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This function consists of piecewise cubic polynomials on \ each interval between the x coordinates of the data points and is constructed \ to be continuous with continuous first and second derivatives. The natural \ cubic spline also has its second derivative equal zero at the x coordinate of \ the first and last data point. 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FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->16], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->16], StyleBox["n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->16, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[ "\n\nAn example shows how to use this function to calculate the \ coefficients and then build the natural cubic spline functions automatically \ as Sp[1,x], ..., Sp[n-1,x] and then plot these functions with the data. \ Execute the bold face ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->16], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->16, FontSlant->"Italic"], StyleBox[" commands as you read the notebook.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->16] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->16], Cell[TextData[ "NSplineCoef[S_,a_,b_,c_,d_,n_,t_]:=Module[{f,alpha,mu,z,L,h},\n\t\ n=Length[S];\n\tf=Table[S[[i,2]],{i,1,n}];\n\tt=Table[S[[i,1]],{i,1,n}]; \n\t\ a=Table[0,{i,1,n}];\n\tb=Table[0,{i,1,n}];\n\tc=Table[0,{i,1,n}];\n\t\ d=Table[0,{i,1,n}];\n\talpha=Table[0,{i,1,n}];\n\tmu=Table[0,{i,1,n}];\n\t\ z=Table[0,{i,1,n}];\n\tL=Table[0,{i,1,n}];\n\th=Table[0,{i,1,n}];\n\t\ L[[1]]=1;\n\tmu[[1]]=0;\n\tz[[1]]=0;\n\th[[1]]=t[[2]]-t[[1]];\n\t\ For[i=2,i0,j--,\n\t\t\ c[[j]]=z[[j]]-mu[[j]]*c[[j+1]];\n\t\tb[[j]]=(f[[j+1]]-f[[j]])/h[[j]]-h[[j]]*\n\ \t\t\t(c[[j+1]]+2*c[[j]])/3;\n\t\td[[j]]=(c[[j+1]]-c[[j]])/(3*h[[j]]);\n\t\t\ a[[j]]=f[[j]]\n\t\t];\n\tPrint[\"finished\"]]"], "Input", AspectRatioFixed->True], Cell[TextData[StyleBox["Let's consider the following data points:", Evaluatable->False, AspectRatioFixed->True, FontSize->16]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "B={{46,40},{49,50},{51,55},{52,63},{54,72},{56,70},{57,77},\n\ {58,73},{59,90},{60,93},{61,96},{62,88},{63,99},{64,110},\n\ {66,113},{67,120},{68,127},{71,137},{72,132}}"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "We shall now determine the coefficients of the spline functions in the \ vectors ", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontSize->16, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["b", Evaluatable->False, AspectRatioFixed->True, FontSize->16, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["c", Evaluatable->False, AspectRatioFixed->True, FontSize->16, FontWeight->"Bold"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["d", Evaluatable->False, AspectRatioFixed->True, FontSize->16, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True, FontSize->16] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["NSplineCoef[B,a,b,c,d,n,t]\na\nb\nc\nd"], "Input", AspectRatioFixed->True], Cell[TextData[StyleBox[ "The following creates the natural cubic spline functions automatically as \ Sp[1,x], ..., Sp[n-1,x]", Evaluatable->False, AspectRatioFixed->True, FontSize->16]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[Sp]\nFor[i=1,iTrue], Cell[TextData[StyleBox[ "What is the function value when x=47? Since 47 is between 46 and 49, we use \ the cubic spline function for the first interval and find:", Evaluatable->False, AspectRatioFixed->True, FontSize->16]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["N[Sp[1,47]]"], "Input", AspectRatioFixed->True], Cell[TextData[StyleBox[ "What is the function value when x=70? Since 70 is between 68 and 71, we use \ the 17th spline function and find:", Evaluatable->False, AspectRatioFixed->True, FontSize->16]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["N[Sp[17,70]]"], "Input", AspectRatioFixed->True], Cell[TextData[StyleBox[ "The following code will plot each spline function separately and then plot \ them together with the data.", Evaluatable->False, AspectRatioFixed->True, FontSize->16]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "For[i=1,iPointSize[.015]]\nSP=Show[Table[P[i],{i,1,n-1}]]\n\ Show[B1,SP]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ " The use of the natural cubic spline function to calculate the derivative or \ integral of a function which interpolates the data points"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The natural cubic spline function can be used to estimate the derivative \ or integral of a function approximated by the set of data points. This can \ be accomplished by differentiating or integrating the natural cubic spline \ function formed from the data points.\n\nFor example, to estimate the \ derivative at x=61.5 of a function approximated by the data points \n", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ "B={{46,40},{49,50},{51,55},{52,63},{54,72},{56,70},{57,77},\n\ {58,73},{59,90},{60,93},{61,96},{62,88},{63,99},{64,110},\n\ {66,113},{67,120},{68,127},{71,137},{72,132}}", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["\n\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["we first note that \n", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox[ "\nSp[11, x_] = \n 96 - (580052348861*(-61 + x))/137154968124 - \n \ (516764741777*(-61 + x)^2)/34288742031 + \n (1549871570977*(-61 + \ x)^3)/137154968124\n\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["describes the natural cubic spline function for the interval 61", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\[LessEqual]62 ", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox[ "and determine the derivative of this function at x=61.5. Thus \n", Evaluatable->False, AspectRatioFixed->True, FontSize->16] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Clear[x,g]\ng[x_]=D[Sp[11,x],x]\ng[61.5]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Thus an estimate for the derivative of a function approximated by the set \ of data points is -10.825. Note from the graph above or from the data that \ the graph dips from x=61 to x=62. This dip causes the derivative to be \ negative at x=61.5. \n\nIn a similar manner, if the integral of a function \ approximated by the set of data points from x=59.5 to 61.5 is to be computed \ then the spline functions \n", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox[ "\nSp[9, x_] = \n 90 + (579693333391*(-59 + x))/45718322708 - \n \ (642801432740*(-59 + x)^2)/34288742031 + \n (1243590635159*(-59 + \ x)^3)/137154968124 for ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["59", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\[LessEqual]60", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox[ "\n \nSp[10, x_] = \n 93 + (163720221865*(-60 + x))/68577484062 + \n \ (1159566174517*(-60 + x)^2)/137154968124 - \n (358513904625*(-60 + \ x)^3)/45718322708 for ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["60", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\[LessEqual]61", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox[ "\n \nSp[11, x_] = \n 96 - (580052348861*(-61 + x))/137154968124 - \n \ (516764741777*(-61 + x)^2)/34288742031 + \n (1549871570977*(-61 + \ x)^3)/137154968124 for ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["61", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox["\[LessEqual]62\n \n", Evaluatable->False, AspectRatioFixed->True, FontSize->16], StyleBox[ "can be integrated to determine an estimate. In particular, the integral \ of Sp[9,x] from 59.5 to 60 + the integral of Sp[10,x] from 60 to 61 + \ integral of Sp[11,x] from 61 to 61.5 would form the estimate of the integral \ of a function approximated by the set of data points from x=59.5 to 61.5. \ This estimate may be computed by the commands below.", Evaluatable->False, AspectRatioFixed->True, FontSize->16] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Integrate[Sp[9,x],{x,59.5,60}]+\nIntegrate[Sp[10,x],{x,60,61}]+\n\ Integrate[Sp[11,x],{x,61,61.5}]"], "Input", AspectRatioFixed->True]}, Open]]}, Open]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{520, 365}, WindowMargins->{{44, Automatic}, {Automatic, 16}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, MacintoshSystemPageSetup->"\<\ AVU/IFiQKFD000000V:^/09R]g0000000OVaH097bCP0AP1Y06`0I@1^0642HZj` 0V:gT0000001nK500TO9>000000000000000009R[[0000000000000000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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