(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 20373, 524] NotebookOptionsPosition[ 18801, 472] NotebookOutlinePosition[ 19383, 494] CellTagsIndexPosition[ 19340, 491] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Practica n\.ba4: Interpolaci\[OAcute]n polinomial \ \>", "Title", CellChangeTimes->{ 3.50634106159375*^9, {3.506341237390625*^9, 3.50634123840625*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Uso del comando interno de ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Subtitle"], Cell[TextData[{ "En esta pr\[AAcute]ctica estudiaremos c\[OAcute]mo construir el polinoimio \ interpolador de un conjunto de datos. ", StyleBox["Mathematica", FontSlant->"Italic"], " posee un comando muy flexible y vers\[AAcute]til, \n\t", StyleBox["InterpolatingPolynomial[", FontWeight->"Bold"], StyleBox["nube,var", FontSlant->"Italic"], StyleBox["]", FontWeight->"Bold"], ", \nque permite construir el polinomio correspondiente:" }], "Text", TextAlignment->Left, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"nube", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "1"}], "}"}]}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"p", "[", "x_", "]"}], "=", RowBox[{"InterpolatingPolynomial", "[", RowBox[{"nube", ",", "x"}], "]"}]}]}], "Input"], Cell[BoxData[ RowBox[{"2", "+", RowBox[{ RowBox[{"(", RowBox[{ FractionBox["3", "2"], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "x"}], ")"}]}]}], ")"}], " ", "x"}]}]], "Output", CellChangeTimes->{3.43711171321875*^9, 3.505743291359375*^9, 3.50574350153125*^9, 3.5063410545*^9}] }, Open ]], Cell["\<\ Observe que el resultado se ofrece en forma de Newton. Para obtener la forma est\[AAcute]ndar, basta expandir el polinomio:\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{"p", "[", "x", "]"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"2", "+", "x", "+", FractionBox[ SuperscriptBox["x", "2"], "4"]}]], "Output", CellChangeTimes->{3.43711171525*^9, 3.505743291375*^9, 3.5057435015625*^9, 3.50634105453125*^9}] }, Open ]], Cell["\<\ Veamos el resultado gr\[AAcute]ficamente:\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"pt", "=", RowBox[{"ListPlot", "[", RowBox[{"nube", ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"PointSize", "[", "0.05", "]"}], "}"}]}]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"pol", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"p", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2.5"}], ",", "2.5"}], "}"}]}], "]"}]}], ";"}], "\n", RowBox[{"Show", "[", RowBox[{"pol", ",", "pt"}], "]"}]}], "Input", 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valor exacto el valor aproximado.\nd) dibujar la funcion diferencia g(x) \ -p(x).\n" }], "Section", CellChangeTimes->{3.506342614859375*^9}, FontFamily->"Arial", FontColor->GrayLevel[0]], Cell[TextData[{ StyleBox["Problema 2 :Polinomio de Lagrange", FontFamily->"Arial"], ". Construcci\[OAcute]n de las funciones de base de Lagrange" }], "Subtitle", CellChangeTimes->{{3.50580004690625*^9, 3.505800072109375*^9}}], Cell[TextData[{ "En caso de necesidad, podemos construir directamente los polinomios b\ \[AAcute]sicos de Lagrange\n ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["L", "i"], " ", RowBox[{"(", "x", ")"}]}], "=", RowBox[{ UnderscriptBox["\[Product]", RowBox[{"j", "\[NotEqual]", "i"}]], FractionBox[ RowBox[{"(", RowBox[{"x", "-", SubscriptBox["x", "j"]}], ")"}], RowBox[{"(", RowBox[{ SubscriptBox["x", "i"], "-", SubscriptBox["x", "j"]}], ")"}]]}]}], TraditionalForm]]], "\nusando las capacidades del ", StyleBox["Mathematica", FontSlant->"Italic"], " para operar con listas:" }], "Text", FontSize->14], Cell[TextData[StyleBox["Sea f(x)=Log[10,Tan(x)].A partir de los \ valores:f(1.00)=0.1924,f(1.05)=0.2414,f(1.10)=0.2933 y f(1.15)=0.3492\nse \ pide:\n1\.ba.-Calcular el polinomio de interpolaci\[OAcute]n de Lagrange.\n\ Para ello, seguir los siguientes pasos.\n- Generar la lista de nodos xi\n- \ Plantear una lista L con elementos unidad donde se almacenar\[AAcute]n los \ Li(x)\n- Calcular los Li(x) y almacenarlos\n- crear un vector y con los \ valores f(xi)\nEl polinomio sera el producto escalar de los vectores y.L\t\n\n\ 2\.ba.-Utilizar dicho polinomio para obtener un valor aproximado de f(1.09)\n\ 4\.ba.-Representar el error cometido, graficamente.", FontFamily->"Arial"]], "Section", CellChangeTimes->{{3.505799986640625*^9, 3.50579998759375*^9}, { 3.505800042984375*^9, 3.505800044390625*^9}, {3.50634263184375*^9, 3.50634263325*^9}}, FontColor->GrayLevel[0]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.505799985140625*^9}], Cell["Problema 3: Polinomios interpoladores de Newton", "Subtitle", FontColor->GrayLevel[0]], Cell[TextData[{ StyleBox["1.- Calcular el polinomio interpolador de Newton para la misma \ funci\[OAcute]n, mediante diferencias finitas.\n\t-Calcular las diferencias \ finitas.\n\t- Plantear el polinomio 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