(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' 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[ 418819, 11449]*) (*NotebookOutlinePosition[ 419454, 11471]*) (* CellTagsIndexPosition[ 419410, 11467]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Create\ a\ list\ representing\ changes\ in\ the\ value\ of\ the\ \ driving\ function\ *) \)\(\[IndentingNewLine]\)\(\(f[x_] := N[2*Sqrt[3*\((1 - x)\)]];\)\[IndentingNewLine] dfValues\ = \ {}; \ amValues = {};\[IndentingNewLine] range = {0, 1}; \ \ segments = 500;\[IndentingNewLine] \(tstep\ = \((range[\([2]\)] - range[\([1]\)])\)/ segments;\)\[IndentingNewLine] \(tolerance = 0.01;\)\[IndentingNewLine] \(Do[R = f[i] - f[ i - tstep]; \ (*\(This\ could\ well\ have\ the\ wrong\ sign \ .. \)\ not\ sure\ *) \[IndentingNewLine]dfValues = Append[dfValues, \ {R, \ tstep}]; \[IndentingNewLine]Rsq\ = \ R\^2; \ amValues = Append[amValues, \ {\(Rsq + 16 tstep\ - R\ \@\(Rsq + \(\(16\) \ \(tstep\)\(\ \)\)\)\)\/\(2\ \((Rsq + 16\ tstep)\)\), \@\(Rsq + \(\(16\) \ \(tstep\)\(\ \)\)\)}], \ {i, range[\([2]\)], range[\([1]\)] + tstep, \(-tstep\)}];\)\[IndentingNewLine] good\ = \ 0; \ iter\ = \ 0; \ \ wlength\ = \ 0; \ \ changed\ = \ 1; \ oldchanged = 1; \ saveddata = {0};\[IndentingNewLine] While[ good \[Equal] 0, \[IndentingNewLine]timing = AbsoluteTime[]\ ; \ \[IndentingNewLine]iter\ = \ iter + 1; \[IndentingNewLine] (*\ Do\ Lowner\ equation\ with\ values\ for\ the\ driving\ function\ in\ \ dfValues\ *) \[IndentingNewLine]Nr\ = \ Length[dfValues]\ ; \[IndentingNewLine]data\ = \ If[oldchanged < changed, \ saveddata, \ {0}]; \[IndentingNewLine]startvalue\ = \ If[oldchanged < changed, \ oldchanged, \ 1]; \[IndentingNewLine]Do[\ a = amValues[\([i, 1]\)]; \ m = amValues[\([i, 2]\)]; \[IndentingNewLine] (*\ We\ now\ do\ these\ calcuations\ when\ the\ R\ and\ t\ values\ are\ \ generated . \ a = \(R\^2 + 16\ t - R\ \@\(R\^2 + 16\ t\)\)\/\(2\ \((R\^2 + \ 16\ t)\)\); m = \@\(R\^2 + 16\ t\);\ \ *) \[IndentingNewLine]\[IndentingNewLine]nF[x_] = m\ \((\(-a\) + x\/m)\)\^a\ \((1 - a + x\/m)\)\^\(1 - a\); \ \[IndentingNewLine]data\ = \ Join[nF[data], \ {0}]; \[IndentingNewLine]If[ i \[Equal] changed - 1, saveddata = data, \ ], \ {i, startvalue, Nr}]; \[IndentingNewLine]h[z_] = {Re[z], Im[z]}; \[IndentingNewLine]ListPlot[Map[h, data], PlotJoined \[Rule] True, AspectRatio \[Rule] \ Automatic]; \[IndentingNewLine]Print["\", \ iter, \ "\<: spent \>", \ AbsoluteTime[] - timing, \ "\< seconds computing curve\>"]; \[IndentingNewLine]\ \[IndentingNewLine] (*\ Add\ more\ values\ to\ the\ driving\ function\ based\ on\ where\ the\ \ curve\ needs\ to\ be\ improved\ *) \[IndentingNewLine]Nr = Length[data] - 1; \[IndentingNewLine]\[IndentingNewLine] (*\ This\ number\ adjusts\ the\ number\ of\ additional\ points\ added\ as\ \ a\ function\ of\ the\ length\ of\ a\ line\ \(\(segment\)\(.\)\)\ \ *) \[IndentingNewLine]refinement = 1/tolerance; \[IndentingNewLine]i = 1; \[IndentingNewLine]newValues\ = \ {}; \ newAmValues = {}; \[IndentingNewLine]wlength\ = \ 0; \ \[IndentingNewLine]totalt = range[\([2]\)]; \[IndentingNewLine]oldchanged\ = \ changed; \ changed\ = \ 1; \[IndentingNewLine]\[IndentingNewLine]Do[\[IndentingNewLine]If[\((\ l\ = \ Abs[data[\([i + 1]\)] - data[\([i]\)]])\) > tolerance, \[IndentingNewLine]\t If[changed \[Equal] 1, \ changed = i, \ ]; \[IndentingNewLine]\t wlength\ = \ Max[l, \ wlength]; \[IndentingNewLine]\t n\ = \ Floor[\((refinement*l)\)]; \[IndentingNewLine]\t ot\ = \ dfValues[\([i, 2]\)]; \ t\ = \ ot; \ R = dfValues[\([i, 1]\)]; \[IndentingNewLine]\t Do[\[IndentingNewLine]\t r = f[t] - f[t - ot/\((n + 1)\)]; \[IndentingNewLine]\t newValues\ = \ Append\ [newValues, \ {r, ot/\((n + 1)\)}\ ]; \ nt = ot/\((n + 1)\); \[IndentingNewLine]\tRsq = r\^2; \ newAmValues = Append[newAmValues, {\ \(Rsq + 16\ nt - r\ \@\(Rsq + 16\ nt\)\)\ \/\(2\ \((Rsq + 16\ nt)\)\), \@\(Rsq + 16\ nt\)}], \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ {t, \ totalt, \ totalt - n*ot/\((n + 1)\), \ \(-ot\)/\((n + 1)\)}], \[IndentingNewLine]\t newValues\ = \ Append[newValues, \ dfValues[\([i]\)]]; \ newAmValues\ = \ Append[newAmValues, amValues[\([i]\)]]]; \[IndentingNewLine]totalt\ -= \ ot, \ {i, \ Nr}]; \ \[IndentingNewLine]If[ Length[dfValues] \[Equal] Length[newValues]\ , \ good\ = \ 1, \ ]; \[IndentingNewLine]Print["\", \ wlength, \ "\<, number of segments = \>", \ Length[dfValues]]; \[IndentingNewLine]Print[ Length[newValues] - Length[dfValues], \ "\< segments to be added\>"]; \ \[IndentingNewLine]dfValues\ = \ newValues; amValues = newAmValues;]\)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .80974 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