(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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). 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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[ 703220, 13559]*) (*NotebookOutlinePosition[ 703887, 13582]*) (* CellTagsIndexPosition[ 703843, 13578]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Complex Orders", "Title"], Cell["Roger Germundsson, Wolfram Research, 1/15/2003", "Subsubtitle"], Cell[CellGroupData[{ Cell["Orders", "Subsection"], Cell[TextData[{ "Some orderings of complex numbers used in ", StyleBox["Mathematica", FontSlant->"Italic"], " ... " }], "Text"], Cell[BoxData[ \(\(EigenSort[\ l_\ ]\ := \ Eigenvalues[\ DiagonalMatrix[\ l\ ]\ ];\)\)], "Input"], Cell["\<\ Also all examples will be generated through the ordering used by Root, so \ that is the default ordering.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(l\ = \ Table[\ N[\ Root[\ #^5 + 11\ #\ + \ 3 &, \ i]], \ {i, 5}]\)], "Input"], Cell[BoxData[ \({\(-0.27259044929139326`\), \(-1.2152213331947168`\) - 1.2933376888072032`\ \[ImaginaryI], \(-1.2152213331947168`\) + 1.2933376888072032`\ \[ImaginaryI], \(\(1.3515165578404134`\)\(\ \[InvisibleSpace]\)\) - 1.2914295544092322`\ \[ImaginaryI], \(\(1.3515165578404134`\)\(\ \[InvisibleSpace]\)\) + 1.2914295544092322`\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Sort[\ l\ ]\)], "Input"], Cell[BoxData[ \({\(-1.2152213331947168`\) - 1.2933376888072032`\ \[ImaginaryI], \(-1.2152213331947168`\) + 1.2933376888072032`\ \[ImaginaryI], \(-0.27259044929139326`\), \ \(\(1.3515165578404134`\)\(\[InvisibleSpace]\)\) - 1.2914295544092322`\ \[ImaginaryI], \(\(1.3515165578404134`\)\(\ \[InvisibleSpace]\)\) + 1.2914295544092322`\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EigenSort[\ l\ ]\)], "Input"], Cell[BoxData[ \({\(\(1.3515165578404134`\)\(\[InvisibleSpace]\)\) + 1.2914295544092322`\ \[ImaginaryI], \(\(1.3515165578404134`\)\(\ \[InvisibleSpace]\)\) - 1.2914295544092322`\ \[ImaginaryI], \(-1.2152213331947168`\) + 1.2933376888072032`\ \[ImaginaryI], \(-1.2152213331947168`\) - 1.2933376888072032`\ \[ImaginaryI], \(-0.27259044929139326`\) + 0.`\ \[ImaginaryI]}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Visualization", "Subsection"], Cell["This will allow us to visualize the order through arrows .. ", "Text"], Cell[BoxData[ \(<< Graphics`Arrow`\)], "Input"], Cell[BoxData[ \(OrderPlot[\ l_\ ]\ := \ \[IndentingNewLine]Module[\ {i = 1}, \[IndentingNewLine]Show\ @\ Graphics[\[IndentingNewLine]{\[IndentingNewLine]Apply[ Arrow[{Re[#1], \ Im[#1]}, \ {Re[#2], \ Im[#2]}] &, Partition[\ l, \ 2, \ 1\ ], \ {1}\ ], \ \[IndentingNewLine]PointSize[\ 0.02], \ \[IndentingNewLine]Map[\ If[\ Im[\ #]\ \[Equal] 0, \ \[IndentingNewLine]{RGBColor[1, 0, 0], Point[\ {Re[#], \ Im[#]}\ ], \ Text[\(i++\), {Re[#], \ Im[#]}, \ {\(-1\), \(-1\)}]}, \ {Point[\ {Re[#], \ Im[#]}\ ], Text[\(i++\), {Re[#], \ Im[#]}, \ {\(-1\), \(-1\)}]}] &, \ l\ ], \ \[IndentingNewLine]Map[\ Circle[\ {0, 0}, \ Abs[#]\ ] &, \ l\ ]\[IndentingNewLine]}, \[IndentingNewLine]Frame\ \[Rule] \ \ True, \ \[IndentingNewLine]AspectRatio\ \[Rule] \ Automatic\[IndentingNewLine]]\[IndentingNewLine]]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Examples", "Subsection"], Cell[CellGroupData[{ Cell["Example 1", "Subsubsection"], Cell["This is more or less a random example ... 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