(************** Content-type: application/mathematica ************** 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[ 26152, 601]*) (*NotebookOutlinePosition[ 26789, 623]*) (* CellTagsIndexPosition[ 26745, 619]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Programming examples", "Title"], Cell[CellGroupData[{ Cell["Set vs SetDelayed", "Section"], Cell["\<\ The purpose of this notebook is to illustrate what can happen when \ you repeatedly run a program in which all of the functions are defined using \ SetDelayed compared to using Set.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Modules or not", "Section"], Cell["\<\ The only reason for using Module is to localize variables. If you \ do not have any entries in the first argument of Module then the Module is \ not doing anything and can actually slow things down. I have used Module here \ instead of Block because of the complexity of the assignments I am making. If \ all you are doing is assigning constant values to symbols then Block going to \ be faster and use less memory.\ \>", "Text"], Cell[BoxData[ \(Clear[x, y, z, t, r, s, a, b, c, bigfn, aa, bb, ab, kk, newfunc, sol2]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Set", "Section"], Cell["\<\ Here is a function in which most of the assignments are immediate. \ The only place we use SetDelayed here is for the FindMinimum function, and \ the only reason we use it there is to be able to vary one of the parameters \ in the function we are trying to minimize.\ \>", "Text"], Cell[BoxData[ \(\(\(\(bigfn1[a_, b_, c_]\)[r_, s_] := Module[{sol, aa, newfunc, bb, ab, kk, sol2}, sol\ = \ NDSolve[{\(x'\)[t] \[Equal] a\ \((y[t] - x[t])\), \(y'\)[t]\ \[Equal] \ x[t] \((b\ - \ z[t])\), \(z'\)[t] \[Equal] \ x[t]\ y[t]\ - \ c\ z[t], x[0] \[Equal] 1, y[0] \[Equal] 1, z[0] \[Equal] 1}, {x, y, z}, {t, 0, 10}]; \[IndentingNewLine]aa = y /. sol[\([1]\)]; \[IndentingNewLine]newfunc[bb_, kk_] = \ r\ aa[bb]\ - \ s/kk; \[IndentingNewLine]sol2[ab_]\ := \ FindMinimum[ newfunc[bb, ab], {bb, 1, 0, 4}]; \[IndentingNewLine]Table[{r1, newfunc[r1, k1], \(sol2[k1]\)[\([1]\)]}, {r1, 9, 10, .5}, {k1, .9, 1.1, .1}]\[IndentingNewLine]]\)\(\[IndentingNewLine]\) \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["SetDelayed", "Section"], Cell["\<\ Here we use SetDelayed for all of the assignments. The consequences \ of this are that every time we evaluate the sol2, we also evaluate newfunc \ which requires that we evaluate aa which requires solving the system of \ differential equations.\ \>", "Text"], Cell[BoxData[ \(\(\(\(bigfn2[a_, b_, c_]\)[r_, s_] := Module[{sol, aa, newfunc, bb, ab, kk, sol2}, sol\ := \ NDSolve[{\(x'\)[t] \[Equal] a\ \((y[t] - x[t])\), \(y'\)[t]\ \[Equal] \ x[t] \((b\ - \ z[t])\), \(z'\)[t] \[Equal] \ x[t]\ y[t]\ - \ c\ z[t], x[0] \[Equal] 1, y[0] \[Equal] 1, z[0] \[Equal] 1}, {x, y, z}, {t, 0, 10}]; \[IndentingNewLine]aa := y /. sol[\([1]\)]; \[IndentingNewLine]newfunc[bb_, kk_] := \ r\ aa[bb]\ - \ s/kk; \[IndentingNewLine]sol2[ab_]\ := \ FindMinimum[ newfunc[bb, ab], {bb, 1, 0, 4}]; \[IndentingNewLine]Table[{r1, newfunc[r1, k1], \(sol2[k1]\)[\([1]\)]}, {r1, 9, 10, .5}, {k1, .9, 1.1, .1}]\[IndentingNewLine]]\)\(\[IndentingNewLine]\) \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["What this leads to:", "Section"], Cell["\<\ First we look at an example of our first function used in a table. \ Note this runs in less than two or three seconds.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t1\ = \ MemoryInUse[]\)], "Input"], Cell[BoxData[ \(1955188\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[Table[\(bigfn1[rr, 10, 8/3]\)[1, 2], {rr, 9, 11, .5}]] // Timing\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{\(1.7899999999999998`\ Second\), ",", InterpretationBox[GridBox[{ {GridBox[{ {"9", "2.6286549041901166`", \(-1.8726038779455165`\)}, {"9", "2.850877126412339`", \(-1.6503816557232942`\)}, {"9", "3.0326953082305206`", \(-1.4685634739051123`\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"9.5`", "3.2193454398375954`", \(-1.8726038779455165`\)}, {"9.5`", "3.4415676620598177`", \(-1.6503816557232942`\)}, {"9.5`", "3.6233858438779993`", \(-1.4685634739051123`\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, 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function. Note what happens to \ the timing. There is also an increase in memory use for this function.\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t2\ = \ MemoryInUse[]\)], "Input"], Cell[BoxData[ \(2688756\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[Table[\(bigfn2[rr, 10, 8/3]\)[1, 2], {rr, 9, 11, .5}]] // Timing\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{\(16.52`\ Second\), ",", InterpretationBox[GridBox[{ {GridBox[{ {"9", "2.6286549041901166`", \(-1.8726038779455165`\)}, {"9", "2.850877126412339`", \(-1.6503816557232942`\)}, {"9", "3.0326953082305206`", \(-1.4685634739051123`\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"9.5`", "3.2193454398375954`", \(-1.8726038779455165`\)}, {"9.5`", "3.4415676620598177`", \(-1.6503816557232942`\)}, {"9.5`", "3.6233858438779993`", \(-1.4685634739051123`\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"10.`", 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