(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[      6610,        225]
NotebookOptionsPosition[      6042,        200]
NotebookOutlinePosition[      6447,        217]
CellTagsIndexPosition[      6404,        214]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["\<\
C\[AAcute]lculo Num\[EAcute]rico                       Febrero  2009\
\>", "Subtitle",
 CellChangeTimes->{{3.4423192508125*^9, 3.44231925578125*^9}},
 TextAlignment->Left,
 TextJustification->0],

Cell[CellGroupData[{

Cell["1er Ejercicio", "Section"],

Cell[TextData[{
 "La funci\[OAcute]n  f(x)=",
 Cell[BoxData[
  FormBox[
   RowBox[{
    FractionBox[
     RowBox[{
      RowBox[{"8", "x"}], "-", "1"}], "x"], "-", 
    SuperscriptBox["\[ExponentialE]", "x"]}], TraditionalForm]]],
 " tiene dos raices.\nSe pretende determinar cada una de ellas mediante \
aproximaciones sucesivas y se dispone de las funciones auxiliares ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SubscriptBox["g", "1"], "(", "x", ")"}], "=", 
    FractionBox[
     RowBox[{"1", "+", 
      RowBox[{"x", "*", 
       SuperscriptBox["\[ExponentialE]", "x"]}]}], "8"]}], TraditionalForm]]],
 " y ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SubscriptBox["g", "2"], "(", "x", ")"}], "=", 
    RowBox[{"Log", "[", 
     FractionBox[
      RowBox[{
       RowBox[{"8", "x"}], "-", "1"}], "x"], "]"}]}], TraditionalForm]]],
 "\n\nDibujar la funci\[OAcute]n para determinar el intervalo en que se \
encuentran las raices.\nEstudiar los criterios de convergencia del \
m\[EAcute]todo de punto fijo para determinar si alguna de las funciones g \
sirven para calcular las ra\[IAcute]ces de f.\nAplicar el m\[EAcute]todo de \
punto fijo y obtener ambas raices con un error menor que ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["10", 
    RowBox[{"-", "5"}]], TraditionalForm]]],
 "."
}], "Text",
 CellChangeTimes->{{3.442319301609375*^9, 3.44231932978125*^9}, {
   3.442319402375*^9, 3.4423194759375*^9}, {3.4423195286875*^9, 
   3.442319732484375*^9}, 3.442661065359375*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell["2\.ba Ejercicio", "Section"],

Cell[TextData[{
 "La soluci\[OAcute]n del sistema lineal :\n",
 Cell[BoxData[{
  FormBox[
   RowBox[{
    RowBox[{
     SubscriptBox["x", "1"], "                           ", "-", " ", 
     SubscriptBox["x", "3"]}], " ", "=", " ", "0.2"}], 
   TraditionalForm], "\[IndentingNewLine]", 
  FormBox[
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"-", "0.5"}], " ", 
      SubscriptBox["x", "1"]}], "+", 
     SubscriptBox["x", "2"], "-", 
     RowBox[{"0.25", " ", 
      SubscriptBox["x", "3"]}]}], " ", "=", " ", 
    RowBox[{"-", "1.425"}]}], TraditionalForm]}]],
 "\n",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SubscriptBox["x", "1"], " ", "-", " ", 
     RowBox[{"0.5", " ", 
      SubscriptBox["x", "2"]}], "        ", "+", "  ", 
     SubscriptBox["x", "3"]}], "  ", "=", " ", "2"}], TraditionalForm]]],
 "\nes ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"0.9", ",", 
      RowBox[{"-", "0.8"}], ",", "0.7"}], ")"}], "t"], TraditionalForm]]],
 ".\n1\.ba.- Calcular el radio espectral de la matriz de paso del \
m\[EAcute]todo de Jacobi.\n2\.ba.- Utilizar el m\[EAcute]todo de Jacobi para \
obtener una soluci\[OAcute]n aproximada del sistema con una precisi\[OAcute]n \
de ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["10", 
    RowBox[{"-", "2"}]], TraditionalForm]]],
 ", partiendo de la aproximaci\[OAcute]n inicial : ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"0", ",", "0", ",", "0"}], ")"}], "t"], " ", ","}], 
   TraditionalForm]]],
 " con un n\[UAcute]mero m\[AAcute]ximo de iteraciones de 300. Utilizar la \
norma eucl\[IAcute]dea.\n3\.ba.- Repetir los apartados anteriores para el m\
\[EAcute]todo de Gauss-Seidel.\n4\.ba.- Comparar el n\.ba de iteraciones \
realizadas con ambos m\[EAcute]todos.\n5\.ba.- Justificar lo que pasar\
\[IAcute]a  si cambiamos el sistema anterior por :\n",
 Cell[BoxData[{
  FormBox[
   RowBox[{
    RowBox[{
     SubscriptBox["x", "1"], "                           ", "-", 
     RowBox[{"2", " ", 
      SubscriptBox["x", "3"]}]}], " ", "=", " ", "0.2"}], 
   TraditionalForm], "\[IndentingNewLine]", 
  FormBox[
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"-", "0.5"}], " ", 
      SubscriptBox["x", "1"]}], "+", 
     SubscriptBox["x", "2"], "-", 
     RowBox[{"0.25", " ", 
      SubscriptBox["x", "3"]}]}], " ", "=", " ", 
    RowBox[{"-", "1.425"}]}], TraditionalForm]}]],
 "\n",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SubscriptBox["x", "1"], " ", "-", " ", 
     RowBox[{"0.5", " ", 
      SubscriptBox["x", "2"]}], "        ", "+", "  ", 
     SubscriptBox["x", "3"]}], "  ", "=", " ", "2"}], TraditionalForm]]],
 "\nNota: Si la Matriz de coeficientes se representa como  A=L+D+U \nLas \
matrices de paso de los m\[EAcute]todos de Jacobi y Gauss-Seidel son \
respectivamente:\n",
 Cell[BoxData[
  FormBox[
   RowBox[{
    SubscriptBox["T", "j"], " ", "=", 
    RowBox[{"-", " ", 
     RowBox[{
      SuperscriptBox["D", 
       RowBox[{"-", "1"}]], "(", 
      RowBox[{"L", " ", "+", " ", "U"}], ")"}], " "}]}], TraditionalForm]]],
 ", y\n",
 Cell[BoxData[
  FormBox[
   SubscriptBox["T", "g"], TraditionalForm]]],
 " = ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"-", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"D", "+", "L"}], ")"}], 
      RowBox[{"-", "1"}]]}], "U"}], TraditionalForm]]]
}], "Text",
 CellChangeTimes->{
  3.442319854421875*^9, {3.4423198929375*^9, 3.442319923640625*^9}, {
   3.442661083109375*^9, 3.442661132875*^9}},
 TextAlignment->Left,
 TextJustification->0]
}, Open  ]]
}, Open  ]]
},
WindowToolbars->{"RulerBar", "EditBar"},
WindowSize->{1085, 907},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
Magnification->1.5,
FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (November 10, 2008)",
StyleDefinitions->"Classic.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[567, 22, 204, 5, 90, "Subtitle"],
Cell[CellGroupData[{
Cell[796, 31, 32, 0, 74, "Section"],
Cell[831, 33, 1514, 43, 244, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2382, 81, 34, 0, 74, "Section"],
Cell[2419, 83, 3595, 113, 459, "Text"]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)