9}, 3.7528535799575787`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"p", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"1", ",", "x", ",", RowBox[{"x", "^", "2"}]}], "}"}], "}"}], ".", "coefs"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"plot1", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"p", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "10"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"plot2", "=", RowBox[{"ListPlot", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"t", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"y", "[", RowBox[{"[", RowBox[{"i", ",", "1"}], "]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"i", ",", "10"}], "}"}]}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Show", "[", RowBox[{"plot1", ",", "plot2"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7528533131058846`*^9, 3.752853371464872*^9}, { 3.7528534043549576`*^9, 3.752853473762431*^9}, {3.752853507735652*^9, 3.752853508207365*^9}, {3.7528535848046246`*^9, 3.752853597222434*^9}}], Cell[BoxData[ GraphicsBox[{{{{}, {}, TagBox[ {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[1.], LineBox[CompressedData[" 1:eJwd1Hk4VVsbAHBDqZBCKRWOzMeQWyrzu0Qp5EppkJBwSJQTiX0qQ5KIY7hI yRBpQoRScrchhAyZMvRRhgzHTGZ969w/9rOf37PWu4Z3vWtJ2l02d+Di4OAo wB/7n3DTjucv+xBdx5oIj4bNDFTWSDE5RXGGwayzlt+x/5QXBVAo3rBr0aNr ADuBvC+xmhIMK5X/VKxgK43Zh/2hhcM94wTqOhEG+i7weGWpNRJa9k4rCWNf 0vF0nZV4AO/0mBtksd80dadO6z8CkwvNYyrYCy4mnRO0REhiOV/ej23wJULm d2sKRO3zMzHEHogzLJ52ToVds9wvTLFD7ZetphbTIFSi5akFduOiU/S4xHMI T3e6aYd9rUJcdSz7BaitUTjjhL0tqql6RP8VPKhVb3TFLrK+R2M1Z0DJciyL jm1HRdzDtCzIiD6U5oXNtehTN814DZZRBssEdlO3hdj0wWyonI+YvoX9tFzV ZUogB+YrVEICsK+/4iuYbM0B1vK24jvYRpH9PJNJbyDp4cmYYOwd14tPTDjn wtbkH7yh2KPnHqWM786DYwKF4mHYpL7X+NhiHhzt6WsIx45UMNcdK8sHXoqN dAS2/Qbl0NH7byG4TE4kEnvfzJr2kZPvoLoe0tle0/FTbkSiAOoM0lvZ/kZ+ 9GQNFEA0xfq/9hdP40qHs9+DmL39f/GM0KuCwz4fgLbh/X/jm9JNbYb0C+Hq Tpuv7PkppxUyBvk/gl7CcQn2+iZ1Vi0ONH+EhluxfOz1l0l1HR54XATSVdRY 9v5i1r2P+UX7F9QZ60vY+8/1MqJf3UqCQAyEsvPzaGi/6BSDhPQtxTPs/AWe kyHpP0jg4oxcYefXrV6INnmwGMTNc56y839Kn0OA/qIYlnl3jrLPB+WP5E4I lECi4o8m9vkJParkmmgtAbnUo77s814UyH9+RbsUBJYWDKywe/2emI0nlcK7 D7PP2PWRT7v5eMy5DMzoXabsekpsdzl4ubYMkHMVE7DvHj3DGt39CSYrVtHZ 9We5R01zdPETPCqkCbHrVf+p5A9X23KINmc27MBWEt1wd6SsHEzN1qqy63tl ZbCZdb8C1gb0V7HvQ3JVovuwxGeI/XT2Ovs+TcflUbtCP0NDwEGZMuxDtOqe r/Of4cKlENt8bBb3rMX7xiq4X6jo/wBbXcdUMzioBpZPF7lbYofw2U8xpmtg 2jk44gj2/9q8X105/wX0016DOvbta2nip7Vq4Ve8sOEm7PqsJS65sTow9lfk qtiE8wXrrNer1ANHf2t0DraWbLRmkH098Ht05yVgh00/n/JpqAeFDTxP6Nhq Ec0Odi8bwF/vicIW7FtVSkZ/2TRCYozMgJEwAz3Lfiv78p9G+JIYbbsL+2vc AW6ZmkY43mhDE8aWpp0u3KrRBI4FvJR2IQaq4r6twiHcDGl+OTp22CI6HUJ1 5S0QxWVEPSeIz1facezwcgtoZxN3NbGd+SaqS/a0gta9dZ5bsAvbeALzk1rh wAeN+bqN+L5e2z2b4PMNnEVZGVrYmVnBHZdU2kGuPK5yWYCBhoSGQo3Ot0PX k01NLdhy14x05aPbYT6NwcjCTtLmS+6Zb4czWwRTbbAjP4c6Wn7qgJJMY9MP 6/H70RM+YWj1HTRfr9lgy4/rha86QPBgN5Qvx+vZrGMgXR3Cyte6G1i0h0OK 2J1uinvHvLrh2u8g1bm1DLT1a0h/zYtuyPWViQrHjogzOXJ34w9oO0J/U7CG gQJkagX+dPyAqkxz28XVDESDhgcseg8Ylqq4qHExEEfK7/cdIT0gv/n55Awn A8Wt2tFZldoDNzsHtr3FrvzsKP68pQfCq98dVceWP7GU4qDZC+qK8dr7OPB7 6ST36n+cfZA1bgeUFQJdjLxRVM/sB/lo/rZX8wTimknp+vd5P1RH7/W2w44/ VcmRVdIPfbzS77dgV+0Q1r8/3Q8+p46J+80RSDH9WcWR079AiT5y+e9ZArE+ NNaXSAzAmwxF9fZpArn1KfTkZg6CmQ5p9WyMQN3WbWuLPw/CayVqsRH28W93 Vb70DsKFwcA81iiB1Kt/efdtG4LyKiHTXdjc2WkbRYKGwEvZXzeLRaA4hqSu l80whNPf7ogfJFCxsOgDjY0jwFwkarf3EIhX8rZcl/o4tK51XVPXRKBK7Yhd Nw+Ow7eamOBz2HdOP94vZj4OTGu5F8ONeL/Md4ZnXcZBfphRzoO9sMKitSaM w+yzeRf1BgINd1o8q+OaAMG3GheDvxCoJk5OgayZABWxpj0/PxEobGM1Ndl2 CoQF6YtueQQq2C+WZ+o2BXOpPkFtuQTqtb4MS8QUqBr0JOlja2UInzgVOwUP UpsIkTcEGjhidWN97RSYjOz/mv+aQPoBI3XeWtOg8ZiD8+dLAs3NbPA0F5kB 6s14x5/JBLLrOPEvd81vcNs3fiwklEDun27bXZZbgGMjt1zdzhEo+wr/8a1H liEwLSvlgCKBkgmXIdVTHIj28RdVcNQH+ScuHZtmcSKucXELZqYP4mjsRoU+ 3Gi2X8bC4qIPkvVwvH6CuhrV6S5LPxL1Qb0zsSf5PvIggQXJQM5Kb2TUd/FG pPtatHY2ReuNkzcy7WUqiovzIp6AwEi1Vd5oqNTOtzKDDzHBRM846TpSEWwR 07dZj7z3ixsqK19HvDBzvnNKAClv80u98MELGY5Yzu602IgKUuO9VTS80LcS e05LJUFUeJVL0jLnGsq7W2ZNmRFEpXfaosxVriHH1Z7dtbVCaGmuotwqzhNd oBN5/PHCKPMmR8rUak+0i6ZYa3J+Eyr7pb3O5YoHqhNS4vPU2Iz4ZamriYqr 6EOm/0MlbhG01eKQ1h2pq4jgpJrVfBdBqzpY3NMX6Yi30+ewW+4W1OswOVqd 6Y76nkpGJzK2osP6J26JzF9B0fKiqwosRNFKH7mgp3wFMeMdK85TtiGeWPsu 3QY3ZBl9rpb/1zakopxuXa3viuYM0uYTi7Yj+WfuNxSCXFBP34HrJUE7kINH RnJ5hzNK0flr+5fzYkj7uO/26hUaygj0FKQpiyPbMPflMDVHFGRxwZh7Rhyd dA1zNqDZo72dyc+Dvkigm2pbb7s62KE4hYcZX4wpqEjrrbuaty26Rd6PMy6g II/unePcr6zQ0e0WaVXSkmiATj5IP3ca6blVnBUMlkT8NnHSNHQCJe6wk66Z kUQUBysDC+u/kXRpeozp2Z0ITYoZCqYeQkzlEurXop1o/PLA9pxqLeQ4tarh M1UKITdXn3t1FGSXPUkJDpdC5I6/Q/IUpMlL/seXjCKkkO3oG82gemnS83hu K38Ubo+v5XfykiHvzniGMWOkkO9tGdFTn2TJTI35xdgE7Ednvg6dVyDni/+0 PH2J4zvbywcSVEjmV/77ZRV4vjqKVSJTjYx/4up057MUUh1QtVnoVSOfeNTq H67G8aMPp6w09pL5IsyF6lo8X6l/vfrPvWSnpbBTYzMejxllaay2n5TrEdX/ 2YPbs4uzOto0SNVcH/HUPhyf4luZpaJJagZ2zDv8kkIUv3h6SIAmaSKX8Hpw CMfzdPmcUdYi6S4U8YkJKWS25+VZMz9tkqHtN58zJYWuMPX+2LZok4HrfzZ5 zEih+vbdat6KOmRc1pOQuTncX13WnGzWIVN8V9HeL+D+xvJaY1Rd8uUxhwOM JSk0bhyYJ+OrS+buLBfTXcHxGjrFF5p1yaIp2fk/f3D/zWbWz6lA/h9eamhM "]]}, Annotation[#, "Charting`Private`Tag$2815#1"]& ]}, {}, {}}, {{}, {{}, {RGBColor[0.368417, 0.506779, 0.709798], PointSize[ 0.012833333333333334`], AbsoluteThickness[1.6], PointBox[{{0., 450.}, {1., 445.}, {2., 431.}, {3., 408.}, {4., 375.}, { 5., 332.}, {6., 279.}, {7., 216.}, {8., 143.}, {9., 61.}}]}, {}}, {}, {}, {{}, {}}}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{ "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 10}, {-31.16664685683412, 449.4108501943528}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{{3.752853404655151*^9, 3.752853442762293*^9}, 3.7528534755716114`*^9, {3.7528535866466985`*^9, 3.752853597792929*^9}}] }, Open ]], Cell[BoxData[""], "Input", CellChangeTimes->{{3.752853372781348*^9, 3.7528534020341587`*^9}}] }, Open ]] }, Open ]] }, WindowSize->{1366, 685}, WindowMargins->{{-8, Automatic}, {Automatic, -8}}, FrontEndVersion->"11.0 for Microsoft Windows (64-bit) (July 28, 2016)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[579, 22, 97, 1, 70, "Section"], Cell[679, 25, 4575, 108, 513, "Input"], Cell[CellGroupData[{ Cell[5279, 137, 526, 14, 30, "Input"], Cell[5808, 153, 681, 24, 72, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6526, 182, 153, 3, 30, "Input"], Cell[6682, 187, 372, 10, 87, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[7103, 203, 109, 1, 70, "Section"], Cell[CellGroupData[{ Cell[7237, 208, 99, 1, 49, "Subsection"], Cell[7339, 211, 3383, 91, 456, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[10759, 307, 133, 1, 49, "Subsection"], Cell[CellGroupData[{ Cell[10917, 312, 1637, 49, 145, "Input"], Cell[12557, 363, 473, 10, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[13067, 378, 1330, 39, 107, "Input"], Cell[14400, 419, 5741, 108, 263, "Output"] }, Open ]], Cell[20156, 530, 94, 1, 30, "Input"] }, Open ]] }, Open ]] } ] *)