(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 153007, 7435]*) (*NotebookOutlinePosition[ 153906, 7464]*) (* CellTagsIndexPosition[ 153862, 7460]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ A general implicit finite difference method for the heat equation\ \ \>", "Section"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ StyleBox["We", "Text"], StyleBox[" ", "Text"], StyleBox["study", "Text"], StyleBox[" ", "Text"], StyleBox["numerical", "Text"], StyleBox[" ", "Text"], StyleBox["methods", "Text"], StyleBox[" ", "Text"], StyleBox["for", "Text"], StyleBox[" ", "Text"], StyleBox["solving", "Text"], StyleBox[" ", "Text"], StyleBox["the", "Text"], StyleBox[" ", "Text"], StyleBox["heat", "Text"], StyleBox[" ", "Text"], StyleBox["equation", "Text"], StyleBox[" ", "Text"], StyleBox[\(u\_t\), "InlineFormula"]}], StyleBox["=", "InlineFormula"], RowBox[{ StyleBox[\(u\_xx\), "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox["using", "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox["space", "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox["steps", "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox["\[CapitalDelta]x", "InlineFormula"], " ", "and", " ", "time", " ", "steps", " ", StyleBox["\[CapitalDelta]t", "InlineFormula"]}]}], ",", \(and\ recurrences\ of\ the\ form\)}], TextForm]], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["+", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(a\_0\)}]}], ")"}], "u", RowBox[{"(", RowBox[{"x", StyleBox[",", "InlineFormula"], StyleBox["t", "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}], "-", RowBox[{\(a\_0\), RowBox[{"(", RowBox[{ RowBox[{"u", RowBox[{"(", RowBox[{ RowBox[{"x", "-", StyleBox["\[CapitalDelta]x", "InlineFormula"]}], StyleBox[",", "InlineFormula"], StyleBox["t", "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}], "+", StyleBox[" ", "InlineFormula"], RowBox[{"u", RowBox[{"(", RowBox[{ RowBox[{"x", "+", StyleBox["\[CapitalDelta]x", "InlineFormula"]}], StyleBox[",", "InlineFormula"], StyleBox["t", "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}]}], StyleBox[")", "InlineFormula"]}]}]}], StyleBox["=", "InlineFormula"], StyleBox[" ", "InlineFormula"], RowBox[{ RowBox[{ RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["-", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(\[Sum]\+\(j = 1\)\%k a\_j\)}]}], ")"}], "u", RowBox[{"(", RowBox[{"x", StyleBox[",", "InlineFormula"], RowBox[{ StyleBox["t", "InlineFormula"], StyleBox["-", "InlineFormula"], "\[CapitalDelta]t"}]}], StyleBox[")", "InlineFormula"]}]}], "+", RowBox[{\(\[Sum]\+\(j = 1\)\%k\), RowBox[{\(a\_j\), StyleBox[" ", "InlineFormula"], RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ RowBox[{"u", RowBox[{"(", RowBox[{ RowBox[{"x", "-", RowBox[{"j", " ", StyleBox["\[CapitalDelta]x", "InlineFormula"]}]}], StyleBox[",", "InlineFormula"], StyleBox[\(t - \[CapitalDelta]t\), "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}], "+", StyleBox[" ", "InlineFormula"], RowBox[{"u", RowBox[{"(", RowBox[{ RowBox[{"x", "+", RowBox[{"j", " ", StyleBox["\[CapitalDelta]x", "InlineFormula"]}]}], StyleBox[",", "InlineFormula"], StyleBox[\(t - \[CapitalDelta]t\), "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}]}], StyleBox[")", "InlineFormula"]}]}]}]}]}], TextForm]], "DisplayFormula"], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{"for", " ", "parameters", " ", StyleBox[\(a\_0\), "InlineFormula"]}], StyleBox[",", "InlineFormula"], StyleBox[\(a\_1\), "InlineFormula"], StyleBox[",", "InlineFormula"], StyleBox["...", "InlineFormula"], StyleBox[",", "InlineFormula"], RowBox[{ RowBox[{ StyleBox[\(a\_k\), "InlineFormula"], ".", " ", "To"}], " ", "defer", " ", "consideration", " ", "of", " ", "boundary", " ", "issues"}], ",", " ", \(we\ study\ the\ case\ where\ the\ space\ coordinate\ x\ wraps\ around\ in\ a\ circle\), ",", " ", RowBox[{"with", " ", "a", " ", "period", " ", "of", " ", StyleBox["n", "InlineFormula"], " ", "steps"}], ",", " ", RowBox[{"for", " ", StyleBox["n", "InlineFormula"], " ", \(even . \ Our\), " ", "recurrence", " ", "then", " ", "takes", " ", "the", " ", "general", " ", "form"}]}], TextForm], FormBox[ RowBox[{ RowBox[{ StyleBox[\(A\ u \((t)\)\), "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox["=", "InlineFormula"], StyleBox[" ", "InlineFormula"], RowBox[{ StyleBox["B", "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox["u", "InlineFormula"], RowBox[{ StyleBox[\((t - \[CapitalDelta]t)\), "InlineFormula"], StyleBox[".", "InlineFormula"], StyleBox[" "], StyleBox["For", "InlineFormula"]}], StyleBox[" ", "InlineFormula"], StyleBox["example", "InlineFormula"]}]}], StyleBox[",", "InlineFormula"], StyleBox[" ", "InlineFormula"], RowBox[{ StyleBox[\(if\ \ k\), "InlineFormula"], StyleBox["=", "InlineFormula"], RowBox[{ RowBox[{ StyleBox["2", "InlineFormula"], StyleBox[" ", "Text"], StyleBox["and", "Text"], StyleBox[" ", "Text"], StyleBox["n", "InlineFormula"]}], StyleBox["=", "InlineFormula"], StyleBox[\(6\ \ we\ have\), "InlineFormula"]}]}]}], TextForm]}], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(k\ = \ 2\), ";", " ", \(n\ = 6\), ";", "\n", RowBox[{\(Aij[i_, j_, n_]\), " ", ":=", " ", RowBox[{"Switch", "[", RowBox[{\(Mod[j - i, n]\), ",", " ", "0", ",", RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["+", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(a\_0\)}]}], ")"}], ",", "1", ",", \(-a\_0\), ",", \(n - 1\), ",", \(-a\_0\), ",", "_", ",", " ", "0"}], "]"}]}], ";", "\n", \(A[n_]\ := \ Table[Aij[i, j, n]\ , {i, \ n}, \ {j, \ n}]\), ";", "\n", \(Ax\ = \ A[n]\), ";", "\n", \(Print["\", MatrixForm[Ax]]\)}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"A = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", GridBox[{ {\(1 + 2\ a\_0\), \(-a\_0\), "0", "0", "0", \(-a\_0\)}, {\(-a\_0\), \(1 + 2\ a\_0\), \(-a\_0\), "0", "0", "0"}, {"0", \(-a\_0\), \(1 + 2\ a\_0\), \(-a\_0\), "0", "0"}, {"0", "0", \(-a\_0\), \(1 + 2\ a\_0\), \(-a\_0\), "0"}, {"0", "0", "0", \(-a\_0\), \(1 + 2\ a\_0\), \(-a\_0\)}, {\(-a\_0\), "0", "0", "0", \(-a\_0\), \(1 + 2\ a\_0\)} }], ")"}], (MatrixForm[ #]&)]}], SequenceForm[ "A = ", MatrixForm[ {{ Plus[ 1, Times[ 2, Subscript[ a, 0]]], Times[ -1, Subscript[ a, 0]], 0, 0, 0, Times[ -1, Subscript[ a, 0]]}, { Times[ -1, Subscript[ a, 0]], Plus[ 1, Times[ 2, Subscript[ a, 0]]], Times[ -1, Subscript[ a, 0]], 0, 0, 0}, {0, Times[ -1, Subscript[ a, 0]], Plus[ 1, Times[ 2, Subscript[ a, 0]]], Times[ -1, Subscript[ a, 0]], 0, 0}, {0, 0, Times[ -1, Subscript[ a, 0]], Plus[ 1, Times[ 2, Subscript[ a, 0]]], Times[ -1, Subscript[ a, 0]], 0}, {0, 0, 0, Times[ -1, Subscript[ a, 0]], Plus[ 1, Times[ 2, Subscript[ a, 0]]], Times[ -1, Subscript[ a, 0]]}, { Times[ -1, Subscript[ a, 0]], 0, 0, 0, Times[ -1, Subscript[ a, 0]], Plus[ 1, Times[ 2, Subscript[ a, 0]]]}}]], Editable->False]], "Print"] }, Open ]], Cell["and", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{\(Bij[i_, j_, k_, n_]\), " ", ":=", "\n", "\t", RowBox[{"Module", "[", RowBox[{\({l}\), ",", RowBox[{ \(l = Min[Mod[i - j, n], Mod[j - i, n]]\), ";", "\n", "\t", RowBox[{"If", "[", RowBox[{\(i == j\), ",", RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["-", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(\[Sum]\+\(l = 1\)\%k a\_l\)}]}], ")"}], ",", \(If[l \[LessEqual] k, a\_l, 0]\)}], "]"}]}]}], "]"}]}], ";", "\n", \(B[k_, n_]\ := \ Table[Bij[i, j, k, n]\ , {i, \ n}, \ {j, \ n}]\), ";", "\n", \(Bx\ = \ B[k, n]\), ";", "\n", \(Print["\", MatrixForm[Bx]]\)}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"B = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", GridBox[{ {\(1 - 2\ \((a\_1 + a\_2)\)\), \(a\_1\), \(a\_2\), "0", \(a\_2\), \(a\_1\)}, {\(a\_1\), \(1 - 2\ \((a\_1 + a\_2)\)\), \(a\_1\), \(a\_2\), "0", \(a\_2\)}, {\(a\_2\), \(a\_1\), \(1 - 2\ \((a\_1 + a\_2)\)\), \(a\_1\), \(a\_2\), "0"}, {"0", \(a\_2\), \(a\_1\), \(1 - 2\ \((a\_1 + a\_2)\)\), \(a\_1\), \(a\_2\)}, {\(a\_2\), "0", \(a\_2\), \(a\_1\), \(1 - 2\ \((a\_1 + a\_2)\)\), \(a\_1\)}, {\(a\_1\), \(a\_2\), "0", \(a\_2\), \(a\_1\), \(1 - 2\ \((a\_1 + a\_2)\)\)} }], ")"}], (MatrixForm[ #]&)]}], SequenceForm[ "B = ", MatrixForm[ {{ Plus[ 1, Times[ -2, Plus[ Subscript[ a, 1], Subscript[ a, 2]]]], Subscript[ a, 1], Subscript[ a, 2], 0, Subscript[ a, 2], Subscript[ a, 1]}, { Subscript[ a, 1], Plus[ 1, Times[ -2, Plus[ Subscript[ a, 1], Subscript[ a, 2]]]], Subscript[ a, 1], Subscript[ a, 2], 0, Subscript[ a, 2]}, { Subscript[ a, 2], Subscript[ a, 1], Plus[ 1, Times[ -2, Plus[ Subscript[ a, 1], Subscript[ a, 2]]]], Subscript[ a, 1], Subscript[ a, 2], 0}, {0, Subscript[ a, 2], Subscript[ a, 1], Plus[ 1, Times[ -2, Plus[ Subscript[ a, 1], Subscript[ a, 2]]]], Subscript[ a, 1], Subscript[ a, 2]}, { Subscript[ a, 2], 0, Subscript[ a, 2], Subscript[ a, 1], Plus[ 1, Times[ -2, Plus[ Subscript[ a, 1], Subscript[ a, 2]]]], Subscript[ a, 1]}, { Subscript[ a, 1], Subscript[ a, 2], 0, Subscript[ a, 2], Subscript[ a, 1], Plus[ 1, Times[ -2, Plus[ Subscript[ a, 1], Subscript[ a, 2]]]]}}]], Editable->False]], "Print"] }, Open ]], Cell[BoxData[ FormBox[ RowBox[{ \(Both\ of\ these\ matrices\ are\ recognized\ as\ convolutions . \ In\ general\), ",", \(the\ Fourier\ transform\ turns\ convolution\ into\ ordinary\ multiplication . \ In\ our\ discrete\ setting\), ",", \(this\ means\ that\ we\ can\ choose\ a\ coordinate\ system\ in\ which \ all\ convolutions\ can\ be\ represented\ as\ diagonal\ matrices . \ In\ other\ words\), ",", " ", \(all\ convolution\ matrices\ can\ be\ simultaneously\ diagonalized \), ",", " ", RowBox[{ "with", " ", "respect", " ", "to", " ", "a", " ", "coordinate", " ", "system", " ", "given", " ", "by", " ", "sin", " ", "and", " ", "cosine", " ", "functions", " ", "whose", " ", "periods", " ", "evenly", " ", "divide", " ", "our", " ", "interval", " ", "of", StyleBox[" ", "InlineFormula"], StyleBox["n", "InlineFormula"], StyleBox[" ", "InlineFormulaF"], "increments", " ", "of", " ", "\[CapitalDelta]x", " ", \(each . \)}]}], TextForm]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(basis[n_]\ := \ Table[Cos[ 2 Floor[j/2] i\ \[Pi]/n + If[j == 1, 0, Mod[j, 2]]\ \[Pi]/2], \ {i, n}, \ {j, n}]; \nCx\ = \ basis[n]; \ Cxi\ = \ Inverse[Cx]; \n Print["\", MatrixForm[Cx]]; \n Print[\*"\"\\"", MatrixForm[FullSimplify[Cxi . Ax . Cx]]]; \n Print[\*"\"\\"", MatrixForm[FullSimplify[Cxi . Bx . Cx]]]\n\)\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"C = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", GridBox[{ {"1", \(1\/2\), \(-\(\@3\/2\)\), \(-\(1\/2\)\), \(-\(\@3\/2\)\), \(-1\)}, {"1", \(-\(1\/2\)\), \(-\(\@3\/2\)\), \(-\(1\/2\)\), \(\@3\/2\), "1"}, {"1", \(-1\), "0", "1", "0", \(-1\)}, {"1", \(-\(1\/2\)\), \(\@3\/2\), \(-\(1\/2\)\), \(-\(\@3\/2\)\), "1"}, {"1", \(1\/2\), \(\@3\/2\), \(-\(1\/2\)\), \(\@3\/2\), \(-1\)}, {"1", "1", "0", "1", "0", "1"} }], ")"}], (MatrixForm[ #]&)]}], SequenceForm[ "C = ", MatrixForm[ {{1, Rational[ 1, 2], Times[ Rational[ -1, 2], Power[ 3, Rational[ 1, 2]]], Rational[ -1, 2], Times[ Rational[ -1, 2], Power[ 3, Rational[ 1, 2]]], -1}, {1, Rational[ -1, 2], Times[ Rational[ -1, 2], Power[ 3, Rational[ 1, 2]]], Rational[ -1, 2], Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]]], 1}, {1, -1, 0, 1, 0, -1}, {1, Rational[ -1, 2], Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]]], Rational[ -1, 2], Times[ Rational[ -1, 2], Power[ 3, Rational[ 1, 2]]], 1}, {1, Rational[ 1, 2], Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]]], Rational[ -1, 2], Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]]], -1}, {1, 1, 0, 1, 0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"C A \\!\\(C\\^\\(-1\\)\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"0", \(1 + a\_0\), "0", "0", "0", "0"}, {"0", "0", \(1 + a\_0\), "0", "0", "0"}, {"0", "0", "0", \(1 + 3\ a\_0\), "0", "0"}, {"0", "0", "0", "0", \(1 + 3\ a\_0\), "0"}, {"0", "0", "0", "0", "0", \(1 + 4\ a\_0\)} }], ")"}], (MatrixForm[ #]&)]}], SequenceForm[ "C A \!\(C\^\(-1\)\) = ", MatrixForm[ {{1, 0, 0, 0, 0, 0}, {0, Plus[ 1, Subscript[ a, 0]], 0, 0, 0, 0}, {0, 0, Plus[ 1, Subscript[ a, 0]], 0, 0, 0}, {0, 0, 0, Plus[ 1, Times[ 3, Subscript[ a, 0]]], 0, 0}, {0, 0, 0, 0, Plus[ 1, Times[ 3, Subscript[ a, 0]]], 0}, {0, 0, 0, 0, 0, Plus[ 1, Times[ 4, Subscript[ a, 0]]]}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"C B \\!\\(C\\^\\(-1\\)\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"0", \(1 - a\_1 - 3\ a\_2\), "0", "0", "0", "0"}, {"0", "0", \(1 - a\_1 - 3\ a\_2\), "0", "0", "0"}, {"0", "0", "0", \(1 - 3\ a\_1 - 3\ a\_2\), "0", "0"}, {"0", "0", "0", "0", \(1 - 3\ a\_1 - 3\ a\_2\), "0"}, {"0", "0", "0", "0", "0", \(1 - 4\ a\_1\)} }], ")"}], (MatrixForm[ #]&)]}], SequenceForm[ "C B \!\(C\^\(-1\)\) = ", MatrixForm[ {{1, 0, 0, 0, 0, 0}, {0, Plus[ 1, Times[ -1, Subscript[ a, 1]], Times[ -3, Subscript[ a, 2]]], 0, 0, 0, 0}, {0, 0, Plus[ 1, Times[ -1, Subscript[ a, 1]], Times[ -3, Subscript[ a, 2]]], 0, 0, 0}, {0, 0, 0, Plus[ 1, Times[ -3, Subscript[ a, 1]], Times[ -3, Subscript[ a, 2]]], 0, 0}, {0, 0, 0, 0, Plus[ 1, Times[ -3, Subscript[ a, 1]], Times[ -3, Subscript[ a, 2]]], 0}, {0, 0, 0, 0, 0, Plus[ 1, Times[ -4, Subscript[ a, 1]]]}}]], Editable->False]], "Print"] }, Open ]], Cell[BoxData[ FormBox[ RowBox[{\(In\ general\), ",", " ", RowBox[{ RowBox[{ "we", " ", "can", " ", "solve", " ", "for", " ", "these", " ", "eigenvalues", " ", "by", " ", "applying", " ", "our", " ", "recurrences", " ", "to", " ", StyleBox["u", "InlineFormula"]}], StyleBox[" ", "InlineFormula"], StyleBox["=", "InlineFormula"], StyleBox[" ", "InlineFormula"], RowBox[{ StyleBox[\(Cos[\[Phi]]\), "InlineFormula"], "using", " ", "increments", " ", "of", " ", "\[CapitalDelta]\[Phi]"}]}], ",", " ", RowBox[{ RowBox[{"and", " ", "then", " ", "substituting", " ", StyleBox[ RowBox[{"\[CapitalDelta]", StyleBox["\[Phi]", "InlineFormula"]}]]}], StyleBox[" ", "InlineFormula"], StyleBox["=", "InlineFormula"], StyleBox[" ", "InlineFormula"], RowBox[{ RowBox[{ StyleBox[\(\(2 \[Pi]\ m\)\/n\), "InlineFormula"], " ", "for", " ", StyleBox["m", "InlineFormula"]}], StyleBox[" ", "InlineFormula"], StyleBox["=", "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox["1", "InlineFormula"]}]}], StyleBox[",", "InlineFormula"], StyleBox["...", "InlineFormula"], StyleBox[",", "InlineFormula"], StyleBox[\(\(n\/2\) : \), "InlineFormula"]}], TextForm]], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{\(\[Lambda]\_0\), " ", StyleBox[\(Cos[\[Phi]]\), "InlineFormula"]}], "=", " ", RowBox[{ RowBox[{ RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["+", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(a\_0\)}]}], ")"}], StyleBox[\(Cos[\[Phi]]\), "InlineFormula"]}], "-", RowBox[{\(a\_0\), RowBox[{"(", RowBox[{ StyleBox[\(Cos[\[Phi] - \ \[CapitalDelta]\[Phi]]\), "InlineFormula"], "+", StyleBox[" ", "InlineFormula"], StyleBox[\(Cos[\[Phi] + \ \[CapitalDelta]\[Phi]]\), "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}]}]}], RowBox[{ RowBox[{\(\[Lambda]\_0\), " ", StyleBox[\(Cos[\[Phi]]\), "InlineFormula"]}], "=", RowBox[{ RowBox[{ RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["+", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(a\_0\)}]}], ")"}], StyleBox[\(Cos[\[Phi]]\), "InlineFormula"]}], "-", RowBox[{"2", \(a\_0\), StyleBox[\(Cos[\[Phi]]\), "InlineFormula"], StyleBox[\(Cos[\[CapitalDelta]\[Phi]]\), "InlineFormula"]}]}]}], RowBox[{ RowBox[{\(\[Lambda]\_0\), "=", RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["+", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(a\_0\), RowBox[{"(", RowBox[{"1", "-", StyleBox[\(Cos[\[CapitalDelta]\[Phi]]\), "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}]}]}], "\n"}], RowBox[{ RowBox[{\(\[Lambda]\_1\), " ", StyleBox[\(Cos[\[Phi]]\), "InlineFormula"]}], "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["-", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(\[Sum]\+\(j = 1\)\%k a\_j\)}]}], ")"}], StyleBox[\(Cos[\[Phi]]\), "InlineFormula"]}], StyleBox["+", "InlineFormula"], RowBox[{\(\[Sum]\+\(j = 1\)\%k\), RowBox[{\(a\_j\), RowBox[{"(", RowBox[{ StyleBox[\(Cos[\[Phi] - \ j\ \[CapitalDelta]\[Phi]]\), "InlineFormula"], "+", StyleBox[" ", "InlineFormula"], StyleBox[\(Cos[\[Phi] + \ j\ \[CapitalDelta]\[Phi]]\), "InlineFormula"]}], StyleBox[")", "InlineFormula"]}], "\n", \(\[Lambda]\_1\), " ", StyleBox[\(Cos[\[Phi]]\), "InlineFormula"]}]}]}], "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{ StyleBox["(", "InlineFormula"], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["-", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(\[Sum]\+\(j = 1\)\%k a\_j\)}]}], ")"}], StyleBox[\(Cos[\[Phi]]\), "InlineFormula"]}], StyleBox["+", "InlineFormula"], RowBox[{\(\(2 \[Sum]\)\+\(j = 1\)\%k\), RowBox[{\(a\_j\), StyleBox[\(Cos[\[Phi]]\), "InlineFormula"], StyleBox[\(Cos[j\ \[CapitalDelta]\[Phi]]\), "InlineFormula"], "\n", \(\[Lambda]\_1\)}]}]}], "=", RowBox[{ RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["-", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], RowBox[{\(\[Sum]\+\(j = 1\)\%k\), RowBox[{\(a\_j\), RowBox[{"(", RowBox[{"1", "-", StyleBox[\(Cos[j\ \[CapitalDelta]\[Phi]]\), "InlineFormula"]}], StyleBox[")", "InlineFormula"]}], "\n", "\[Lambda]"}]}]}]}], " ", "=", " ", RowBox[{\(\[Lambda]\_1\/\[Lambda]\_0\), "=", FractionBox[ RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["-", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], RowBox[{\(\[Sum]\+\(j = 1\)\%k\), RowBox[{\(a\_j\), RowBox[{"(", RowBox[{"1", "-", StyleBox[\(Cos[j\ \[CapitalDelta]\[Phi]]\), "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}]}]}]}], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["+", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], \(a\_0\), RowBox[{"(", RowBox[{"1", "-", StyleBox[\(Cos[\[CapitalDelta]\[Phi]]\), "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}]}]]}]}]}]}]}]}], "DisplayFormula"], Cell[CellGroupData[{ Cell[BoxData[ \(eigenval[k_, \[CapitalDelta]\[Phi]_]\ := \((1 - \ \[Sum]\+\(j = 1\)\%k 2\ \(a\_j\) \((1\ - \ Cos[j\ \[CapitalDelta]\[Phi]])\))\)/ \((1 + \ 2\ \(a\_0\) \((1\ - \ Cos[\[CapitalDelta]\[Phi]])\))\); \n Print[eigenval[3, \[CapitalDelta]\[Phi]]]; \n Table[eigenval[k, 2 \[Pi]\ m/n], \ {m, n/2}]\)], "Input"], Cell[BoxData[ \(\(1 - 2\ \((1 - Cos[\[CapitalDelta]\[Phi]])\)\ a\_1 - 2\ \((1 - Cos[2\ \[CapitalDelta]\[Phi]])\)\ a\_2 - 2\ \((1 - Cos[3\ \[CapitalDelta]\[Phi]])\)\ a\_3\)\/\(1 + 2\ \((1 - Cos[\[CapitalDelta]\[Phi]])\)\ a\_0\)\)], "Print"], Cell[BoxData[ \({\(1 - a\_1 - 3\ a\_2\)\/\(1 + a\_0\), \(1 - 3\ a\_1 - 3\ a\_2\)\/\(1 + 3\ a\_0\), \(1 - 4\ a\_1\)\/\(1 + 4\ a\_0\)}\)], "Output"] }, Open ]], Cell[TextData[{ "On the other hand, what should these eigenvalues be? We want to study the \ heat equation for the initial condition u(x,0) = Cos[", Cell[BoxData[ \(TraditionalForm\`\[Omega]\/\[CapitalDelta]x\)]], "x], choosing \[Omega] = ", Cell[BoxData[ \(TraditionalForm\`\(2 \[Pi]\ m\)\/\(\ n\ \)\)]], " so Cos[", Cell[BoxData[ \(TraditionalForm\`\[Omega]\/\[CapitalDelta]x\)]], "x] completes m cycles in the interval ", StyleBox[ "n \[CapitalDelta]x corresponding to our circular grid. This has the \ general solution", "InlineFormula"] }], "Text"], Cell[BoxData[ \(TraditionalForm \`u(x, t)\ = \ E\^\(\(-\((\[Omega]\/\[CapitalDelta]x)\)\^2\) t\)\ Cos[\[Omega]\ \[CapitalDelta]x\ x]\)], "DisplayFormula"], Cell[TextData[{ "In particular, for a time increment of ", StyleBox["\[CapitalDelta]t we have the ", "InlineFormula"], StyleBox["exact", "InlineFormula", FontSlant->"Italic"], StyleBox[" solution", "InlineFormula"] }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "(", RowBox[{"x", ",", StyleBox[ RowBox[{ StyleBox["\[CapitalDelta]", "InlineFormula"], "t"}]]}], ")"}], " ", "=", " ", RowBox[{ \(\(E\^\(\(-w\^2\) p\)\) \(u(x, 0)\)\ \ where\ \ p\), " ", "=", " ", FractionBox[ StyleBox["\[CapitalDelta]t", "InlineFormula "], \(\[CapitalDelta]x\^2\)]}]}], TraditionalForm]], "DisplayFormula"], Cell[TextData[{ "In other words, the desired eigenvalues of our recurrence are ", Cell[BoxData[ \(TraditionalForm\`E\^\(\(-w\^2\) p\)\)]], " for ", Cell[BoxData[ FormBox[ StyleBox[\(m\ = \ 1, ... , n\/2\), "InlineFormula"], TextForm]]], ". We have \[CapitalDelta]\[Phi] = ", Cell[BoxData[ \(TraditionalForm\`\(\[Omega]\ \)\)]], ".\nFor specific values of m we want choose our parameters ", Cell[BoxData[ FormBox[ StyleBox[\(a\_0, a\_1, ... , a\_k\), "InlineFormula"], TextForm]]], "so" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(E\^\(\(-w\^2\) p\)\), " ", "=", " ", FractionBox[ RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["-", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], RowBox[{\(\[Sum]\+\(j = 1\)\%k\), RowBox[{\(a\_j\), "(", RowBox[{"1", "-", StyleBox[ RowBox[{"Cos", "[", RowBox[{"j", FormBox["\[Omega]", "TraditionalForm"]}], "]"}], "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}]}]}], RowBox[{ StyleBox["1", "InlineFormula"], StyleBox["+", "InlineFormula"], RowBox[{ StyleBox["2", "InlineFormula"], RowBox[{\(a\_0\), "(", RowBox[{"1", "-", StyleBox[ RowBox[{"Cos", "[", FormBox["\[Omega]", "TraditionalForm"], "]"}], "InlineFormula"]}], StyleBox[")", "InlineFormula"]}]}]}]]}]}], TraditionalForm]], "DisplayFormula"], Cell["\<\ or in other words we want solve a system of equations, each of the \ form\ \>", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ StyleBox["2", "InlineFormula"], StyleBox[" ", "InlineFormula"], RowBox[{\(E\^\(\(-w\^2\) p\)\), "(", RowBox[{"1", "-", StyleBox[ RowBox[{"Cos", "[", FormBox["\[Omega]", "TraditionalForm"], "]"}], "InlineFormula"]}], StyleBox[")", "InlineFormula"]}], " ", \(a\_0\)}], " ", "+", " ", RowBox[{"2", " ", RowBox[{\(\[Sum]\+\(j = 1\)\%k\), RowBox[{ RowBox[{"(", RowBox[{"1", "-", StyleBox[ RowBox[{"Cos", "[", RowBox[{"j", FormBox["\[Omega]", "TraditionalForm"]}], "]"}], "InlineFormula"]}], StyleBox[")", "InlineFormula"]}], " ", \(a\_j\)}]}]}]}], " ", "=", " ", RowBox[{ \(1 - E\^\(\(-w\^2\) p\)\ \ \ \ for\ \ \ \ p\), " ", "=", " ", FractionBox[ StyleBox["\[CapitalDelta]t", "InlineFormula "], \(\[CapitalDelta]x\^2\)]}]}], ",", " ", RowBox[{"\[Omega]", " ", "=", " ", RowBox[{ FormBox[\(\(2 \[Pi]\ m\)\/\(\ n\ \)\), "TraditionalForm"], "."}]}]}], TraditionalForm]], "DisplayFormula"], Cell[BoxData[ FormBox[ RowBox[{\(For\ compatibility\ with\ other\ methods\), ",", " ", RowBox[{ "in", " ", "all", " ", "that", " ", "follows", " ", "we", " ", "rewrite", " ", "each", " ", FormBox[ StyleBox[\(\ a\_j\), "InlineFormula"], "TraditionalForm"], "as", StyleBox[ RowBox[{" ", StyleBox[" ", "InlineFormula"]}]], StyleBox[\(a\_j\), "InlineFormula"], StyleBox["p", "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox["to"], StyleBox[" "], StyleBox["make"], StyleBox[" "], StyleBox["the"], StyleBox[" "], StyleBox["dependence"], StyleBox[" "], StyleBox["on"], StyleBox[" "], StyleBox["p", "InlineFormula"], StyleBox[" ", "InlineFormula"], StyleBox[\(explicit . \)]}]}], TextForm]], "Text"], Cell[BoxData[ RowBox[{\(idealcoefs[n_, p_, m_]\), " ", ":=", "\n", "\t", RowBox[{"Module", "[", "\n", "\t\t", RowBox[{\({a, b, i, j, k, x, z}\), ",", "\n", "\t\t", RowBox[{ \(k\ = \ Length[m] - 1\), ";", "\n", "\t\t", \(x\ = \ Table[N[E^ \((\(-p\) \((2 \[Pi]\ m\[LeftDoubleBracket]i\[RightDoubleBracket]/n) \)^2)\)], \ {i, k + 1}]\), ";", "\n", "\t\t", \(a\ = \ Table[\n\t\t\t\t If[j == 1, \n\t\t\t\t\t x\[LeftDoubleBracket]i\[RightDoubleBracket] N[2 p \(( 1 - Cos[ 2 \[Pi]\ m\[LeftDoubleBracket]i\[RightDoubleBracket] j/n])\)], \n\t\t\t\t\t N[2 p \(( 1 - Cos[ 2 \[Pi]\ m\[LeftDoubleBracket]i\[RightDoubleBracket] \((j - 1)\)/n])\)]], \n \t\t\t\t{i, k + 1}, {j, k + 1}]\), ";", "\n", "\t\t", \(b\ = \ 1 - x\), ";", "\n", "\t\t", StyleBox[\(LinearSolve[N[a], N[b]]\), "MR"]}]}], "\n", "\t\t", "]"}]}]], "Input"], Cell[BoxData[ \(desired[n_, p_, k_]\ := \n\t Table[N[E^\((\(-p\) \((2 \[Pi]\ m/n)\)^2)\)], \ {m, k}]\)], "Input"], Cell[BoxData[ RowBox[{\(actual[n_, p_, k_, a_]\), " ", ":=", "\n", "\t", RowBox[{"Table", "[", RowBox[{ RowBox[{"N", "[", RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"2", RowBox[{"(", RowBox[{ UnderoverscriptBox["\[Sum]", \(j = 2\), StyleBox[\(Length[a]\), "TI\[CapitalUAcute]"]], StyleBox[ \(a\[LeftDoubleBracket]j\[RightDoubleBracket]\ p \((1 - Cos[2 \[Pi]\ m\ \((j - 1)\)/n])\)\), "TI"]}], StyleBox[")", "TI"]}]}]}], StyleBox[")", "TI"]}], StyleBox["/", "TI"], \((1 + 2 a\[LeftDoubleBracket]1\[RightDoubleBracket]\ p \((1 - Cos[2 \[Pi]\ m/n])\))\)}], "]"}], ",", "\n", "\t\t", \({m, k}\)}], "]"}]}]], "Input"], Cell[BoxData[ \(<< Graphics`MultipleListPlot`\)], "Input"], Cell[BoxData[ \(graphbasic[n_, p_, k_]\ := \n\t Module[\n\t\t{a, b, c, d, f}, \n\t\t a\ = \ {desired[n, p, k], \ "\"}; \n\t\t f[z_]\ := \ {actual[n, p, k, z], \ z}; \n\t\tb\ = \ f[{0, 1}]; \n \t\tc\ = \ f[{1, 0}]; \n\t\td\ = \ f[{ .5, .5}]; \n\t\t MultipleListPlot[\n\t\t\ta\[LeftDoubleBracket]1\[RightDoubleBracket], \ b\[LeftDoubleBracket]1\[RightDoubleBracket], \ c\[LeftDoubleBracket]1\[RightDoubleBracket], \ d\[LeftDoubleBracket]1\[RightDoubleBracket], \n\t\t\t PlotJoined\ -> \ True, \ SymbolShape\ -> \ None, AspectRatio\ -> 1.5, \n\t\t\t PlotLegend\ -> \ {"\", "\", \ "\", "\"}]; \n \t]\)], "Input"], Cell[BoxData[ \(graphbasicerror[n_, p_, k_, t_]\ := \n\t Module[\n\t\t{a, b, c, d, f}, \n\t\ta\ = \ desired[n, p, k]; \n\t\t f[z_]\ := \ {actual[n, p, k, z]^t - a^t, \ z}; \n\t\t b\ = \ f[{0, 1}]; \n\t\tc\ = \ f[{1, 0}]; \n\t\t d\ = \ f[{ .5, .5}]; \n\t\t MultipleListPlot[\n\t\t\tb\[LeftDoubleBracket]1\[RightDoubleBracket], \ c\[LeftDoubleBracket]1\[RightDoubleBracket], \ d\[LeftDoubleBracket]1\[RightDoubleBracket], \n\t\t\t PlotJoined\ -> \ True, \ SymbolShape\ -> \ None, AspectRatio\ -> 1.5, PlotRange\ -> \ All, AxesOrigin\ -> \ {1, 0}, \n\t\t\t PlotLegend\ -> \ {"\", \ "\", "\"}]; \n \t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(graphbasic[50, .4, 25]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .86131 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.297242 0.410149 0.430656 0.410149 [ [ 0 0 0 0 ] [ 1 .86131 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath % Start of sub-graphic p 0.0238095 0.0205074 0.570674 0.840805 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0380952 0.573536 0.896369 [ [.21429 .56104 -3 -9 ] [.21429 .56104 3 0 ] [.40476 .56104 -6 -9 ] [.40476 .56104 6 0 ] [.59524 .56104 -6 -9 ] [.59524 .56104 6 0 ] [.78571 .56104 -6 -9 ] [.78571 .56104 6 0 ] [.97619 .56104 -6 -9 ] [.97619 .56104 6 0 ] [.01131 .12535 -24 -4.5 ] [.01131 .12535 0 4.5 ] [.01131 .34944 -30 -4.5 ] [.01131 .34944 0 4.5 ] [.01131 .79763 -24 -4.5 ] [.01131 .79763 0 4.5 ] [.01131 1.02172 -18 -4.5 ] [.01131 1.02172 0 4.5 ] [.01131 1.24581 -24 -4.5 ] [.01131 1.24581 0 4.5 ] [.01131 1.4699 -6 -4.5 ] [.01131 1.4699 0 4.5 ] [ 0 0 0 0 ] [ 1 1.5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .57354 m .21429 .57979 L s [(5)] .21429 .56104 0 1 Mshowa .40476 .57354 m .40476 .57979 L s [(10)] .40476 .56104 0 1 Mshowa .59524 .57354 m .59524 .57979 L s [(15)] .59524 .56104 0 1 Mshowa .78571 .57354 m .78571 .57979 L s [(20)] .78571 .56104 0 1 Mshowa .97619 .57354 m .97619 .57979 L s [(25)] .97619 .56104 0 1 Mshowa .125 Mabswid .0619 .57354 m .0619 .57729 L s .1 .57354 m .1 .57729 L s .1381 .57354 m .1381 .57729 L s .17619 .57354 m .17619 .57729 L s .25238 .57354 m .25238 .57729 L s .29048 .57354 m .29048 .57729 L s .32857 .57354 m .32857 .57729 L s .36667 .57354 m .36667 .57729 L s .44286 .57354 m .44286 .57729 L s .48095 .57354 m .48095 .57729 L s .51905 .57354 m .51905 .57729 L s .55714 .57354 m .55714 .57729 L s .63333 .57354 m .63333 .57729 L s .67143 .57354 m .67143 .57729 L s .70952 .57354 m .70952 .57729 L s .74762 .57354 m .74762 .57729 L s .82381 .57354 m .82381 .57729 L s .8619 .57354 m .8619 .57729 L s .9 .57354 m .9 .57729 L s .9381 .57354 m .9381 .57729 L s .25 Mabswid 0 .57354 m 1 .57354 L s .02381 .12535 m .03006 .12535 L s [(-0.5)] .01131 .12535 1 0 Mshowa .02381 .34944 m .03006 .34944 L s [(-0.25)] .01131 .34944 1 0 Mshowa .02381 .79763 m .03006 .79763 L s [(0.25)] .01131 .79763 1 0 Mshowa .02381 1.02172 m .03006 1.02172 L s [(0.5)] .01131 1.02172 1 0 Mshowa .02381 1.24581 m .03006 1.24581 L s [(0.75)] .01131 1.24581 1 0 Mshowa .02381 1.4699 m .03006 1.4699 L s [(1)] .01131 1.4699 1 0 Mshowa .125 Mabswid .02381 .17017 m .02756 .17017 L s .02381 .21499 m .02756 .21499 L s .02381 .25981 m .02756 .25981 L s .02381 .30463 m .02756 .30463 L s .02381 .39426 m .02756 .39426 L s .02381 .43908 m .02756 .43908 L s .02381 .4839 m .02756 .4839 L s .02381 .52872 m .02756 .52872 L s .02381 .61835 m .02756 .61835 L s .02381 .66317 m .02756 .66317 L s .02381 .70799 m .02756 .70799 L s .02381 .75281 m .02756 .75281 L s .02381 .84245 m .02756 .84245 L s .02381 .88726 m .02756 .88726 L s .02381 .93208 m .02756 .93208 L s .02381 .9769 m .02756 .9769 L s .02381 1.06654 m .02756 1.06654 L s .02381 1.11136 m .02756 1.11136 L s .02381 1.15618 m .02756 1.15618 L s .02381 1.20099 m .02756 1.20099 L s .02381 1.29063 m .02756 1.29063 L s .02381 1.33545 m .02756 1.33545 L s .02381 1.38027 m .02756 1.38027 L s .02381 1.42509 m .02756 1.42509 L s .02381 .08053 m .02756 .08053 L s .02381 .03571 m .02756 .03571 L s .25 Mabswid .02381 0 m .02381 1.5 L s 0 0 m 1 0 L 1 1.5 L 0 1.5 L closepath clip newpath .001 w .0619 1.46426 m .1 1.44754 L .1381 1.42037 L .17619 1.38374 L .21429 1.33897 L .25238 1.28759 L .29048 1.2313 L .32857 1.17184 L .36667 1.11092 L .40476 1.05015 L .44286 .99094 L .48095 .9345 L .51905 .88177 L .55714 .83344 L .59524 .78994 L .63333 .75145 L .67143 .71798 L .70952 .68933 L .74762 .6652 L .78571 .64518 L .82381 .62883 L .8619 .61568 L .9 .60525 L .9381 .59711 L .97619 .59083 L s [ .5 3 ] 0 Mabsdash .0619 1.46425 m .1 1.44738 L .1381 1.41955 L .17619 1.3812 L .21429 1.33295 L .25238 1.27555 L .29048 1.2099 L .32857 1.13705 L .36667 1.05813 L .40476 .9744 L .44286 .88718 L .48095 .79784 L .51905 .70778 L .55714 .61844 L .59524 .53121 L .63333 .44749 L .67143 .36857 L .70952 .29572 L .74762 .23007 L .78571 .17267 L .82381 .12441 L .8619 .08607 L .9 .05824 L .9381 .04137 L .97619 .03571 L s [ .5 4 5 4 ] 0 Mabsdash .0619 1.46429 m .1 1.44793 L .1381 1.42223 L .17619 1.38919 L .21429 1.3511 L .25238 1.31018 L .29048 1.26836 L .32857 1.22718 L .36667 1.18775 L .40476 1.1508 L .44286 1.11676 L .48095 1.08581 L .51905 1.058 L .55714 1.03323 L .59524 1.01138 L .63333 .99228 L .67143 .97574 L .70952 .96158 L .74762 .94966 L .78571 .93982 L .82381 .93193 L .8619 .92591 L .9 .92166 L .9381 .91913 L .97619 .91829 L s [ 5 2 ] 0 Mabsdash .0619 1.46427 m .1 1.44766 L .1381 1.42092 L .17619 1.38539 L .21429 1.34267 L .25238 1.29456 L .29048 1.24284 L .32857 1.18917 L .36667 1.13505 L .40476 1.0817 L .44286 1.03013 L .48095 .98109 L .51905 .93513 L .55714 .89262 L .59524 .85381 L .63333 .81881 L .67143 .78768 L .70952 .76041 L .74762 .73696 L .78571 .71728 L .82381 .70128 L .8619 .68892 L .9 .68013 L .9381 .67488 L .97619 .67313 L s MathSubEnd P % End of sub-graphic 0 g .74841 .2666 m .74841 .59472 L .97619 .59472 L .97619 .2666 L F 1 g .74841 .2666 m .74841 .59472 L .97619 .59472 L .97619 .2666 L F 0 g .001 w [ ] 0 setdash .74841 .2666 m .97619 .2666 L .97619 .59472 L .74841 .59472 L .74841 .2666 L s % Start of sub-graphic p 0.748405 0.266597 0.97619 0.594716 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0297619 0.297619 0.0297619 0.297619 [ [ 0 0 0 0 ] [ 1 1.44048 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1.44048 L 0 1.44048 L closepath clip newpath % Start of sub-graphic p 0.0535714 0.0535714 0.35119 0.35119 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .00935 w [ 5 2 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .20238 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (C) show 69.000000 13.000000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (-) show 75.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (N) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Math1Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math1Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math1Mono %%+ font Courier grestore % Start of sub-graphic p 0.0535714 0.39881 0.35119 0.696429 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .00935 w [ .5 4 5 4 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .54762 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (implicit) show 111.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 0.744048 0.35119 1.04167 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .00935 w [ .5 3 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .89286 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (explicit) show 111.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 1.08929 0.35119 1.3869 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .00935 w [ ] 0 setdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 1.2381 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (true) show 87.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore MathSubEnd P % End of sub-graphic 0 0 m 1 0 L 1 .86131 L 0 .86131 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 248}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(graphbasicerror[50, .4, 10, 20]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .80093 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.278073 0.381395 0.400464 0.381395 [ [ 0 0 0 0 ] [ 1 .80093 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath % Start of sub-graphic p 0.0238095 0.0190697 0.532336 0.781859 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0820106 0.10582 0.4607 52.7831 [ [.12963 .4482 -3 -9 ] [.12963 .4482 3 0 ] [.34127 .4482 -3 -9 ] [.34127 .4482 3 0 ] [.55291 .4482 -3 -9 ] [.55291 .4482 3 0 ] [.76455 .4482 -3 -9 ] [.76455 .4482 3 0 ] [.97619 .4482 -6 -9 ] [.97619 .4482 6 0 ] [.01131 .19678 -36 -4.5 ] [.01131 .19678 0 4.5 ] [.01131 .72462 -30 -4.5 ] [.01131 .72462 0 4.5 ] [.01131 .98853 -24 -4.5 ] [.01131 .98853 0 4.5 ] [.01131 1.25245 -30 -4.5 ] [.01131 1.25245 0 4.5 ] [ 0 0 0 0 ] [ 1 1.5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .12963 .4607 m .12963 .46695 L s [(2)] .12963 .4482 0 1 Mshowa .34127 .4607 m .34127 .46695 L s [(4)] .34127 .4482 0 1 Mshowa .55291 .4607 m .55291 .46695 L s [(6)] .55291 .4482 0 1 Mshowa .76455 .4607 m .76455 .46695 L s [(8)] .76455 .4482 0 1 Mshowa .97619 .4607 m .97619 .46695 L s [(10)] .97619 .4482 0 1 Mshowa .125 Mabswid .18254 .4607 m .18254 .46445 L s .23545 .4607 m .23545 .46445 L s .28836 .4607 m .28836 .46445 L s .39418 .4607 m .39418 .46445 L s .44709 .4607 m .44709 .46445 L s .5 .4607 m .5 .46445 L s .60582 .4607 m .60582 .46445 L s .65873 .4607 m .65873 .46445 L s .71164 .4607 m .71164 .46445 L s .81746 .4607 m .81746 .46445 L s .87037 .4607 m .87037 .46445 L s .92328 .4607 m .92328 .46445 L s .07672 .4607 m .07672 .46445 L s .25 Mabswid 0 .4607 m 1 .4607 L s .02381 .19678 m .03006 .19678 L s [(-0.005)] .01131 .19678 1 0 Mshowa .02381 .72462 m .03006 .72462 L s [(0.005)] .01131 .72462 1 0 Mshowa .02381 .98853 m .03006 .98853 L s [(0.01)] .01131 .98853 1 0 Mshowa .02381 1.25245 m .03006 1.25245 L s [(0.015)] .01131 1.25245 1 0 Mshowa .125 Mabswid .02381 .24957 m .02756 .24957 L s .02381 .30235 m .02756 .30235 L s .02381 .35513 m .02756 .35513 L s .02381 .40792 m .02756 .40792 L s .02381 .51348 m .02756 .51348 L s .02381 .56627 m .02756 .56627 L s .02381 .61905 m .02756 .61905 L s .02381 .67183 m .02756 .67183 L s .02381 .7774 m .02756 .7774 L s .02381 .83018 m .02756 .83018 L s .02381 .88296 m .02756 .88296 L s .02381 .93575 m .02756 .93575 L s .02381 1.04131 m .02756 1.04131 L s .02381 1.0941 m .02756 1.0941 L s .02381 1.14688 m .02756 1.14688 L s .02381 1.19966 m .02756 1.19966 L s .02381 .144 m .02756 .144 L s .02381 .09122 m .02756 .09122 L s .02381 .03843 m .02756 .03843 L s .02381 1.30523 m .02756 1.30523 L s .02381 1.35801 m .02756 1.35801 L s .02381 1.4108 m .02756 1.4108 L s .02381 1.46358 m .02756 1.46358 L s .25 Mabswid .02381 0 m .02381 1.5 L s .001 w .02381 .44984 m .12963 .34086 L .23545 .13551 L .34127 .03571 L .44709 .13328 L .55291 .29957 L .65873 .40836 L .76455 .44935 L .87037 .45904 L .97619 .46053 L s [ .5 3 ] 0 Mabsdash .02381 .48687 m .12963 .74433 L .23545 1.21931 L .34127 1.46429 L .44709 1.29015 L .55291 .94392 L .65873 .67634 L .76455 .53985 L .87037 .48629 L .97619 .46848 L s [ .5 4 5 4 ] 0 Mabsdash .02381 .46841 m .12963 .54449 L .23545 .68408 L .34127 .75034 L .44709 .68816 L .55291 .58057 L .65873 .5057 L .76455 .47323 L .87037 .46337 L .97619 .46115 L s 0 0 m 1 0 L 1 1.5 L 0 1.5 L closepath clip newpath MathSubEnd P % End of sub-graphic 0 g .69761 .24791 m .69761 .55302 L .97619 .55302 L .97619 .24791 L F 1 g .69761 .24791 m .69761 .55302 L .97619 .55302 L .97619 .24791 L F 0 g .001 w [ ] 0 setdash .69761 .24791 m .97619 .24791 L .97619 .55302 L .69761 .55302 L .69761 .24791 L s % Start of sub-graphic p 0.697607 0.247906 0.97619 0.553022 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0297619 0.297619 0.0297619 0.297619 [ [ 0 0 0 0 ] [ 1 1.09524 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1.09524 L 0 1.09524 L closepath clip newpath % Start of sub-graphic p 0.0535714 0.0535714 0.35119 0.35119 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .00701 w [ .5 4 5 4 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .20238 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (C) show 69.000000 13.000000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (-) show 75.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (N) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Math1Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math1Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math1Mono %%+ font Courier grestore % Start of sub-graphic p 0.0535714 0.39881 0.35119 0.696429 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .00701 w [ .5 3 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .54762 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (implicit) show 111.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 0.744048 0.35119 1.04167 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .00701 w [ ] 0 setdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .89286 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (explicit) show 111.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore MathSubEnd P % End of sub-graphic 0 0 m 1 0 L 1 .80093 L 0 .80093 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 230.625}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[BoxData[ \(\(graphsub[f_, n_, p_, m2_, m3_, m4_, k_]\ := \n\t Module[\n\t\t{z2, z3, z4, y}, \n\t\tz2\ = \ idealcoefs[n, p, m2]; \n \t\tz3\ = \ idealcoefs[n, p, m3]; \n\t\t z4\ = \ idealcoefs[n, p, m4]; \n \t\t{f[{ .5, .5, 0, 0}], f[z2], f[z3], f[z4]}\n]\t\)\)], "Input"], Cell[BoxData[ \(graphideal[n_, p_, m2_, m3_, m4_, k_]\ := \n\t Module[\n\t\t{a, b, c, d, e, f}, \n\t\t a\ = \ {desired[n, p, k], \ "\"}; \n\t\t f[z_]\ := \ {actual[n, p, k, z], \ z}; \n \t\t{b, c, d, e}\ = \ graphsub[f, n, p, m2, m3, m4, k]\ ; \n\t\t MultipleListPlot[\n\t\t\ta\[LeftDoubleBracket]1\[RightDoubleBracket], \ b\[LeftDoubleBracket]1\[RightDoubleBracket], \ c\[LeftDoubleBracket]1\[RightDoubleBracket], \ d\[LeftDoubleBracket]1\[RightDoubleBracket], \ e\[LeftDoubleBracket]1\[RightDoubleBracket], \n\t\t\t PlotJoined\ -> \ True, \ SymbolShape\ -> \ None, AspectRatio\ -> 1, \n\t\t\t PlotLegend\ -> \ {"\", \ "\", \ "\<2nd\>", \ "\<3rd\>", \ "\<4th\>"}]; \n\t]\)], "Input"], Cell[BoxData[ \(grapherror[n_, p_, m2_, m3_, m4_, k_, t_]\ := \n\t Module[\n\t\t{a, b, e, f, z2, z3}, \n\t\te\ = desired[n, p, k]; \n\t\t f[z_]\ := \ {actual[n, p, k, z]^t - e^t, \ z}; \n \t\t{a, b, c, d}\ = \ graphsub[f, n, p, m2, m3, m4, k]\ ; \n\t\t ListPlot[a\[LeftDoubleBracket]1\[RightDoubleBracket], \n\t\t\t PlotJoined\ -> \ True, ImageSize\ -> \ {244, 144}, AspectRatio\ -> \ .5, \ PlotRange\ -> \ All, AxesOrigin\ -> \ {1, 0}, \ PlotLabel\ -> N[a\[LeftDoubleBracket]2\[RightDoubleBracket], 4]]; \n\t\tListPlot[b\[LeftDoubleBracket]1\[RightDoubleBracket], \n\t\t\t PlotJoined\ -> \ True, \ ImageSize\ -> \ {244, 144}, AspectRatio\ -> \ .5, PlotRange\ -> \ All, AxesOrigin\ -> \ {1, 0}, \ PlotLabel\ -> N[b\[LeftDoubleBracket]2\[RightDoubleBracket], 4]]; \n\t\tListPlot[c\[LeftDoubleBracket]1\[RightDoubleBracket], \n\t\t\t PlotJoined\ -> \ True, \ ImageSize\ -> \ {244, 144}, AspectRatio\ -> \ .5, PlotRange\ -> \ All, AxesOrigin\ -> \ {1, 0}, \ PlotLabel\ -> N[c\[LeftDoubleBracket]2\[RightDoubleBracket], 4]]; \n\t\tListPlot[d\[LeftDoubleBracket]1\[RightDoubleBracket], \n\t\t\t PlotJoined\ -> \ True, ImageSize\ -> \ {244, 144}, AspectRatio\ -> \ .5, \ PlotRange\ -> \ All, AxesOrigin\ -> \ {1, 0}, \ PlotLabel\ -> N[d\[LeftDoubleBracket]2\[RightDoubleBracket], 4]]; \n\t\t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(graphideal[50, .5, {1, 4}, {1, 3, 5}, {1, 3, 5, 7}, 25]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .78493 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.397585 0.373776 0.392465 0.373776 [ [ 0 0 0 0 ] [ 1 .78493 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath % Start of sub-graphic p 0.0238095 0.0186888 0.771361 0.766241 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0380952 0.175063 0.807469 [ [.21429 .16256 -3 -9 ] [.21429 .16256 3 0 ] [.40476 .16256 -6 -9 ] [.40476 .16256 6 0 ] [.59524 .16256 -6 -9 ] [.59524 .16256 6 0 ] [.78571 .16256 -6 -9 ] [.78571 .16256 6 0 ] [.97619 .16256 -6 -9 ] [.97619 .16256 6 0 ] [.01131 .01357 -24 -4.5 ] [.01131 .01357 0 4.5 ] [.01131 .33656 -18 -4.5 ] [.01131 .33656 0 4.5 ] [.01131 .49805 -18 -4.5 ] [.01131 .49805 0 4.5 ] [.01131 .65954 -18 -4.5 ] [.01131 .65954 0 4.5 ] [.01131 .82104 -18 -4.5 ] [.01131 .82104 0 4.5 ] [.01131 .98253 -6 -4.5 ] [.01131 .98253 0 4.5 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .17506 m .21429 .18131 L s [(5)] .21429 .16256 0 1 Mshowa .40476 .17506 m .40476 .18131 L s [(10)] .40476 .16256 0 1 Mshowa .59524 .17506 m .59524 .18131 L s [(15)] .59524 .16256 0 1 Mshowa .78571 .17506 m .78571 .18131 L s [(20)] .78571 .16256 0 1 Mshowa .97619 .17506 m .97619 .18131 L s [(25)] .97619 .16256 0 1 Mshowa .125 Mabswid .0619 .17506 m .0619 .17881 L s .1 .17506 m .1 .17881 L s .1381 .17506 m .1381 .17881 L s .17619 .17506 m .17619 .17881 L s .25238 .17506 m .25238 .17881 L s .29048 .17506 m .29048 .17881 L s .32857 .17506 m .32857 .17881 L s .36667 .17506 m .36667 .17881 L s .44286 .17506 m .44286 .17881 L s .48095 .17506 m .48095 .17881 L s .51905 .17506 m .51905 .17881 L s .55714 .17506 m .55714 .17881 L s .63333 .17506 m .63333 .17881 L s .67143 .17506 m .67143 .17881 L s .70952 .17506 m .70952 .17881 L s .74762 .17506 m .74762 .17881 L s .82381 .17506 m .82381 .17881 L s .8619 .17506 m .8619 .17881 L s .9 .17506 m .9 .17881 L s .9381 .17506 m .9381 .17881 L s .25 Mabswid 0 .17506 m 1 .17506 L s .02381 .01357 m .03006 .01357 L s [(-0.2)] .01131 .01357 1 0 Mshowa .02381 .33656 m .03006 .33656 L s [(0.2)] .01131 .33656 1 0 Mshowa .02381 .49805 m .03006 .49805 L s [(0.4)] .01131 .49805 1 0 Mshowa .02381 .65954 m .03006 .65954 L s [(0.6)] .01131 .65954 1 0 Mshowa .02381 .82104 m .03006 .82104 L s [(0.8)] .01131 .82104 1 0 Mshowa .02381 .98253 m .03006 .98253 L s [(1)] .01131 .98253 1 0 Mshowa .125 Mabswid .02381 .05394 m .02756 .05394 L s .02381 .09432 m .02756 .09432 L s .02381 .13469 m .02756 .13469 L s .02381 .21544 m .02756 .21544 L s .02381 .25581 m .02756 .25581 L s .02381 .29618 m .02756 .29618 L s .02381 .37693 m .02756 .37693 L s .02381 .4173 m .02756 .4173 L s .02381 .45768 m .02756 .45768 L s .02381 .53842 m .02756 .53842 L s .02381 .5788 m .02756 .5788 L s .02381 .61917 m .02756 .61917 L s .02381 .69992 m .02756 .69992 L s .02381 .74029 m .02756 .74029 L s .02381 .78067 m .02756 .78067 L s .02381 .86141 m .02756 .86141 L s .02381 .90179 m .02756 .90179 L s .02381 .94216 m .02756 .94216 L s .25 Mabswid .02381 0 m .02381 1 L s 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath .001 w .0619 .97618 m .1 .95743 L .1381 .92714 L .17619 .8867 L .21429 .83789 L .25238 .78275 L .29048 .72347 L .32857 .66222 L .36667 .60103 L .40476 .54169 L .44286 .48567 L .48095 .43409 L .51905 .38769 L .55714 .34687 L .59524 .31171 L .63333 .28204 L .67143 .2575 L .70952 .2376 L .74762 .22175 L .78571 .20938 L .82381 .19989 L .8619 .19274 L .9 .18746 L .9381 .18361 L .97619 .18087 L s [ .5 3 ] 0 Mabsdash .0619 .97619 m .1 .95756 L .1381 .92775 L .17619 .88847 L .21429 .84176 L .25238 .7898 L .29048 .73469 L .32857 .67833 L .36667 .62229 L .40476 .56785 L .44286 .51595 L .48095 .46724 L .51905 .42215 L .55714 .38093 L .59524 .34368 L .63333 .31041 L .67143 .28107 L .70952 .25555 L .74762 .23375 L .78571 .21555 L .82381 .20083 L .8619 .18949 L .9 .18146 L .9381 .17666 L .97619 .17506 L s [ .5 4 5 4 ] 0 Mabsdash .0619 .97618 m .1 .95743 L .1381 .92716 L .17619 .8867 L .21429 .83777 L .25238 .78225 L .29048 .72207 L .32857 .65909 L .36667 .595 L .40476 .53128 L .44286 .46913 L .48095 .40953 L .51905 .35323 L .55714 .30078 L .59524 .25255 L .63333 .20879 L .67143 .16965 L .70952 .13519 L .74762 .10543 L .78571 .08035 L .82381 .05991 L .8619 .04408 L .9 .03281 L .9381 .02606 L .97619 .02381 L s [ 5 2 ] 0 Mabsdash .0619 .97618 m .1 .95743 L .1381 .92714 L .17619 .8867 L .21429 .83789 L .25238 .78275 L .29048 .72349 L .32857 .66228 L .36667 .60119 L .40476 .54206 L .44286 .48641 L .48095 .43544 L .51905 .38997 L .55714 .35045 L .59524 .317 L .63333 .28949 L .67143 .2675 L .70952 .25048 L .74762 .23774 L .78571 .22857 L .82381 .22224 L .8619 .21809 L .9 .21556 L .9381 .21422 L .97619 .2138 L s [ .5 4 5 4 .5 3 ] 0 Mabsdash .0619 .97618 m .1 .95743 L .1381 .92714 L .17619 .8867 L .21429 .83789 L .25238 .78275 L .29048 .72347 L .32857 .66222 L .36667 .60103 L .40476 .54168 L .44286 .48566 L .48095 .43407 L .51905 .38766 L .55714 .34681 L .59524 .31161 L .63333 .2819 L .67143 .25731 L .70952 .23737 L .74762 .22154 L .78571 .20925 L .82381 .19999 L .8619 .19329 L .9 .18879 L .9381 .1862 L .97619 .18535 L s MathSubEnd P % End of sub-graphic 0 g .80874 .24295 m .80874 .54198 L .97619 .54198 L .97619 .24295 L F 1 g .80874 .24295 m .80874 .54198 L .97619 .54198 L .97619 .24295 L F 0 g .001 w [ ] 0 setdash .80874 .24295 m .97619 .24295 L .97619 .54198 L .80874 .54198 L .80874 .24295 L s % Start of sub-graphic p 0.808739 0.242954 0.97619 0.541975 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0297619 0.297619 0.0297619 0.297619 [ [ 0 0 0 0 ] [ 1 1.78571 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1.78571 L 0 1.78571 L closepath clip newpath % Start of sub-graphic p 0.0535714 0.0535714 0.35119 0.35119 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ .5 4 5 4 .5 3 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .20238 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (4) show 69.000000 13.000000 moveto (th) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 0.39881 0.35119 0.696429 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ 5 2 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .54762 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (3) show 69.000000 13.000000 moveto (rd) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 0.744048 0.35119 1.04167 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ .5 4 5 4 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .89286 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (2) show 69.000000 13.000000 moveto (nd) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 1.08929 0.35119 1.3869 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ .5 3 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 1.2381 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (C) show 69.000000 13.000000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (-) show 75.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (N) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Math1Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math1Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math1Mono %%+ font Courier grestore % Start of sub-graphic p 0.0535714 1.43452 0.35119 1.73214 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ ] 0 setdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 1.58333 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (true) show 87.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore MathSubEnd P % End of sub-graphic 0 0 m 1 0 L 1 .78493 L 0 .78493 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 226}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(grapherror[50, .5, {1, 4}, {1, 3, 5}, {1, 3, 5, 7}, 10, 20]\)], "Input"], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .5 %%ImageSize: 244 144 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0820106 0.10582 0.0118565 117.194 [ [.12963 -0.00064 -3 -9 ] [.12963 -0.00064 3 0 ] [.34127 -0.00064 -3 -9 ] [.34127 -0.00064 3 0 ] [.55291 -0.00064 -3 -9 ] [.55291 -0.00064 3 0 ] [.76455 -0.00064 -3 -9 ] [.76455 -0.00064 3 0 ] [.97619 -0.00064 -6 -9 ] [.97619 -0.00064 6 0 ] [.01131 .12905 -30 -4.5 ] [.01131 .12905 0 4.5 ] [.01131 .24625 -30 -4.5 ] [.01131 .24625 0 4.5 ] [.01131 .36344 -30 -4.5 ] [.01131 .36344 0 4.5 ] [.01131 .48063 -30 -4.5 ] [.01131 .48063 0 4.5 ] [.5 .5125 -46.3438 0 ] [.5 .5125 46.3438 12 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .12963 .01186 m .12963 .01811 L s [(2)] .12963 -0.00064 0 1 Mshowa .34127 .01186 m .34127 .01811 L s [(4)] .34127 -0.00064 0 1 Mshowa .55291 .01186 m .55291 .01811 L s [(6)] .55291 -0.00064 0 1 Mshowa .76455 .01186 m .76455 .01811 L s [(8)] .76455 -0.00064 0 1 Mshowa .97619 .01186 m .97619 .01811 L s [(10)] .97619 -0.00064 0 1 Mshowa .125 Mabswid .18254 .01186 m .18254 .01561 L s .23545 .01186 m .23545 .01561 L s .28836 .01186 m .28836 .01561 L s .39418 .01186 m .39418 .01561 L s .44709 .01186 m .44709 .01561 L s .5 .01186 m .5 .01561 L s .60582 .01186 m .60582 .01561 L s .65873 .01186 m .65873 .01561 L s .71164 .01186 m .71164 .01561 L s .81746 .01186 m .81746 .01561 L s .87037 .01186 m .87037 .01561 L s .92328 .01186 m .92328 .01561 L s .07672 .01186 m .07672 .01561 L s .25 Mabswid 0 .01186 m 1 .01186 L s .02381 .12905 m .03006 .12905 L s [(0.001)] .01131 .12905 1 0 Mshowa .02381 .24625 m .03006 .24625 L s [(0.002)] .01131 .24625 1 0 Mshowa .02381 .36344 m .03006 .36344 L s [(0.003)] .01131 .36344 1 0 Mshowa .02381 .48063 m .03006 .48063 L s [(0.004)] .01131 .48063 1 0 Mshowa .125 Mabswid .02381 .0353 m .02756 .0353 L s .02381 .05873 m .02756 .05873 L s .02381 .08217 m .02756 .08217 L s .02381 .10561 m .02756 .10561 L s .02381 .15249 m .02756 .15249 L s .02381 .17593 m .02756 .17593 L s .02381 .19937 m .02756 .19937 L s .02381 .22281 m .02756 .22281 L s .02381 .26968 m .02756 .26968 L s .02381 .29312 m .02756 .29312 L s .02381 .31656 m .02756 .31656 L s .02381 .34 m .02756 .34 L s .02381 .38688 m .02756 .38688 L s .02381 .41032 m .02756 .41032 L s .02381 .43376 m .02756 .43376 L s .02381 .45719 m .02756 .45719 L s .25 Mabswid .02381 0 m .02381 .5 L s gsave .5 .5125 -107.344 -4 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 63.000000 13.000000 moveto (8) show 69.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.5) show 87.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 96.562500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.5) show 114.562500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 124.125000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0) show 130.125000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 139.687500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 145.687500 13.000000 moveto (<) show 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Math2Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math2Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math2Mono %%+ font Courier grestore .5 Mabswid .02381 .03256 m .12963 .21572 L .23545 .47322 L .34127 .4881 L .44709 .29129 L .55291 .11516 L .65873 .03739 L .76455 .01624 L .87037 .0124 L .97619 .0119 L s 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{244, 144}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .5 %%ImageSize: 244 144 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0820106 0.10582 0.231386 3228.87 [ [.12963 .21889 -3 -9 ] [.12963 .21889 3 0 ] [.34127 .21889 -3 -9 ] [.34127 .21889 3 0 ] [.55291 .21889 -3 -9 ] [.55291 .21889 3 0 ] [.76455 .21889 -3 -9 ] [.76455 .21889 3 0 ] [.97619 .21889 -6 -9 ] [.97619 .21889 6 0 ] [.01131 .03765 -48 -4.5 ] [.01131 .03765 0 4.5 ] [.01131 .10223 -48 -4.5 ] [.01131 .10223 0 4.5 ] [.01131 .16681 -48 -4.5 ] [.01131 .16681 0 4.5 ] [.01131 .29596 -42 -4.5 ] [.01131 .29596 0 4.5 ] [.01131 .36054 -42 -4.5 ] [.01131 .36054 0 4.5 ] [.01131 .42512 -42 -4.5 ] [.01131 .42512 0 4.5 ] [.01131 .4897 -42 -4.5 ] [.01131 .4897 0 4.5 ] [.5 .5125 -48.7812 0 ] [.5 .5125 48.7812 12 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .12963 .23139 m .12963 .23764 L s [(2)] .12963 .21889 0 1 Mshowa .34127 .23139 m .34127 .23764 L s [(4)] .34127 .21889 0 1 Mshowa .55291 .23139 m .55291 .23764 L s [(6)] .55291 .21889 0 1 Mshowa .76455 .23139 m .76455 .23764 L s [(8)] .76455 .21889 0 1 Mshowa .97619 .23139 m .97619 .23764 L s [(10)] .97619 .21889 0 1 Mshowa .125 Mabswid .18254 .23139 m .18254 .23514 L s .23545 .23139 m .23545 .23514 L s .28836 .23139 m .28836 .23514 L s .39418 .23139 m .39418 .23514 L s .44709 .23139 m .44709 .23514 L s .5 .23139 m .5 .23514 L s .60582 .23139 m .60582 .23514 L s .65873 .23139 m .65873 .23514 L s .71164 .23139 m .71164 .23514 L s .81746 .23139 m .81746 .23514 L s .87037 .23139 m .87037 .23514 L s .92328 .23139 m .92328 .23514 L s .07672 .23139 m .07672 .23514 L s .25 Mabswid 0 .23139 m 1 .23139 L s .02381 .03765 m .03006 .03765 L s [(-0.00006)] .01131 .03765 1 0 Mshowa .02381 .10223 m .03006 .10223 L s [(-0.00004)] .01131 .10223 1 0 Mshowa .02381 .16681 m .03006 .16681 L s [(-0.00002)] .01131 .16681 1 0 Mshowa .02381 .29596 m .03006 .29596 L s [(0.00002)] .01131 .29596 1 0 Mshowa .02381 .36054 m .03006 .36054 L s [(0.00004)] .01131 .36054 1 0 Mshowa .02381 .42512 m .03006 .42512 L s [(0.00006)] .01131 .42512 1 0 Mshowa .02381 .4897 m .03006 .4897 L s [(0.00008)] .01131 .4897 1 0 Mshowa .125 Mabswid .02381 .0538 m .02756 .0538 L s .02381 .06994 m .02756 .06994 L s .02381 .08609 m .02756 .08609 L s .02381 .11838 m .02756 .11838 L s .02381 .13452 m .02756 .13452 L s .02381 .15066 m .02756 .15066 L s .02381 .18295 m .02756 .18295 L s .02381 .1991 m .02756 .1991 L s .02381 .21524 m .02756 .21524 L s .02381 .24753 m .02756 .24753 L s .02381 .26367 m .02756 .26367 L s .02381 .27982 m .02756 .27982 L s .02381 .31211 m .02756 .31211 L s .02381 .32825 m .02756 .32825 L s .02381 .3444 m .02756 .3444 L s .02381 .37669 m .02756 .37669 L s .02381 .39283 m .02756 .39283 L s .02381 .40897 m .02756 .40897 L s .02381 .44126 m .02756 .44126 L s .02381 .45741 m .02756 .45741 L s .02381 .47355 m .02756 .47355 L s .02381 .02151 m .02756 .02151 L s .02381 .00537 m .02756 .00537 L s .25 Mabswid .02381 0 m .02381 .5 L s gsave .5 .5125 -109.781 -4 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 63.000000 13.000000 moveto (8) show 69.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.3423) show 105.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 114.562500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.6578) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 150.562500 13.000000 moveto (<) show 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Math2Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math2Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math2Mono %%+ font Courier grestore .5 Mabswid .02381 .23139 m .12963 .39303 L .23545 .4881 L .34127 .23139 L .44709 .0119 L .55291 .05178 L .65873 .1612 L .76455 .21546 L .87037 .22916 L .97619 .23119 L s 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{244, 144}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .5 %%ImageSize: 244 144 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0820106 0.10582 0.241528 699804 [ [.12963 .22903 -3 -9 ] [.12963 .22903 3 0 ] [.34127 .22903 -3 -9 ] [.34127 .22903 3 0 ] [.55291 .22903 -3 -9 ] [.55291 .22903 3 0 ] [.76455 .22903 -3 -9 ] [.76455 .22903 3 0 ] [.97619 .22903 -6 -9 ] [.97619 .22903 6 0 ] [.01131 .03159 -39 -6.25 ] [.01131 .03159 0 6.25 ] [.01131 .10157 -39 -6.25 ] [.01131 .10157 0 6.25 ] [.01131 .17155 -39 -6.25 ] [.01131 .17155 0 6.25 ] [.01131 .31151 -33 -6.25 ] [.01131 .31151 0 6.25 ] [.01131 .38149 -33 -6.25 ] [.01131 .38149 0 6.25 ] [.01131 .45147 -33 -6.25 ] [.01131 .45147 0 6.25 ] [.5 .5125 -74.5625 0 ] [.5 .5125 74.5625 12 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .12963 .24153 m .12963 .24778 L s [(2)] .12963 .22903 0 1 Mshowa .34127 .24153 m .34127 .24778 L s [(4)] .34127 .22903 0 1 Mshowa .55291 .24153 m .55291 .24778 L s [(6)] .55291 .22903 0 1 Mshowa .76455 .24153 m .76455 .24778 L s [(8)] .76455 .22903 0 1 Mshowa .97619 .24153 m .97619 .24778 L s [(10)] .97619 .22903 0 1 Mshowa .125 Mabswid .18254 .24153 m .18254 .24528 L s .23545 .24153 m .23545 .24528 L s .28836 .24153 m .28836 .24528 L s .39418 .24153 m .39418 .24528 L s .44709 .24153 m .44709 .24528 L s .5 .24153 m .5 .24528 L s .60582 .24153 m .60582 .24528 L s .65873 .24153 m .65873 .24528 L s .71164 .24153 m .71164 .24528 L s .81746 .24153 m .81746 .24528 L s .87037 .24153 m .87037 .24528 L s .92328 .24153 m .92328 .24528 L s .07672 .24153 m .07672 .24528 L s .25 Mabswid 0 .24153 m 1 .24153 L s .02381 .03159 m .03006 .03159 L s gsave .01131 .03159 -39 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (-3) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-7) show grestore .02381 .10157 m .03006 .10157 L s gsave .01131 .10157 -39 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (-2) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-7) show grestore .02381 .17155 m .03006 .17155 L s gsave .01131 .17155 -39 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (-1) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-7) show grestore .02381 .31151 m .03006 .31151 L s gsave .01131 .31151 -33 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (1) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-7) show grestore .02381 .38149 m .03006 .38149 L s gsave .01131 .38149 -33 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (2) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-7) show grestore .02381 .45147 m .03006 .45147 L s gsave .01131 .45147 -33 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (3) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-7) show grestore .125 Mabswid .02381 .04558 m .02756 .04558 L s .02381 .05958 m .02756 .05958 L s .02381 .07358 m .02756 .07358 L s .02381 .08757 m .02756 .08757 L s .02381 .11556 m .02756 .11556 L s .02381 .12956 m .02756 .12956 L s .02381 .14356 m .02756 .14356 L s .02381 .15755 m .02756 .15755 L s .02381 .18554 m .02756 .18554 L s .02381 .19954 m .02756 .19954 L s .02381 .21354 m .02756 .21354 L s .02381 .22753 m .02756 .22753 L s .02381 .25552 m .02756 .25552 L s .02381 .26952 m .02756 .26952 L s .02381 .28352 m .02756 .28352 L s .02381 .29751 m .02756 .29751 L s .02381 .3255 m .02756 .3255 L s .02381 .3395 m .02756 .3395 L s .02381 .3535 m .02756 .3535 L s .02381 .36749 m .02756 .36749 L s .02381 .39548 m .02756 .39548 L s .02381 .40948 m .02756 .40948 L s .02381 .42348 m .02756 .42348 L s .02381 .43747 m .02756 .43747 L s .02381 .01759 m .02756 .01759 L s .02381 .00359 m .02756 .00359 L s .02381 .46547 m .02756 .46547 L s .02381 .47946 m .02756 .47946 L s .02381 .49346 m .02756 .49346 L s .25 Mabswid .02381 0 m .02381 .5 L s gsave .5 .5125 -135.562 -4 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 63.000000 13.000000 moveto (8) show 69.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.1363) show 105.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 114.562500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.4695) show 150.562500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 160.125000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.09857) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 202.125000 13.000000 moveto (<) show 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Math2Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math2Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math2Mono %%+ font Courier grestore .5 Mabswid .02381 .24153 m .12963 .36735 L .23545 .24153 L .34127 .0119 L .44709 .24153 L .55291 .4881 L .65873 .43527 L .76455 .31224 L .87037 .25644 L .97619 .2435 L s 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{244, 144}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .5 %%ImageSize: 244 144 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0820106 0.10582 0.283344 1.67246e+08 [ [.12963 .27084 -3 -9 ] [.12963 .27084 3 0 ] [.34127 .27084 -3 -9 ] [.34127 .27084 3 0 ] [.55291 .27084 -3 -9 ] [.55291 .27084 3 0 ] [.76455 .27084 -3 -9 ] [.76455 .27084 3 0 ] [.97619 .27084 -6 -9 ] [.97619 .27084 6 0 ] [.01131 .03247 -51 -6.25 ] [.01131 .03247 0 6.25 ] [.01131 .1161 -39 -6.25 ] [.01131 .1161 0 6.25 ] [.01131 .19972 -43.5 -6.25 ] [.01131 .19972 0 6.25 ] [.01131 .36697 -37.5 -6.25 ] [.01131 .36697 0 6.25 ] [.01131 .45059 -33 -6.25 ] [.01131 .45059 0 6.25 ] [.5 .5125 -103.344 0 ] [.5 .5125 103.344 12 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .12963 .28334 m .12963 .28959 L s [(2)] .12963 .27084 0 1 Mshowa .34127 .28334 m .34127 .28959 L s [(4)] .34127 .27084 0 1 Mshowa .55291 .28334 m .55291 .28959 L s [(6)] .55291 .27084 0 1 Mshowa .76455 .28334 m .76455 .28959 L s [(8)] .76455 .27084 0 1 Mshowa .97619 .28334 m .97619 .28959 L s [(10)] .97619 .27084 0 1 Mshowa .125 Mabswid .18254 .28334 m .18254 .28709 L s .23545 .28334 m .23545 .28709 L s .28836 .28334 m .28836 .28709 L s .39418 .28334 m .39418 .28709 L s .44709 .28334 m .44709 .28709 L s .5 .28334 m .5 .28709 L s .60582 .28334 m .60582 .28709 L s .65873 .28334 m .65873 .28709 L s .71164 .28334 m .71164 .28709 L s .81746 .28334 m .81746 .28709 L s .87037 .28334 m .87037 .28709 L s .92328 .28334 m .92328 .28709 L s .07672 .28334 m .07672 .28709 L s .25 Mabswid 0 .28334 m 1 .28334 L s .02381 .03247 m .03006 .03247 L s gsave .01131 .03247 -51 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (-1.5) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-9) show grestore .02381 .1161 m .03006 .1161 L s gsave .01131 .1161 -39 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (-1) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-9) show grestore .02381 .19972 m .03006 .19972 L s gsave .01131 .19972 -43.5 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (-5) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-10) show grestore .02381 .36697 m .03006 .36697 L s gsave .01131 .36697 -37.5 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (5) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-10) show grestore .02381 .45059 m .03006 .45059 L s gsave .01131 .45059 -33 -4.25 Mabsadd m 1 1 Mabs scale /Courier findfont 10 scalefont setfont (1) show /Math1Mono findfont 10 scalefont setfont (\\264) show /Courier findfont 10 scalefont setfont (10) show 0 5.25 rmoveto /Courier findfont 7.5 scalefont setfont (-9) show grestore .125 Mabswid .02381 .0492 m .02756 .0492 L s .02381 .06592 m .02756 .06592 L s .02381 .08265 m .02756 .08265 L s .02381 .09937 m .02756 .09937 L s .02381 .13282 m .02756 .13282 L s .02381 .14955 m .02756 .14955 L s .02381 .16627 m .02756 .16627 L s .02381 .183 m .02756 .183 L s .02381 .21645 m .02756 .21645 L s .02381 .23317 m .02756 .23317 L s .02381 .24989 m .02756 .24989 L s .02381 .26662 m .02756 .26662 L s .02381 .30007 m .02756 .30007 L s .02381 .31679 m .02756 .31679 L s .02381 .33352 m .02756 .33352 L s .02381 .35024 m .02756 .35024 L s .02381 .38369 m .02756 .38369 L s .02381 .40042 m .02756 .40042 L s .02381 .41714 m .02756 .41714 L s .02381 .43387 m .02756 .43387 L s .02381 .01575 m .02756 .01575 L s .02381 .46731 m .02756 .46731 L s .02381 .48404 m .02756 .48404 L s .25 Mabswid .02381 0 m .02381 .5 L s gsave .5 .5125 -164.344 -4 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 63.000000 13.000000 moveto (8) show 69.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.05464) show 111.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 120.562500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.4864) show 156.562500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 166.125000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.09999) show 208.125000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (,) show 217.687500 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (0.00656) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 259.687500 13.000000 moveto (<) show 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Math2Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math2Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math2Mono %%+ font Courier grestore .5 Mabswid .02381 .28334 m .12963 .4881 L .23545 .28334 L .34127 .0119 L .44709 .28334 L .55291 .39612 L .65873 .28334 L .76455 .24716 L .87037 .26744 L .97619 .28009 L s 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{244, 144}, ImageCacheValid->False] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(graphideal[100, 2, {1, 3}, {1.5, 3.5, 25}, {1, 2.7, 4.5, 35}, 50]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .78493 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.397585 0.373776 0.392465 0.373776 [ [ 0 0 0 0 ] [ 1 .78493 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath % Start of sub-graphic p 0.0238095 0.0186888 0.771361 0.766241 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0190476 0.416271 0.564356 [ [.21429 .40377 -6 -9 ] [.21429 .40377 6 0 ] [.40476 .40377 -6 -9 ] [.40476 .40377 6 0 ] [.59524 .40377 -6 -9 ] [.59524 .40377 6 0 ] [.78571 .40377 -6 -9 ] [.78571 .40377 6 0 ] [.97619 .40377 -6 -9 ] [.97619 .40377 6 0 ] [.01131 .13409 -24 -4.5 ] [.01131 .13409 0 4.5 ] [.01131 .27518 -30 -4.5 ] [.01131 .27518 0 4.5 ] [.01131 .55736 -24 -4.5 ] [.01131 .55736 0 4.5 ] [.01131 .69845 -18 -4.5 ] [.01131 .69845 0 4.5 ] [.01131 .83954 -24 -4.5 ] [.01131 .83954 0 4.5 ] [.01131 .98063 -6 -4.5 ] [.01131 .98063 0 4.5 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .41627 m .21429 .42252 L s [(10)] .21429 .40377 0 1 Mshowa .40476 .41627 m .40476 .42252 L s [(20)] .40476 .40377 0 1 Mshowa .59524 .41627 m .59524 .42252 L s [(30)] .59524 .40377 0 1 Mshowa .78571 .41627 m .78571 .42252 L s [(40)] .78571 .40377 0 1 Mshowa .97619 .41627 m .97619 .42252 L s [(50)] .97619 .40377 0 1 Mshowa .125 Mabswid .0619 .41627 m .0619 .42002 L s .1 .41627 m .1 .42002 L s .1381 .41627 m .1381 .42002 L s .17619 .41627 m .17619 .42002 L s .25238 .41627 m .25238 .42002 L s .29048 .41627 m .29048 .42002 L s .32857 .41627 m .32857 .42002 L s .36667 .41627 m .36667 .42002 L s .44286 .41627 m .44286 .42002 L s .48095 .41627 m .48095 .42002 L s .51905 .41627 m .51905 .42002 L s .55714 .41627 m .55714 .42002 L s .63333 .41627 m .63333 .42002 L s .67143 .41627 m .67143 .42002 L s .70952 .41627 m .70952 .42002 L s .74762 .41627 m .74762 .42002 L s .82381 .41627 m .82381 .42002 L s .8619 .41627 m .8619 .42002 L s .9 .41627 m .9 .42002 L s .9381 .41627 m .9381 .42002 L s .25 Mabswid 0 .41627 m 1 .41627 L s .02381 .13409 m .03006 .13409 L s [(-0.5)] .01131 .13409 1 0 Mshowa .02381 .27518 m .03006 .27518 L s [(-0.25)] .01131 .27518 1 0 Mshowa .02381 .55736 m .03006 .55736 L s [(0.25)] .01131 .55736 1 0 Mshowa .02381 .69845 m .03006 .69845 L s [(0.5)] .01131 .69845 1 0 Mshowa .02381 .83954 m .03006 .83954 L s [(0.75)] .01131 .83954 1 0 Mshowa .02381 .98063 m .03006 .98063 L s [(1)] .01131 .98063 1 0 Mshowa .125 Mabswid .02381 .02122 m .02756 .02122 L s .02381 .04944 m .02756 .04944 L s .02381 .07766 m .02756 .07766 L s .02381 .10588 m .02756 .10588 L s .02381 .16231 m .02756 .16231 L s .02381 .19053 m .02756 .19053 L s .02381 .21875 m .02756 .21875 L s .02381 .24696 m .02756 .24696 L s .02381 .3034 m .02756 .3034 L s .02381 .33162 m .02756 .33162 L s .02381 .35984 m .02756 .35984 L s .02381 .38805 m .02756 .38805 L s .02381 .44449 m .02756 .44449 L s .02381 .47271 m .02756 .47271 L s .02381 .50092 m .02756 .50092 L s .02381 .52914 m .02756 .52914 L s .02381 .58558 m .02756 .58558 L s .02381 .6138 m .02756 .6138 L s .02381 .64201 m .02756 .64201 L s .02381 .67023 m .02756 .67023 L s .02381 .72667 m .02756 .72667 L s .02381 .75488 m .02756 .75488 L s .02381 .7831 m .02756 .7831 L s .02381 .81132 m .02756 .81132 L s .02381 .86776 m .02756 .86776 L s .02381 .89597 m .02756 .89597 L s .02381 .92419 m .02756 .92419 L s .02381 .95241 m .02756 .95241 L s .25 Mabswid .02381 0 m .02381 1 L s 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath .001 w .04286 .97619 m .0619 .96308 L .08095 .94192 L .1 .91365 L .11905 .87953 L .1381 .841 L .15714 .79956 L .17619 .75675 L .19524 .71399 L .21429 .67251 L .23333 .63336 L .25238 .59731 L .27143 .56488 L .29048 .53635 L .30952 .51177 L .32857 .49104 L .34762 .47389 L .36667 .45998 L .38571 .4489 L .40476 .44026 L .42381 .43362 L .44286 .42863 L .4619 .42493 L .48095 .42225 L .5 .42033 L .51905 .41898 L .5381 .41806 L .55714 .41743 L .57619 .41701 L .59524 .41673 L .61429 .41656 L .63333 .41644 L .65238 .41638 L .67143 .41633 L .69048 .41631 L .70952 .41629 L .72857 .41628 L .74762 .41628 L .76667 .41627 L .78571 .41627 L .80476 .41627 L .82381 .41627 L .84286 .41627 L .8619 .41627 L .88095 .41627 L .9 .41627 L .91905 .41627 L .9381 .41627 L .95714 .41627 L .97619 .41627 L Mistroke Mfstroke [ .5 3 ] 0 Mabsdash .04286 .97619 m .0619 .9631 L .08095 .94201 L .1 .9139 L .11905 .87999 L .1381 .84163 L .15714 .80014 L .17619 .75678 L .19524 .71264 L .21429 .66866 L .23333 .62555 L .25238 .58386 L .27143 .54399 L .29048 .50618 L .30952 .47058 L .32857 .43724 L .34762 .40616 L .36667 .37728 L .38571 .35052 L .40476 .32577 L .42381 .30293 L .44286 .28186 L .4619 .26246 L .48095 .24459 L .5 .22815 L .51905 .21304 L .5381 .19914 L .55714 .18637 L .57619 .17465 L .59524 .16388 L .61429 .15401 L .63333 .14497 L .65238 .13669 L .67143 .12913 L .69048 .12223 L .70952 .11595 L .72857 .11025 L .74762 .10511 L .76667 .10047 L .78571 .09633 L .80476 .09265 L .82381 .08941 L .84286 .08659 L .8619 .08418 L .88095 .08216 L .9 .08053 L .91905 .07927 L .9381 .07837 L .95714 .07784 L .97619 .07766 L Mistroke Mfstroke [ .5 4 5 4 ] 0 Mabsdash .04286 .97619 m .0619 .96308 L .08095 .94192 L .1 .91362 L .11905 .87935 L .1381 .84041 L .15714 .79808 L .17619 .7536 L .19524 .70808 L .21429 .66246 L .23333 .6175 L .25238 .57379 L .27143 .53177 L .29048 .49172 L .30952 .45382 L .32857 .41818 L .34762 .38482 L .36667 .35369 L .38571 .32475 L .40476 .2979 L .42381 .27303 L .44286 .25004 L .4619 .2288 L .48095 .20919 L .5 .19112 L .51905 .17446 L .5381 .15912 L .55714 .145 L .57619 .13201 L .59524 .12007 L .61429 .10911 L .63333 .09905 L .65238 .08983 L .67143 .0814 L .69048 .07371 L .70952 .0667 L .72857 .06033 L .74762 .05457 L .76667 .04939 L .78571 .04475 L .80476 .04062 L .82381 .03699 L .84286 .03383 L .8619 .03113 L .88095 .02887 L .9 .02703 L .91905 .02562 L .9381 .02461 L .95714 .02401 L .97619 .02381 L Mistroke Mfstroke [ 5 2 ] 0 Mabsdash .04286 .97619 m .0619 .96308 L .08095 .94192 L .1 .91364 L .11905 .87947 L .1381 .84078 L .15714 .79899 L .17619 .75551 L .19524 .71166 L .21429 .6686 L .23333 .6273 L .25238 .58857 L .27143 .55301 L .29048 .52107 L .30952 .49302 L .32857 .46902 L .34762 .44912 L .36667 .43326 L .38571 .42132 L .40476 .41314 L .42381 .40851 L .44286 .40717 L .4619 .40888 L .48095 .41336 L .5 .42033 L .51905 .42951 L .5381 .44063 L .55714 .45341 L .57619 .46758 L .59524 .48289 L .61429 .49908 L .63333 .51592 L .65238 .53318 L .67143 .55064 L .69048 .56808 L .70952 .58533 L .72857 .60219 L .74762 .61849 L .76667 .63408 L .78571 .64881 L .80476 .66254 L .82381 .67515 L .84286 .68654 L .8619 .69661 L .88095 .70526 L .9 .71244 L .91905 .71809 L .9381 .72216 L .95714 .72461 L .97619 .72543 L Mistroke Mfstroke [ .5 4 5 4 .5 3 ] 0 Mabsdash .04286 .97619 m .0619 .96308 L .08095 .94192 L .1 .91365 L .11905 .87953 L .1381 .841 L .15714 .79959 L .17619 .75684 L .19524 .7142 L .21429 .673 L .23333 .63434 L .25238 .59911 L .27143 .56793 L .29048 .54121 L .30952 .5191 L .32857 .50155 L .34762 .48834 L .36667 .47909 L .38571 .47333 L .40476 .47047 L .42381 .46991 L .44286 .47099 L .4619 .47308 L .48095 .47557 L .5 .47789 L .51905 .47951 L .5381 .48 L .55714 .47898 L .57619 .47618 L .59524 .47138 L .61429 .46447 L .63333 .45542 L .65238 .44429 L .67143 .43119 L .69048 .41631 L .70952 .3999 L .72857 .38227 L .74762 .36376 L .76667 .34473 L .78571 .32559 L .80476 .30673 L .82381 .28856 L .84286 .27146 L .8619 .2558 L .88095 .24193 L .9 .23014 L .91905 .22069 L .9381 .21379 L .95714 .20959 L .97619 .20818 L Mistroke Mfstroke MathSubEnd P % End of sub-graphic 0 g .80874 .24295 m .80874 .54198 L .97619 .54198 L .97619 .24295 L F 1 g .80874 .24295 m .80874 .54198 L .97619 .54198 L .97619 .24295 L F 0 g .001 w [ ] 0 setdash .80874 .24295 m .97619 .24295 L .97619 .54198 L .80874 .54198 L .80874 .24295 L s % Start of sub-graphic p 0.808739 0.242954 0.97619 0.541975 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0297619 0.297619 0.0297619 0.297619 [ [ 0 0 0 0 ] [ 1 1.78571 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1.78571 L 0 1.78571 L closepath clip newpath % Start of sub-graphic p 0.0535714 0.0535714 0.35119 0.35119 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ .5 4 5 4 .5 3 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .20238 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (4) show 69.000000 13.000000 moveto (th) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 0.39881 0.35119 0.696429 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ 5 2 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .54762 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (3) show 69.000000 13.000000 moveto (rd) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 0.744048 0.35119 1.04167 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ .5 4 5 4 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 .89286 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (2) show 69.000000 13.000000 moveto (nd) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore % Start of sub-graphic p 0.0535714 1.08929 0.35119 1.3869 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ .5 3 ] 0 Mabsdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 1.2381 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (C) show 69.000000 13.000000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (-) show 75.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (N) show 81.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Math1Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math1Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math1Mono %%+ font Courier grestore % Start of sub-graphic p 0.0535714 1.43452 0.35119 1.73214 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.5 0.5 0.5 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .01169 w [ ] 0 setdash 0 .5 m 1 .5 L s MathSubEnd P % End of sub-graphic 0 g gsave .38988 1.58333 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20 translate 1 -1 scale gsave 0.000000 0.000000 0.000000 setrgbcolor 1.000000 setlinewidth gsave newpath 61.000000 16.000000 moveto 558.000000 16.000000 lineto 558.000000 4.000000 lineto 61.000000 4.000000 lineto 61.000000 16.000000 lineto closepath clip newpath 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 63.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (true) show 87.000000 13.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 rmoveto 1.000000 setlinewidth grestore grestore %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore MathSubEnd P % End of sub-graphic 0 0 m 1 0 L 1 .78493 L 0 .78493 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 226}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(grapherror[100, 2, {1, 3}, {1.5, 3.5, 25}, {1, 2.7, 4.5, 35}, 10, 30] \)], "Input"], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .5 %%ImageSize: 244 144 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0820106 0.10582 0.0119052 739.375 [ [.12963 -0.00059 -3 -9 ] [.12963 -0.00059 3 0 ] [.34127 -0.00059 -3 -9 ] [.34127 -0.00059 3 0 ] [.55291 -0.00059 -3 -9 ] [.55291 -0.00059 3 0 ] [.76455 -0.00059 -3 -9 ] [.76455 -0.00059 3 0 ] [.97619 -0.00059 -6 -9 ] [.97619 -0.00059 6 0 ] [.01131 .08584 -36 -4.5 ] [.01131 .08584 0 4.5 ] [.01131 .15978 -36 -4.5 ] [.01131 .15978 0 4.5 ] [.01131 .23372 -36 -4.5 ] [.01131 .23372 0 4.5 ] [.01131 .30766 -36 -4.5 ] [.01131 .30766 0 4.5 ] [.01131 .38159 -36 -4.5 ] [.01131 .38159 0 4.5 ] [.01131 .45553 -36 -4.5 ] [.01131 .45553 0 4.5 ] [.5 .5125 -46.3438 0 ] [.5 .5125 46.3438 12 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .12963 .01191 m .12963 .01816 L s [(2)] .12963 -0.00059 0 1 Mshowa .34127 .01191 m .34127 .01816 L s [(4)] .34127 -0.00059 0 1 Mshowa .55291 .01191 m .55291 .01816 L s [(6)] .55291 -0.00059 0 1 Mshowa .76455 .01191 m .76455 .01816 L s [(8)] .76455 -0.00059 0 1 Mshowa .97619 .01191 m .97619 .01816 L s [(10)] .97619 -0.00059 0 1 Mshowa .125 Mabswid .18254 .01191 m .18254 .01566 L s .23545 .01191 m .23545 .01566 L s .28836 .01191 m .28836 .01566 L s .39418 .01191 m .39418 .01566 L s .44709 .01191 m .44709 .01566 L s .5 .01191 m .5 .01566 L s .60582 .01191 m .60582 .01566 L s .65873 .01191 m .65873 .01566 L s .71164 .01191 m .71164 .01566 L s .81746 .01191 m .81746 .01566 L s .87037 .01191 m .87037 .01566 L s .92328 .01191 m .92328 .01566 L s .07672 .01191 m .07672 .01566 L s .25 Mabswid 0 .01191 m 1 .01191 L s .02381 .08584 m .03006 .08584 L s [(0.0001)] .01131 .08584 1 0 Mshowa .02381 .15978 m .03006 .15978 L s [(0.0002)] .01131 .15978 1 0 Mshowa .02381 .23372 m .03006 .23372 L s [(0.0003)] .01131 .23372 1 0 Mshowa .02381 .30766 m .03006 .30766 L s [(0.0004)] .01131 .30766 1 0 Mshowa .02381 .38159 m .03006 .38159 L s [(0.0005)] .01131 .38159 1 0 Mshowa .02381 .45553 m .03006 .45553 L s [(0.0006)] .01131 .45553 1 0 Mshowa .125 Mabswid .02381 .02669 m .02756 .02669 L s .02381 .04148 m .02756 .04148 L s .02381 .05627 m .02756 .05627 L s .02381 .07106 m .02756 .07106 L s .02381 .10063 m .02756 .10063 L s .02381 .11542 m .02756 .11542 L s .02381 .13021 m .02756 .13021 L s .02381 .14499 m .02756 .14499 L