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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[ 79463, 2523]*) (*NotebookOutlinePosition[ 80197, 2549]*) (* CellTagsIndexPosition[ 80153, 2545]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ L'Hospital's Rule, Min-Max (local & absolute) Curve Graphing, Newton's Method\ \>", "Subtitle", TextAlignment->Left, TextJustification->0], Cell[TextData[{ " ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Subtitle", TextAlignment->Left, TextJustification->0], Cell["\<\ Dr. Andrzej Korzeniowski Department of Mathematics, UTA\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Lab 4", 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To find the positive solution take the initial guess as 9:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(x0 = 9\), ";", "\n", RowBox[{\(f[x_]\), ":=", RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}]}]}], "\n", \(\(f'\)[x]\)}], "Input"], Cell[BoxData[ \(3 - 2\ x - 30\ x\^2 + 4\ x\^3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x1 = x0 - f[x0]\/\(f'\)[x0]\)], "Input"], Cell[BoxData[ \(5072\/471\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x1 // N\)], "Input"], Cell[BoxData[ \(10.7685774946921442`\)], "Output"] }, Open ]], Cell["\<\ To streamline computations define a new function g[x] which \ evaluates numerically the first 25 digits of approximation sequence:\ \>", "Text"], Cell[BoxData[ \(g[x_] := N[x - f[x]\/\(f'\)[x], 25]\)], "Input"], Cell["Then", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x1 = g[x0]\)], "Input"], Cell[BoxData[ \(10.76857749469214437367303609341825902336`25\)], "Output"] }, Open ]], Cell["\<\ To compute n iterations starting at x0 one uses the NestList[g, \ x0, n] as follows:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(NestList[g, x0, 6]\)\(//\)\(TableForm\)\(\n\)\)\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {"9"}, {"10.76857749469214437367303609341825902336`25"}, {"10.216955306206155083503611723443462405`24.1064"}, {"10.12058299258635588182831319643859182795`23.2099"}, {"10.117806239312983280218683208819809`22.3144"}, {"10.11780397413295562844228212398523`21.4188"}, {"10.117803974131449032521296575722705`20.5233"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {9, 10.76857749469214437367304`25, 10.21695530620615508350361`24.1064, 10.1205829925863558818283`23.2099, 10.117806239312983280219`22.3144, 10.11780397413295562844`21.4188, 10.1178039741314490325`20.5233}]]], "Output"] }, Open ]], Cell[TextData[{ " Compare the above results to built-in functions such as NRoots[expr[x] = \ = 0, x], NSolve[ expr[x] = = 0], FindRoot[expr[x] = = 0,{x, ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], "}],\n where ", Cell[BoxData[ \(TraditionalForm\`x\_\(\(0\)\(\ \ \)\)\)]], "is the initial approximation chosen in the vicinity of the exact \ solution." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NRoots", "[", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}], "==", "0"}], ",", "x"}], "]"}]], "Input"], Cell[BoxData[ \(x == \(-1.70506413927755069`\) || x == \(\(0.79363008257305081`\)\(\[InvisibleSpace]\)\) - 1.50613717733980889`\ I || x == \(\(0.79363008257305081`\)\(\[InvisibleSpace]\)\) + 1.50613717733980889`\ I || x == 10.1178039741314496`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NSolve", "[", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}], "==", "0"}], ",", "x"}], "]"}]], "Input"], Cell[BoxData[ \({{x \[Rule] \(-1.70506413927755069`\)}, {x \[Rule] \ \(\(0.79363008257305081`\)\(\[InvisibleSpace]\)\) - 1.50613717733980889`\ I}, {x \[Rule] \(\(0.79363008257305081`\)\(\ \[InvisibleSpace]\)\) + 1.50613717733980889`\ I}, {x \[Rule] 10.1178039741314496`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}], "==", "0"}], ",", \({x, 9}\)}], "]"}]], "Input"], Cell[BoxData[ \({x \[Rule] 10.117803974132955`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Options[FindRoot]\)], "Input"], Cell[BoxData[ \({AccuracyGoal \[Rule] Automatic, Compiled \[Rule] True, DampingFactor \[Rule] 1, Jacobian \[Rule] Automatic, MaxIterations \[Rule] 15, WorkingPrecision \[Rule] 16}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}], "==", "0"}], ",", \({x, 9}\), ",", \(AccuracyGoal \[Rule] 25\), ",", \(WorkingPrecision \[Rule] 25\)}], "]"}]], "Input"], Cell[BoxData[ \({x \[Rule] 10.11780397413144903252129657601282799502`25}\)], "Output"] }, Open ]], Cell["\<\ As one can see the 4-th iteration in Newton's scheme agrees up to \ the first 7 digits with the exact solution, while the 6-th iteration provides the exact first 21 digits! Numerical \ analysts, \"Go to dust!\"\ \>", "Text"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{666, 599}, WindowMargins->{{4, Automatic}, {Automatic, 4}}, Magnification->1 ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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