(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 202364, 3469]*) (*NotebookOutlinePosition[ 203495, 3509]*) (* CellTagsIndexPosition[ 203399, 3502]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Parallel Computing Toolkit Examples\ \>", "Title", CellTags->"i:1"], Cell["A Minimal Surface", "Subsubtitle"], Cell[TextData[{ "This example shows the use of ", StyleBox["ParallelParametricPlot3D", "MR"], " to compute a visual representation of an incomplete minimally immersed \ Klein bottle into ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^3\)]], " with total curvature ", Cell[BoxData[ \(TraditionalForm\`\(-4\) \[Pi]\)]], ". The example was provided by Jeff Gaynor, Wolfram Research, Inc." }], "Text"], Cell["\<\ Reference: J. J. Gaynor, Minimal Immersions of K into R^3, dissertation, \ Ruprecht-Karls-Universit\[ADoubleDot]t, Heidelberg Germany (July, 1998)\ \>", \ "Text"], Cell[CellGroupData[{ Cell["Init", "Section"], Cell[CellGroupData[{ Cell["Parallel Processing Toolkit", "Subsection"], Cell[TextData[{ "Load the ", StyleBox["Parallel Computing Toolkit", FontSlant->"Italic"], " and the package with the parallel commands." }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\n", \(Needs["\"]\)}], "Input", CellLabel->"In[16]:=", InitializationCell->True], Cell[TextData[{ "Now, start any number of remote kernels. This command assumes that you \ have set up the list ", StyleBox["$AvailableMachines", "MR"], " to list all normally available remote computers." }], "Text"], Cell[BoxData[ \(LaunchSlaves[] // TableForm\)], "Input", CellLabel->"In[18]:="], Cell["Find the number of available parallel kernels.", "Text"], Cell[BoxData[ \(Length[$Slaves]\)], "Input", CellLabel->"In[19]:="], Cell["Optionally enable latency hiding.", "Text"], Cell[BoxData[ \(\(\($LoadFactor = 2;\)\(\ \)\( (*\ latency\ hiding\ *) \)\)\)], "Input",\ CellLabel->"In[21]:="] }, Open ]], Cell[CellGroupData[{ Cell["The Surface Definitions", "Subsection"], Cell[TextData[{ "These auxiliary definitions are used in the function ", StyleBox["OneEnded", "MR"], " that gives the coordinates of surface points depending on two \ parameters." }], "Text"], Cell[BoxData[{ \(\n\(nalphas[s_] := N[Exp[4*s]\ + \ Exp[2*s] + 1];\)\), "\n", \(\(nbetas[s_] := N[4*\((\(-2\)*Exp[6*s] - 3*Exp[4*s] + 3*Exp[2*s] + 2)\)];\)\), "\n", \(\(ners[s_] := N[Exp[2\ s] - 1];\)\), "\n", \(\(nArs[u_, s_] := N[WeierstrassP[u, nalphas[s], nbetas[s]]\ - \ ners[s]];\)\), "\n", \(\(nomps[s_] := N[EllipticK[N[\(-Exp[2\ s]\)]]/Sqrt[3]];\)\), "\n", \(\(nomqs[s_] := I*N[EllipticK[1 + N[Exp[2*s]]]/Sqrt[3]];\)\)}], "Input", CellLabel->"In[9]:=", InitializationCell->True], Cell[BoxData[ \(\(\(OneEnded[u_, \ s_]\ := \ \n\ \ \ Module[{Ar, \ hOverg2, \ hgOver2, \ h}, \ \n\ \ \ \ Ar\ = \ nArs[u, \ s]; \ hOverg2\ = \ \n\ \ \ \ \ \ N[\((\((27*I)\)/2*E^\((3*s)\))\)/ Ar\ - \ \((\((81*I)\)/2*E^\((4*s)\))\)/ Ar^2\ + \ \n\ \ \ \ \ \ \ \ \((\((81*I)\)/2*E^\((5*s)\))\)/ Ar^3]; \ \n\ \ \ \ \ hgOver2\ = \ \n\ \ \ \ \ \ N[\(-I\)/2* Ar^2\ - \ \((I/18*Ar^3)\)/E^s\ - \ \((3*I)\)/2*Ar* E^s]; \ \n\ \ \ \ \ h\ = \ N[\(-3\)*I*Ar*E^s\ - \ \((27*I*E^\((3*s)\))\)/ Ar]; \ \n\ \ \ \ \ Re[{hOverg2\ - \ hgOver2, \ I*\((hOverg2\ + \ hgOver2)\), \ h}]];\)\(\ \)\)\)], "Input", CellLabel->"In[15]:=", InitializationCell->True], Cell["Finally, send the surface definitions to all remote kernels.", "Text"], Cell[BoxData[ \(ExportEnvironment[{nalphas, nbetas, ners, nArs, nomps, nomqs, OneEnded}]\)], "Input", CellLabel->"In[20]:="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Parametric Plots", "Section", CellTags->"i:2"], Cell[CellGroupData[{ Cell["The Moebius Strip Inside the Surface", "Subsection"], Cell["\<\ The surface contains a Moebius strip, which is made visible by \ restricting the parameter ranges suitable.\ \>", "Text"], Cell["\<\ Here is a sequential preview using a small number of plot points.\ \ \>", "Text"], Cell[BoxData[ \(\(ParametricPlot3D[ OneEnded[ x\ + \ I*y, \ \(-1\)], \ \n\ \ \ \ {x, \ \(-nomps[\(-1\)]\), \ nomps[\(-1\)]}, \ \n\ \ \ \ {y, \ Im[nomqs[\(-1\)]]/2\ - \ 0.05, \ Im[nomqs[\(-1\)]]/2\ + \ 0.05}, \ \n\ \ \ \ PlotPoints\ -> \ {15, \ 6}, Compiled \[Rule] False];\)\)], "Input", CellLabel->"In[22]:="], Cell["Here is a low-resolution parallel rendering.", "Text"], Cell[BoxData[ \(\(ParallelParametricPlot3D[ OneEnded[ x\ + \ I*y, \ \(-1\)], \ \n\ \ \ 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