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\(punktplot = ListPlot[Table[{xp[k], yp[k]}, {k, 0, m}], PlotJoined\ -> \ False, \t PlotRange -> {{\(-5\), 5}, {\(-4\), 4}}, \n\tPlotStyle -> Blue, Prolog\ -> \ AbsolutePointSize[5], AspectRatio -> 0.5, AxesLabel -> {"\<> x\>", "\< ^ y\>"}]\)], "Input"], Cell[BoxData[ \(linienplot = ListPlot[Table[{xp[k], yp[k]}, {k, 0, m}], PlotJoined\ -> \ True, \t PlotRange -> {{\(-5\), 5}, {\(-4\), 4}}, \n\tPlotStyle -> Blue, Prolog\ -> \ AbsolutePointSize[5], AspectRatio -> 0.5, AxesLabel -> {"\<> x\>", "\< ^ y\>"}]\)], "Input"], Cell[BoxData[ \(Show[ellipsplot, linienplot, punktplot, Prolog\ -> \ AbsolutePointSize[5]]\)], "Input"], Cell[BoxData[ RowBox[{" ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Newton\ - \ Interpolation\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), "Section", FontColor->RGBColor[1, 0, 0]]}]], "Input", Background->RGBColor[0, 1, 0]], Cell[BoxData[ RowBox[{" ", StyleBox[\( (*\ \ \ \ \ \ \ \ Funktionsunterprogramm\ zur\ Berechnung\ \ der\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), "Subsubtitle", FontColor->RGBColor[1, 0, 1]], "\[IndentingNewLine]", " ", StyleBox[\( (*\ \ \ \ \ Dividierte\ Differenzen\ f\[UDoubleDot]r\ die\ \ Newton - Interpolation\ \ \ \ \ \ \ \ \ \ \ *) \), "Subsubtitle", FontColor->RGBColor[1, 0, 1]], " "}]], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(FunkDivDiff[xpform_, ypform_] := \ \((\[IndentingNewLine]Do[ DivDiffret[k, 1] = \((ypform[k + 1] - ypform[k])\)/\((xpform[k + 1] - xpform[k])\), {k, 0, m - 1}]; \[IndentingNewLine]Do[ Do[DivDiffret[k, j] = \((DivDiffret[k + 1, j - 1] - DivDiffret[k, j - 1])\)/\((xpform[k + j] - xpform[k])\), {k, 0, m - j}], {j, 2, m}]; DivDiffmatret = Table[DivDiffret[0, j], {j, 1, m}]; {DivDiffmatret})\)\)], "Input"], Cell[BoxData[ \(\(DivDiffxmat = FunkDivDiff[tp, xp]\ ;\)\)], "Input"], Cell[BoxData[ RowBox[{\(DivDiffx = DivDiffxmat[\([1]\)];\), StyleBox[ RowBox[{" ", StyleBox[" ", FontColor->RGBColor[1, 0, 1]]}]], StyleBox[\( (*\ \ Dividierte\ Differenzen\ f\[UDoubleDot]r\ die\ x - Werte\ \ \ *) \), FontColor->RGBColor[1, 0, 1]]}]], "Input"], Cell[BoxData[ RowBox[{\(DivDiffymat = FunkDivDiff[tp, yp]\ ;\), StyleBox[" ", FontColor->RGBColor[1, 0, 1]]}]], "Input"], Cell[BoxData[ RowBox[{\(DivDiffy = DivDiffymat[\([1]\)];\), " ", StyleBox[\( (*\ \ Dividierte\ Differenzen\ f\[UDoubleDot]r\ die\ y - Werte\ \ \ *) \), FontColor->RGBColor[1, 0, 1]]}]], "Input"], Cell[BoxData[ RowBox[{" ", StyleBox[\( (*\ \ \ \ St\[UDoubleDot]tzstellen\ xj\ und\ yj\ f\ \[UDoubleDot]r\ den\ Graph\ der\ Newton - Interpolation\ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]]}]], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(Do[{xnew[j] = xp[0], prox = tj[j] - tp[0], \[IndentingNewLine]Do[{xnew[j] = xnew[j] + prox*DivDiffx[\([i]\)], prox = prox*\((tj[j] - tp[i])\)}, {i, 1, m}], \[IndentingNewLine]ynew[j] = yp[0], proy = tj[j] - tp[0], \n\t Do[{ynew[j] = ynew[j] + proy*DivDiffy[\([i]\)], proy = proy*\((tj[j] - tp[i])\)}, {i, 1, m}]}, {j, 0, nd}]\)], "Input"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{" ", StyleBox[" ", FontColor->RGBColor[1, 0, 1]]}]], StyleBox[\( (*\ \ \ \ Parametrische\ Darstellung\ der\ Newton - Interpolation\ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]]}]], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(newtonplot = ListPlot[Table[{xnew[j], ynew[j]}, {j, 0, nd}], PlotJoined\ -> \ True, \n\t PlotRange -> {{\(-5\), 5}, {\(-4\), 4}}, \tPlotStyle -> Red, AspectRatio -> 0.5, AxesLabel -> {"\<> x\>", "\< ^ y\>"}]\)], "Input"], Cell[BoxData[ \(\(\(Show[newtonplot, ellipsplot, linienplot, punktplot, Prolog\ -> \ AbsolutePointSize[5]]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[BoxData[ RowBox[{" ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ Nat\[UDoubleDot]rliche\ Spline - Interpolation\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \ \), "Section", FontColor->RGBColor[1, 0, 0]]}]], "Input", Background->RGBColor[0, 1, 0]], Cell[BoxData[ RowBox[{" ", StyleBox[\( (*\ \ \ \ \ \ \ \ Funktionsunterprogramm\ zur\ Berechnung\ \ der\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), "Subsubtitle", FontColor->RGBColor[1, 0, 1]], "\[IndentingNewLine]", " ", StyleBox[\( (*\ \ \ \ \ Koeffizienten\ f\[UDoubleDot]r\ die\ nat\ \[UDoubleDot]rlichen\ Spline - Interpolation\ \ \ \ \ \ \ \ \ *) \), "Subsubtitle", FontColor->RGBColor[1, 0, 1]], " "}]], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ RowBox[{\(FunkNatSpl[xpform_, ypform_]\), ":=", " ", RowBox[{ "(", "\[IndentingNewLine]", " ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Erstellen\ \ der\ Tridiagonalmatrix\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), "Subsubtitle", FontColor->RGBColor[1, 0, 1]], "\n", " ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ \ \((\ Haupt - \ und\ Nebendiagonale\ )\)\ \ \ \ \ \ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]], "\[IndentingNewLine]", RowBox[{\(Du[0] = xpform[1] - xpform[0]\), ";", "\n", " ", \(Do[{Du[k] = xpform[k + 1] - xpform[k], Dh[k] = 2 \((Du[k - 1] + Du[k])\)}, {k, 1, m - 1}]\), ";", "\[IndentingNewLine]", " ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ Cholesky - Zerlegung\ der\ Tridiagonalmatrix\ \ \ \ \ \ \ \ \ \ \ *) \), "Subsubtitle", FontColor->RGBColor[1, 0, 1]], "\[IndentingNewLine]", \(Ch[1] = Sqrt[Dh[1]]\), ";", "\n", " ", \(Do[{Cn[k - 1] = Du[k - 1]/Ch[k - 1], Ch[k] = Sqrt[Dh[k] - Cn[k - 1]^2]}, {k, 2, m - 1}]\), ";", "\[IndentingNewLine]", " ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ \ Vorw\[ADoubleDot]rtsrechnung\ \ "\"\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]], StyleBox["\n", FontColor->RGBColor[1, 0, 1]], " ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ \ und\ Erstellen\ der\ rechten\ \ Seite\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]], "\[IndentingNewLine]", \(Dv[0] = ypform[1] - ypform[0]\), ";", " ", \(Dv[1] = ypform[2] - ypform[1]\), ";", "\n", " ", \(Dr[1] = 3 \((Dv[1]/Du[1] - Dv[0]/Du[0])\)\), ";", "\n", " ", \(Z[1] = Dr[1]/Ch[1]\), ";", "\n", " ", \(Do[{Dv[k] = ypform[k + 1] - ypform[k], \n\t\tDr[k] = 3 \((Dv[k]/Du[k] - Dv[k - 1]/Du[k - 1])\), \n\t\tZ[ k] = \((Dr[k] - Z[k - 1]*Cn[k - 1])\)/Ch[k]}, \n\ \ \ {k, 2, m - 1}]\), ";", "\[IndentingNewLine]", " ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R\[UDoubleDot]ckw\ \[ADoubleDot]rtsrechnung\ "\"\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]], StyleBox["\n", FontColor->RGBColor[1, 0, 1]], " ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ \ \tBerechnung\ der\ \ Koeffizienten\ B\_k\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]], "\[IndentingNewLine]", \(B[m] = 0\), ";", " ", \(B[m - 1] = Z[m - 1]/Ch[m - 1]\), ";", "\n", " ", \(Do[ B[k] = \((Z[k] - B[k + 1]*Cn[k])\)/Ch[k], {k, m - 2, 1, \(-1\)}]\), ";", " ", \(B[0] = 0\), ";", "\[IndentingNewLine]", " ", StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ \ \ \ Berechnung\ der\ \ Koeffizienten\ A\_\(\(k\)\(\ \)\), \ C\_k\ , \ D\_k\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]], "\[IndentingNewLine]", \(Do[{A[ k] = \((B[k + 1] - B[k])\)/\((3 Du[k])\), \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ Cc[k] = Dv[k]/Du[k] - Du[k]*\((B[k + 1] + 2 B[k])\)/3, \ Dc[k] = ypform[k]}, {k, 0, m - 1}]\), ";", "\[IndentingNewLine]", " ", \(Aret = Table[A[k], \ {k, 0, m - 1}]\), ";", " ", \(Bret = Table[B[k], {k, 0, m}]\), ";", " ", "\[IndentingNewLine]", " ", \(Cret = Table[Cc[k], {k, 0, m - 1}]\), ";", " ", \(Dret = Table[Dc[k], {k, 0, m - 1}]\), ";", "\[IndentingNewLine]", \({Aret, Bret, Cret, Dret}\)}], ")"}]}]], "Input"], Cell[BoxData[ \(\(ABCDxmat = FunkNatSpl[tp, xp]\ \ ;\)\)], "Input"], Cell[BoxData[ RowBox[{\(Anatx = ABCDxmat[\([1]\)]\ ;\), " ", StyleBox[\( (*\ Koeffizienen\ Ak\ \ \(f \[UDoubleDot]r\)\ die\ x - Werte\ *) \), FontColor->RGBColor[1, 0, 1], Background->None]}]], "Input"], Cell[BoxData[ RowBox[{\(Bnatx = ABCDxmat[\([2]\)];\), " ", RowBox[{ StyleBox["(*", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], RowBox[{ RowBox[{ StyleBox["Koeffizienen", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["Bk", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox[\(f \[UDoubleDot]r\), FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["die", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["x", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox["-", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["Werte", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["*)", FontColor->RGBColor[1, 0, 1]]}]}]], "Input"], Cell[BoxData[ RowBox[{\(Cnatx = ABCDxmat[\([3]\)];\), " ", RowBox[{ StyleBox["(*", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], RowBox[{ RowBox[{ StyleBox["Koeffizienen", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["Ck", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox[\(f \[UDoubleDot]r\), FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["die", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["x", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox["-", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["Werte", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["*)", FontColor->RGBColor[1, 0, 1]]}]}]], "Input"], Cell[BoxData[ RowBox[{\(Dnatx = ABCDxmat[\([4]\)];\), " ", RowBox[{ StyleBox["(*", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], RowBox[{ RowBox[{ StyleBox["Koeffizienen", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["Dk", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox[\(f \[UDoubleDot]r\), FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["die", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["x", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox["-", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["Werte", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["*)", FontColor->RGBColor[1, 0, 1]]}]}]], "Input"], Cell[BoxData[ \(\(ABCDymat = FunkNatSpl[tp, yp]\ ;\)\)], "Input"], Cell[BoxData[ RowBox[{\(Anaty = ABCDymat[\([1]\)];\), " ", StyleBox[\( (*\ Koeffizienen\ Ak\ \ \(f \[UDoubleDot]r\)\ die\ x - Werte\ *) \), FontColor->RGBColor[1, 0, 1], Background->None]}]], "Input"], Cell[BoxData[ RowBox[{\(Bnaty = ABCDymat[\([2]\)];\), " ", RowBox[{ StyleBox["(*", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], RowBox[{ RowBox[{ StyleBox["Koeffizienen", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["Bk", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox[\(f \[UDoubleDot]r\), FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["die", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["x", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox["-", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["Werte", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["*)", FontColor->RGBColor[1, 0, 1]]}]}]], "Input"], Cell[BoxData[ RowBox[{\(Cnaty = ABCDymat[\([3]\)];\), " ", RowBox[{ StyleBox["(*", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], RowBox[{ RowBox[{ StyleBox["Koeffizienen", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["Ck", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox[\(f \[UDoubleDot]r\), FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["die", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["x", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox["-", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["Werte", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["*)", FontColor->RGBColor[1, 0, 1]]}]}]], "Input"], Cell[BoxData[ RowBox[{\(Dnaty = ABCDymat[\([4]\)]\ ;\), " ", RowBox[{ StyleBox["(*", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], RowBox[{ RowBox[{ StyleBox["Koeffizienen", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["Dk", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox[\(f \[UDoubleDot]r\), FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["die", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox[" ", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["x", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox["-", FontColor->RGBColor[1, 0, 1], Background->None], StyleBox["Werte", FontColor->RGBColor[1, 0, 1], Background->None]}], StyleBox[" ", FontColor->RGBColor[1, 0, 1]], StyleBox["*)", FontColor->RGBColor[1, 0, 1]]}]}]], "Input"], Cell[BoxData[ RowBox[{" ", StyleBox[\( (*\ \ \ \ St\[UDoubleDot]tzstellen\ xj\ und\ yj\ f\ \[UDoubleDot]r\ den\ Graph\ der\ Spline - Interpolation\ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]]}]], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(Do[{tint = tp[0], \n\t Do[{Dnt[k] = tp[k + 1] - tp[k], If[\((tj[j] \[GreaterEqual] tint\ )\)\ \[And] \ \ \((tj[j] \[LessEqual] tint + Dnt[k]\ )\), {knt = k, Break[]}\ , tint = tint + Dnt[k]]}, {k, 0, m - 1}], \n\t Dntmj = tj[j] - tp[knt], \[IndentingNewLine]\ \ \ \ \ xnat[j] = Anatx[\([knt + 1]\)]*Dntmj^3 + Bnatx[\([knt + 1]\)]*Dntmj^2 + Cnatx[\([knt + 1]\)]*Dntmj + Dnatx[\([knt + 1]\)]\ , \n\t ynat[j] = Anaty[\([knt + 1]\)]*Dntmj^3 + Bnaty[\([knt + 1]\)]*Dntmj^2 + Cnaty[\([knt + 1]\)]*Dntmj + \ \ Dnaty[\([knt + 1]\)]\ }, \n\t{j, 0, nd}]\)], "Input"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{" ", StyleBox[" ", FontColor->RGBColor[1, 0, 1]]}]], StyleBox[\( (*\ \ \ \ \ \ \ \ \ \ Graph\ der\ nat . \ Spline - Funktion\ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \), FontColor->RGBColor[1, 0, 1]]}]], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(natsplplot = ListPlot[Table[{xnat[j], ynat[j]}, {j, 0, nd}], PlotJoined\ -> \ True, \n\t PlotRange -> {{\(-5\), 5}, {\(-4\), 4}}, \tPlotStyle -> 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