(*********************************************************************** 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[ 17045, 607]*) (*NotebookOutlinePosition[ 29903, 1078]*) (* CellTagsIndexPosition[ 29859, 1074]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Analytic vs. Numerical", FontSlant->"Italic"]], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Exact and Approximate results: the function N", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will attempt to give exact answers for formulae which involve only \ integers, rational numbers and symbols. 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Arithmetic between \ expressions of different precision yields a lower-precision answer, as it \ should." }], "Text"], Cell[BoxData[{ \(pi16\ = \ N[Pi]\), \(pi45\ = \ N[Pi, \ 45]\n\), \(pi45\ - \ pi16\)}], "Input"], Cell[TextData[{ "The function ", StyleBox["N", FontWeight->"Bold"], " can be tacked on to the end of a formula using ", StyleBox["postfix notation", FontSlant->"Italic"], "; a double slash followed by the ", StyleBox["N", FontWeight->"Bold"], ". (Postfix notation can be used with any function which takes a single \ parameter.)\nE, the base of natural logarithms (2.71828...), is capitalized \ like all other built-in functions." }], "Text"], Cell[BoxData[{ \(E^3\ + \ \(5!\)/7\), \(E^3\ + \ \(5!\)/7\ \ \ // \ N\n\), \(E^3\ + \ \(5!\)/7\ \ \ // \ Sqrt\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Numerical Artifacts and \"Chop\"", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Numerical computations sometimes yield a tiny real or imaginary \ component which is just an artifact of finite-precision arithmetic:\ \>", "Text"], Cell[BoxData[{ \(Sin[2.0\ Pi]\), \(Sqrt[\ \ \(-1.44\)\ ]\)}], "Input"], Cell[TextData[{ "If you wish to assume that values less than ", Cell[BoxData[ \(TraditionalForm\`10\^\(-10\)\)]], " should be rounded to zero, use ", StyleBox["Chop", FontWeight->"Bold"], ". 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For example, to find \ a complex root it may be necessary to give a complex starting value:\ \>", "Text"], Cell[BoxData[{ \(FindRoot[\ \ Sin[x]\ == \ 2, \ \ {x, 1}]\ \ \ \ \ \ \ \ (*\ \(fails!\)\ *) \), \(FindRoot[\ \ Sin[x]\ == \ 2, \ \ {x, I}]\)}], "Input"], Cell[TextData[{ "\nThe first parameter can be just a formula, not an equation, in which \ case ", StyleBox["FindRoot", FontWeight->"Bold"], " assumes \"==0\": " }], "Text"], Cell["FindRoot[ Cos[x]-x, {x,0}]", "Input"], Cell["\<\ It may make sense to specify two initial points (to make use of \ the \"Secant Method\"), or give a starting values and lower/upper \ bounds.\ \>", "Text"], Cell["\<\ FindRoot[Zeta[I*t - 1/3], {t, {3,4}}] FindRoot[ Sin[Exp[z]]==0.7, {z, 3, 2.7, 3.3}]\ \>", "Input"], Cell[TextData[{ "Two of the options which can be used with ", StyleBox["FindRoot", FontWeight->"Bold"], " are WorkingPrecision (the default is 16 digits), and MaxIterations (the \ default is 15). When you increase the former, it is often necessary to \ increase the latter as well." }], "Text"], Cell[BoxData[{ \(FindRoot[\ \((13 x - 17)\)^2, \ \ {x, 1}, \ WorkingPrecision -> 30]\n \), \(FindRoot[\ \((13 x - 17)\)^2, \ \ {x, 1}, \ WorkingPrecision -> 30, \n \t\t\t\t\t\t\t\t\t\t\t\tMaxIterations -> 40]\)}], "Input"], Cell[TextData[{ StyleBox["FindRoot", FontWeight->"Bold"], " works with simultaneous equations in several variables. Give an initial \ value for each variable." }], "Text"], Cell[BoxData[ \(eq1\ \ = \ \ Sin[x] == Log[y^2 + 1]; \neq2\ \ = \ \ x + y == z; \n eq3\ \ = \ \ y^2 + z^2 == 1; \n\n FindRoot[\ {eq1, eq2, eq3}, \ {x, 1}, {y, 1}, {z, 1}]\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Integrate and NIntegrate", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["NIntegrate", FontWeight->"Bold"], " will succeed in integrating (numerically) functions which ", StyleBox["Integrate", FontWeight->"Bold"], " cannot handle. 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Occasionally there are 2 or more solutions.\nTo extract a solution, \ even if there is only one, use index notation (e.g., soln[[1]]), and use \ \"slash dot\" syntax to pull out the part after the arrow." }], "Text"], Cell["\<\ solution = DSolve[y'[t]==A y[t]-3, y[t], t]; soln1 = y[t] /. solution[[1]]\ \>", "Input"], Cell["\<\ Ordinary differential equations of several functions can also be \ solved as long as the functions are all of the same single independent \ variable.\ \>", "Text"], Cell["\<\ d=DSolve[{y[t]==x'[t], y'[t]-x[t]==6t}, {x[t],y[t]}, t] fx[t_] = x[t] /. d[[1]]; (* get x(t) *) fy[t_] = y[t] /. d[[1]]; (* get y(t) *) fy[ FOO ] fy[ FOO ] /. {C[1]->0, C[2]->1}\ \>", "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Numerical solution, NDSolve: InterpolatingFunction", "Subsection"], Cell[TextData[{ "The analogous numerical solver, ", StyleBox["NDSolve", FontWeight->"Bold"], ", requires that the independent variable be given lower and upper bounds, \ since the answer will be an \"interpolating function\" -- rather like a huge \ table of numerical ", Cell[BoxData[ \(TraditionalForm\`\((t, y)\)\)]], " pairs. 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Extrapolation outside this region is unreliable." }], "Text"], Cell["\<\ NDSolve[{y'[t]==7y[t]-3, y[2]==3}, y[t], {t,-1,1}]; f[t_] = y[t] /. %[[1]] (* define function f *) { f[0.9], f[1.0], f[1.1] }\ \>", "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Graphing the solution", "Subsection"], Cell["\<\ The function Plot can be used to graph the interpolating function, \ over a subrange of the function's interval of definition.\ \>", "Text"], Cell["\<\ soln = NDSolve[ {y'[t]==2Sin[2y[t]]-1, y[1.1]==1.5}, \t\t\t\t\t\t\ty[t], {t,-1,1}]; f[t_] = y[t] /. soln[[1]]; (* define function f *) Plot[ f[t], {t,-1,1}];\ \>", "Input"], Cell["\<\ Partial differential equations take longer to compute, when they can be done \ at all. Have patience with the following example! (30 seconds on a \ SparcStation5)\ \>", "Text"], Cell["\<\ (* Nonlinear Sine-Gorden equation (PDE) *) Y = y[x,t]; sge = D[Y,t,t] == D[Y,x,x] + Sin[Y]; ic1 = y[x,0] == Exp[-x^2]; ic2 = Derivative[0,1][y][x,0] == 0; bc = y[-5,t] == y[5,t]; soln = NDSolve[ {sge,ic1,ic2,bc}, Y, {x,-5,5}, {t,0,5}] Plot3D[ Y /. soln[[1]], {x,-5,5}, {t,0,5}];\ \>", "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Other uses of an interpolating function", "Subsection"], Cell["\<\ One can also get numerical values of the interpolating function \ anywhere in the region of definition, including derivatives, though the \ numerical derivatives may be suspect.\ \>", "Text"], Cell[BoxData[{ \(sol\ \ = \ \ NDSolve[\ {\(y'\)[t] == 2 t, \ y[0] == 0}, \ y[t], \ {t, 0, 5}]; \n\n square[t_]\ = \ y[t]\ /. \ sol[\([1]\)]\n\t\), \({\ \ square[3], \ square[5], \ \(square'\)[5], \ \ \(\(square'\)'\)[5]\ }\)}], "Input"], Cell[TextData[{ StyleBox["Table", FontWeight->"Bold"], " and ", StyleBox["TableForm", FontWeight->"Bold"], " can be used to make a nice table of values for the function:" }], "Text"], Cell[BoxData[ \(TableForm[ Table[\ \ \ {w, square[w], \(square'\)[w]}, \ \ {w, 1, 5, 0.5}]]\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["NSum and NProduct", "Section"], Cell[TextData[{ "Analogous to the functions ", StyleBox["Sum", FontWeight->"Bold"], " and ", StyleBox["Product", FontWeight->"Bold"], " are the functions which find numerical approximations to the sum/product \ of an infinite sequence." }], "Text"], Cell[BoxData[{ \(\ \ Sum[\ \ Log[n + 1]/n^3, \ \ {n, 1, Infinity}\ ]\), \(NSum[\ \ Log[n + 1]/n^3, \ \ {n, 1, Infinity}\ ]\), \(NSum[\ \ Log[n + 1]/n^3, \ \ {n, 1, Infinity}, WorkingPrecision -> 25] \)}], "Input"], Cell[BoxData[{ \(\ \ Product[\ 1 + Sin[n/100]/n^2, \ \ {n, 1, Infinity}]\), \(NProduct[\ 1 + Sin[n/100]/n^2, \ \ {n, 1, Infinity}]\)}], "Input"] }, Closed]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1152}, {0, 900}}, WindowToolbars->{}, CellGrouping->Automatic, WindowSize->{658, 600}, WindowMargins->{{178, Automatic}, {Automatic, 102}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. 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