(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 12862, 465] NotebookOptionsPosition[ 11234, 413] NotebookOutlinePosition[ 11703, 431] CellTagsIndexPosition[ 11660, 428] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Practica n\.ba3: Raices de Ecuaciones no lineales\ \>", "Title", CellChangeTimes->{{3.506341229125*^9, 3.506341229859375*^9}}], Cell[TextData[StyleBox["Resoluci\[OAcute]n de una ecuaci\[OAcute]n no lineal", FontVariations->{"Underline"->True}]], "Subtitle"], Cell[CellGroupData[{ Cell[TextData[{ "Calcular la raiz de la ecuacion f(x)=(1+x)Sin[x]-1 en el intervalo ( 0. , \ 1. )\nUtilizando el m\[EAcute]todo de Bisecci\[OAcute]n, Punto fijo, \ Newton-Raphson y Secante\nCon un error menor que ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "6"}]], TraditionalForm]]], "\n\n" }], "Section"], Cell[TextData[StyleBox["Biseccion", FontSize->14, FontWeight->"Bold"]], "Subsection"], Cell[TextData[StyleBox["Newton-Raphson", FontSize->14, FontWeight->"Bold"]], "Subsection"], Cell[TextData[StyleBox["Secante", FontSize->14, FontWeight->"Bold"]], "Subsection"] }, Open ]], Cell[TextData[{ "Calcular la raiz de la ecuacion f(x)=(1+x)Sin[x]-1 en el intervalo ( 0. , \ 1. )\nUtilizando el m\[EAcute]todo de Punto fijo, estudiando las condiciones \ de convergencia para la funci\[OAcute]n g(x) utilizada.\nResolver el problema \ con un error menor que ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "6"}]], TraditionalForm]]], "\n\n" }], "Section", CellChangeTimes->{{3.50574230003125*^9, 3.50574234421875*^9}}], Cell["Punto fijo", "Subsection", CellChangeTimes->{3.50574236115625*^9}], Cell["Funci\[OAcute]n utilizada", "Subsubsection", CellChangeTimes->{3.5057423714375*^9}], Cell[BoxData[ RowBox[{ RowBox[{"g", "[", "x_", "]"}], ":=", RowBox[{"ArcSin", "[", FractionBox["1", RowBox[{"1", "+", "x"}]], "]"}]}]], "Input", CellChangeTimes->{3.5057423841875*^9}], Cell["Condiciones de convergencia", "Subsubsection", CellChangeTimes->{3.505742393140625*^9}], Cell["Proceso iterativo", "Subsubsection", CellChangeTimes->{{3.506342426828125*^9, 3.506342437734375*^9}}], Cell[TextData[StyleBox["Raices M\[UAcute]ltiples", FontVariations->{"Underline"->True}]], "Subtitle"], Cell[CellGroupData[{ Cell[TextData[{ "Sea p(x) el polinomio de grado 4,\n\tp(x)=", Cell[BoxData[ RowBox[{ StyleBox["7.14292992000000026`", StyleBoxAutoDelete->True, PrintPrecision->9], "\[InvisibleSpace]", "-", RowBox[{"20.047104`", " ", "x"}], "+", RowBox[{ StyleBox["19.095600000000001`", StyleBoxAutoDelete->True, PrintPrecision->6], " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"7.45000000000000017`", " ", SuperscriptBox["x", "3"]}], "+", SuperscriptBox["x", "4"]}]]], ",\nSe pide:\n1) Representar gr\[AAcute]ficamente p(x).\n2) Utilizar el \ comando Factor para obtener sus raices.\n3) Utilizar el comando FindRoot para \ obtener sus raices dando distintas aproximaciones iniciales. \ \[DownQuestion]Qu\[EAcute] ocurre cuando \[EAcute]stas est\[AAcute]n cercanas \ a la raiz m\[UAcute]ltiple?, \[DownQuestion]Y si est\[AAcute]n \ pr\[OAcute]ximas a las otras raices?\n4) Utilizar el mdo. de Newton-Raphson \ con una aproximaci\[OAcute]n inicial x0=1.4, una precisi\[OAcute]n de ", Cell[BoxData[ SuperscriptBox["10", RowBox[{"-", "6"}]]]], " y un n\.ba m\[AAcute]ximo de iteraciones igual a 25. Hacer los \ c\[AAcute]lculos con una precisi\[OAcute]n de 12 d\[IAcute]gitos ( N[ \ expr,12]).\n\tLa f\[OAcute]rmula de recursi\[OAcute]n es:", Cell[BoxData[ SubscriptBox["x", RowBox[{"i", "+", "1"}]]]], " = ", Cell[BoxData[ SubscriptBox["x", "i"]]], " - p(", Cell[BoxData[ SubscriptBox["x", "i"]]], ") / p\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], ")\n5) Repetir el apdo. anterior con el mdo. de NewtonRaphson acelerado \ (valor de m=2).\n\tLa f\[OAcute]rmula de recursi\[OAcute]n es:", Cell[BoxData[ SubscriptBox["x", RowBox[{"i", "+", "1"}]]]], " = ", Cell[BoxData[ SubscriptBox["x", "i"]]], " - m . p(", Cell[BoxData[ SubscriptBox["x", "i"]]], ") / p\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], ")\n6) Idem. con el mdo. de Halley.\n\tLa f\[OAcute]rmula de \ recursi\[OAcute]n es:\n", Cell[BoxData[ SubscriptBox["x", RowBox[{"i", "+", "1"}]]]], " = ", Cell[BoxData[ SubscriptBox["x", "i"]]], " - 2. p(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], ") /( 2 . p\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], ") - p(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p\[CloseCurlyQuote]\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], "))\n", "7) Idem. con el mdo. de Newton-Raphson modificado para raices \ m\[UAcute]ltiples.\n\tLa f\[OAcute]rmula de recursi\[OAcute]n es:\n", Cell[BoxData[ SubscriptBox["x", RowBox[{"i", "+", "1"}]]]], " = ", Cell[BoxData[ SubscriptBox["x", "i"]]], " - p(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], ") /( p\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], ") - p(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p\[CloseCurlyQuote]\[CloseCurlyQuote](", Cell[BoxData[ SubscriptBox["x", "i"]]], "))\n\t", "\n\t" }], "Section", CellChangeTimes->{{3.505799628015625*^9, 3.505799690875*^9}}], Cell["\<\ 1) Representar gr\[AAcute]ficamente p(x).\ \>", "Subsection"], Cell["\<\ 2) Utilizar el comando Factor para obtener sus raices.\ \>", "Subsection", CellChangeTimes->{{3.505799632265625*^9, 3.505799633953125*^9}}], Cell["\<\ 3) Utilizar el comando FindRoot para obtener sus raices dando distintas \ aproximaciones iniciales. \[DownQuestion]Qu\[EAcute] ocurre cuando \ \[EAcute]stas est\[AAcute]n cercanas a la raiz m\[UAcute]ltiple?, \ \[DownQuestion]Y si est\[AAcute]n pr\[OAcute]ximas a las otras raices?\ \>", "Subsection"], Cell[TextData[{ "4) Utilizar el mdo. de Newton-Raphson con una aproximaci\[OAcute]n inicial \ x0=1.4, una precisi\[OAcute]n de ", Cell[BoxData[ SuperscriptBox["10", RowBox[{"-", "6"}]]]], " y un n\.ba m\[AAcute]ximo de iteraciones igual a 25. Hacer los \ c\[AAcute]lculos con una precisi\[OAcute]n de 12 d\[IAcute]gitos ( N[ \ expr,12]).\n\tLa f\[OAcute]rmula de recursi\[OAcute]n es:", Cell[BoxData[ SubscriptBox["x", RowBox[{"i", "+", "1"}]]]], " = ", Cell[BoxData[ SubscriptBox["x", "i"]]], " - p(", Cell[BoxData[ SubscriptBox["x", "i"]]], ") / p'(", Cell[BoxData[ SubscriptBox["x", "i"]]], ")" }], "Subsection"], Cell[TextData[{ "5) Repetir el apdo. anterior con el mdo. de NewtonRaphson acelerado (valor \ de m=2).\n\tLa f\[OAcute]rmula de recursi\[OAcute]n es:", Cell[BoxData[ SubscriptBox["x", RowBox[{"i", "+", "1"}]]]], " = ", Cell[BoxData[ SubscriptBox["x", "i"]]], " - m . p(", Cell[BoxData[ SubscriptBox["x", "i"]]], ") / p'(", Cell[BoxData[ SubscriptBox["x", "i"]]], ")" }], "Subsection"], Cell[TextData[{ "6) Idem. con el mdo. de Halley.\n\tLa f\[OAcute]rmula de recursi\[OAcute]n \ es:\n", Cell[BoxData[ SubscriptBox["x", RowBox[{"i", "+", "1"}]]]], " = ", Cell[BoxData[ SubscriptBox["x", "i"]]], " - 2. p(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p'(", Cell[BoxData[ SubscriptBox["x", "i"]]], ") /( 2 . p'(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p'(", Cell[BoxData[ SubscriptBox["x", "i"]]], ") - p(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p''(", Cell[BoxData[ SubscriptBox["x", "i"]]], "))" }], "Subsection"], Cell[TextData[{ "7) Idem. con el mdo. de Newton-Raphson modificado para raices \ m\[UAcute]ltiples.\n\tLa f\[OAcute]rmula de recursi\[OAcute]n es:\n", Cell[BoxData[ SubscriptBox["x", RowBox[{"i", "+", "1"}]]]], " = ", Cell[BoxData[ SubscriptBox["x", "i"]]], " - p(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p'(", Cell[BoxData[ SubscriptBox["x", "i"]]], ") /( p'(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p'(", Cell[BoxData[ SubscriptBox["x", "i"]]], ") - p(", Cell[BoxData[ SubscriptBox["x", "i"]]], "). p''(", Cell[BoxData[ SubscriptBox["x", "i"]]], "))\n\t\n\t" }], "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Resoluci\[OAcute]n de un sistema de ecuaciones no \ lineales", FontVariations->{"Underline"->True}]], "Subtitle"], Cell[CellGroupData[{ Cell[TextData[{ "Resolver el sistema de tres ecuaciones con tres incognitas:\n3 ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], "-cos[", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], ".", Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], "]-1/2 = 0\n", Cell[BoxData[ FormBox[ SubsuperscriptBox["x", "1", "2"], TraditionalForm]]], "-81.", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ FormBox[ SubscriptBox["x", "2"], TraditionalForm], " ", "+", " ", "0.1"}], ")"}], "2"], TraditionalForm]]], "+sen(", Cell[BoxData[ FormBox[ SubscriptBox["x", "3"], TraditionalForm]]], ")+1.06 = 0\nExp(-", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1."], SubscriptBox["x", "2"]}], TraditionalForm]]], ") + 20.", Cell[BoxData[ FormBox[ SubscriptBox["x", "3"], TraditionalForm]]], " + (10\[Pi] - 3)/6 = 0\nmediante el mdo. de Newton, siguiendo los \ siguientes pasos:\n1\.ba.- Definici\[OAcute]n del sistema\n2\.ba.- Definir la \ matriz Jacobiana\n3\.ba.- Realizando las iteraciones a mano.(ejecutando \ progresivamente los comandos, sin ciclo)\n4\.ba.- Realizando las iteraciones \ autom\[AAcute]ticamente con criterio de parada que el max. valor absoluto de \ la diferencia de las aproximaciones para cada una de las incognitas sea menor \ que ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "5"}]], TraditionalForm]]], "\n5\.ba.-Resolver con el comando FindRoot." }], "Section", CellChangeTimes->{{3.505799743375*^9, 3.505799775046875*^9}}], Cell["1\.ba.- Definici\[OAcute]n del sistema", "Subsection"], Cell["\<\ 2\.ba.- Definir la matriz Jacobiana\ \>", "Subsection"], Cell[TextData[{ "4\.ba.- Realizando las iteraciones autom\[AAcute]ticamente con criterio de \ parada que el max. valor absoluto de la diferencia de las aproximaciones para \ cada una de las incognitas sea menor que ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "5"}]], TraditionalForm]]] }], "Subsection", TextAlignment->Left, TextJustification->1], Cell["5\.ba.-Con el comando FindRoot.", "Subsection"] }, Open ]] }, Open ]] }, Open ]] }, WindowToolbars->"EditBar", CellGrouping->Manual, WindowSize->{1672, 933}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"8.0 for Microsoft Windows (32-bit) (November 7, 2010)", StyleDefinitions->FrontEnd`FileName[{"Report"}, "StandardReport.nb", CharacterEncoding -> "WindowsANSI"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[579, 22, 138, 3, 81, "Title"], Cell[720, 27, 130, 1, 29, "Subtitle"], Cell[CellGroupData[{ Cell[875, 32, 335, 9, 177, "Section"], Cell[1213, 43, 87, 2, 24, "Subsection"], Cell[1303, 47, 92, 2, 24, "Subsection"], Cell[1398, 51, 85, 2, 24, "Subsection"] }, Open ]], Cell[1498, 56, 461, 11, 177, "Section"], Cell[1962, 69, 73, 1, 24, "Subsection"], Cell[2038, 72, 90, 1, 23, "Subsubsection"], Cell[2131, 75, 199, 6, 53, "Input"], Cell[2333, 83, 94, 1, 23, "Subsubsection"], Cell[2430, 86, 108, 1, 23, "Subsubsection"], Cell[2541, 89, 102, 1, 29, "Subtitle"], Cell[CellGroupData[{ Cell[2668, 94, 3318, 111, 573, "Section"], Cell[5989, 207, 71, 2, 24, "Subsection"], Cell[6063, 211, 150, 3, 24, "Subsection"], Cell[6216, 216, 312, 5, 24, "Subsection"], Cell[6531, 223, 645, 22, 43, "Subsection"], Cell[7179, 247, 405, 16, 42, "Subsection"], Cell[7587, 265, 577, 28, 59, "Subsection"], Cell[8167, 295, 629, 28, 93, "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell[8833, 328, 138, 2, 29, "Subtitle"], Cell[CellGroupData[{ Cell[8996, 334, 1632, 54, 335, "Section"], Cell[10631, 390, 60, 0, 24, "Subsection"], Cell[10694, 392, 65, 2, 24, "Subsection"], Cell[10762, 396, 376, 10, 26, "Subsection"], Cell[11141, 408, 53, 0, 24, "Subsection"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)