(* 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[     11540,        355]
NotebookOptionsPosition[     10326,        311]
NotebookOutlinePosition[     10731,        327]
CellTagsIndexPosition[     10688,        324]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["\<\
Math 481A
California State University, Northridge
Picard Iteration\
\>", "Section",
 CellChangeTimes->{{3.404587942926263*^9, 3.404587944715847*^9}}],

Cell[CellGroupData[{

Cell["\[Copyright]2007 BE Shapiro", "Subsection",
 CellChangeTimes->{{3.404587947250783*^9, 3.404587948779574*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{
    RowBox[{"Picard", "[", 
     RowBox[{"f_", ",", "t_", ",", "t0_", ",", "y0_", ",", "n_"}], "]"}], ":=",
     
    RowBox[{"Module", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"i", ",", 
        RowBox[{"y", "=", "y0"}]}], "}"}], ",", " ", "\[IndentingNewLine]", 
      " ", 
      RowBox[{
       RowBox[{"Print", "[", 
        RowBox[{
         RowBox[{"Subscript", "[", 
          RowBox[{"\"\<\[CurlyPhi]\>\"", ",", "0"}], "]"}], ",", "\"\<=\>\"", 
         ",", "y0"}], "]"}], ";", "\[IndentingNewLine]", 
       "\[IndentingNewLine]", 
       RowBox[{"For", "[", 
        RowBox[{
         RowBox[{"i", "=", "0"}], ",", 
         RowBox[{"i", "<", "n"}], ",", 
         RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"ynext", "=", 
           RowBox[{"y0", "+", 
            RowBox[{
             SubsuperscriptBox["\[Integral]", "t0", "t"], 
             RowBox[{
              RowBox[{"(", 
               RowBox[{"f", "[", 
                RowBox[{"s", ",", 
                 RowBox[{"y", "/.", 
                  RowBox[{"{", 
                   RowBox[{"t", "\[Rule]", " ", "s"}], "}"}]}]}], "]"}], 
               ")"}], 
              RowBox[{"\[DifferentialD]", "s"}]}]}]}]}], ";", 
          "\[IndentingNewLine]", 
          RowBox[{"Print", "[", 
           RowBox[{
            RowBox[{"Subscript", "[", 
             RowBox[{"\"\<\[CurlyPhi]\>\"", ",", 
              RowBox[{"i", "+", "1"}]}], "]"}], ",", "\"\<=\>\"", ",", 
            "ynext"}], "]"}], ";", "\[IndentingNewLine]", 
          RowBox[{"If", "[", 
           RowBox[{
            RowBox[{"SameQ", "[", 
             RowBox[{"y", ",", " ", "ynext"}], "]"}], ",", " ", 
            "\[IndentingNewLine]", 
            RowBox[{
             RowBox[{"Print", "[", "\"\<Algorithm has converged.\>\"", "]"}], 
             ";", " ", "\[IndentingNewLine]", 
             RowBox[{"Return", "[", "y", "]"}]}]}], "\[IndentingNewLine]", 
           "]"}], ";", " ", "\[IndentingNewLine]", 
          RowBox[{"y", "=", "ynext"}], ";"}]}], "\[IndentingNewLine]", "]"}], 
       ";", "\n", 
       RowBox[{"Return", "[", "y", "]"}]}]}], "\n", "]"}]}], ";"}], 
  "\n"}]], "Input",
 CellChangeTimes->{{3.404569683236595*^9, 3.404569769337825*^9}, 
   3.4045698987434998`*^9, {3.40457002011335*^9, 3.4045701248364077`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"t_", ",", "y_"}], "]"}], ":=", " ", "y"}]], "Input",
 CellChangeTimes->{{3.404569771348445*^9, 3.404569812966927*^9}, {
  3.404569849890006*^9, 3.404569865933441*^9}, {3.4045699833322287`*^9, 
  3.4045699920561123`*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Picard", "[", 
  RowBox[{"f", ",", " ", "t", ",", " ", "0", ",", " ", "1", ",", " ", "5"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.404569785207412*^9, 3.404569790386941*^9}, {
  3.4045698815940857`*^9, 3.404569887120977*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "0"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", "1"}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 0], "=", 1],
  Editable->False]], "Print",
 CellChangeTimes->{{3.4045697909969797`*^9, 3.404569816365814*^9}, {
   3.404569869064645*^9, 3.404569887746338*^9}, 3.404569994688085*^9, {
   3.40457005817824*^9, 3.404570090018292*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "1"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", 
   RowBox[{"1", "+", "t"}]}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 1], "=", 1 + $CellContext`t],
  Editable->False]], "Print",
 CellChangeTimes->{{3.4045697909969797`*^9, 3.404569816365814*^9}, {
   3.404569869064645*^9, 3.404569887746338*^9}, 3.404569994688085*^9, {
   3.40457005817824*^9, 3.4045700900217323`*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "2"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", 
   RowBox[{"1", "+", "t", "+", 
    FractionBox[
     SuperscriptBox["t", "2"], "2"]}]}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 2], "=", 1 + $CellContext`t + 
   Rational[1, 2] $CellContext`t^2],
  Editable->False]], "Print",
 CellChangeTimes->{{3.4045697909969797`*^9, 3.404569816365814*^9}, {
   3.404569869064645*^9, 3.404569887746338*^9}, 3.404569994688085*^9, {
   3.40457005817824*^9, 3.404570090067202*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "3"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", 
   RowBox[{"1", "+", "t", "+", 
    FractionBox[
     SuperscriptBox["t", "2"], "2"], "+", 
    FractionBox[
     SuperscriptBox["t", "3"], "6"]}]}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 3], "=", 1 + $CellContext`t + 
   Rational[1, 2] $CellContext`t^2 + Rational[1, 6] $CellContext`t^3],
  Editable->False]], "Print",
 CellChangeTimes->{{3.4045697909969797`*^9, 3.404569816365814*^9}, {
   3.404569869064645*^9, 3.404569887746338*^9}, 3.404569994688085*^9, {
   3.40457005817824*^9, 3.404570090070528*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "4"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", 
   RowBox[{"1", "+", "t", "+", 
    FractionBox[
     SuperscriptBox["t", "2"], "2"], "+", 
    FractionBox[
     SuperscriptBox["t", "3"], "6"], "+", 
    FractionBox[
     SuperscriptBox["t", "4"], "24"]}]}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 4], "=", 1 + $CellContext`t + 
   Rational[1, 2] $CellContext`t^2 + Rational[1, 6] $CellContext`t^3 + 
   Rational[1, 24] $CellContext`t^4],
  Editable->False]], "Print",
 CellChangeTimes->{{3.4045697909969797`*^9, 3.404569816365814*^9}, {
   3.404569869064645*^9, 3.404569887746338*^9}, 3.404569994688085*^9, {
   3.40457005817824*^9, 3.4045700901059303`*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "5"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", 
   RowBox[{"1", "+", "t", "+", 
    FractionBox[
     SuperscriptBox["t", "2"], "2"], "+", 
    FractionBox[
     SuperscriptBox["t", "3"], "6"], "+", 
    FractionBox[
     SuperscriptBox["t", "4"], "24"], "+", 
    FractionBox[
     SuperscriptBox["t", "5"], "120"]}]}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 5], "=", 1 + $CellContext`t + 
   Rational[1, 2] $CellContext`t^2 + Rational[1, 6] $CellContext`t^3 + 
   Rational[1, 24] $CellContext`t^4 + Rational[1, 120] $CellContext`t^5],
  Editable->False]], "Print",
 CellChangeTimes->{{3.4045697909969797`*^9, 3.404569816365814*^9}, {
   3.404569869064645*^9, 3.404569887746338*^9}, 3.404569994688085*^9, {
   3.40457005817824*^9, 3.4045700901571207`*^9}}]
}, Open  ]],

Cell[BoxData[
 RowBox[{"1", "+", "t", "+", 
  FractionBox[
   SuperscriptBox["t", "2"], "2"], "+", 
  FractionBox[
   SuperscriptBox["t", "3"], "6"], "+", 
  FractionBox[
   SuperscriptBox["t", "4"], "24"], "+", 
  FractionBox[
   SuperscriptBox["t", "5"], "120"]}]], "Output",
 CellChangeTimes->{{3.404569797283326*^9, 3.404569816466655*^9}, {
   3.4045698691501637`*^9, 3.404569887832151*^9}, 3.404569994876327*^9, {
   3.4045700582120447`*^9, 3.404570090160265*^9}}]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"t_", ",", "y_"}], "]"}], ":=", "t"}]], "Input",
 CellChangeTimes->{{3.4045699966713133`*^9, 3.404570001249915*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Picard", "[", 
  RowBox[{"f", ",", " ", "t", ",", " ", "0", ",", " ", "1", ",", " ", "5"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.4045700027118263`*^9, 3.404570005585664*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "0"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", "1"}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 0], "=", 1],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.404570005997134*^9, {3.404570038299335*^9, 3.404570054421179*^9}, 
   3.404570094996086*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "1"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", 
   RowBox[{"1", "+", 
    FractionBox[
     SuperscriptBox["t", "2"], "2"]}]}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 1], "=", 1 + Rational[1, 2] $CellContext`t^2],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.404570005997134*^9, {3.404570038299335*^9, 3.404570054421179*^9}, 
   3.4045700949981833`*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   SubscriptBox["\<\"\[CurlyPhi]\"\>", "2"], "\[InvisibleSpace]", "\<\"=\"\>",
    "\[InvisibleSpace]", 
   RowBox[{"1", "+", 
    FractionBox[
     SuperscriptBox["t", "2"], "2"]}]}],
  SequenceForm[
   Subscript["\[CurlyPhi]", 2], "=", 1 + Rational[1, 2] $CellContext`t^2],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.404570005997134*^9, {3.404570038299335*^9, 3.404570054421179*^9}, 
   3.4045700950492764`*^9}],

Cell[BoxData["\<\"Algorithm has converged.\"\>"], "Print",
 CellChangeTimes->{
  3.404570005997134*^9, {3.404570038299335*^9, 3.404570054421179*^9}, 
   3.4045700950994883`*^9}]
}, Open  ]],

Cell[BoxData[
 RowBox[{"1", "+", 
  FractionBox[
   SuperscriptBox["t", "2"], "2"]}]], "Output",
 CellChangeTimes->{
  3.40457000620833*^9, {3.4045700387951527`*^9, 3.404570054426622*^9}, 
   3.40457009514944*^9}]
}, Open  ]]
}, Open  ]]
}, Open  ]]
},
WindowSize->{640, 750},
WindowMargins->{{4, Automatic}, {Automatic, 4}},
FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (June 19, 2007)",
StyleDefinitions->FrontEnd`FileName[{"Creative"}, "NaturalColor.nb", 
  CharacterEncoding -> "UTF-8"]
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[590, 23, 159, 5, 112, "Section"],
Cell[CellGroupData[{
Cell[774, 32, 115, 1, 34, "Subsection"],
Cell[892, 35, 2393, 61, 314, "Input"],
Cell[3288, 98, 277, 6, 37, "Input"],
Cell[CellGroupData[{
Cell[3590, 108, 252, 5, 37, "Input"],
Cell[CellGroupData[{
Cell[3867, 117, 431, 10, 22, "Print"],
Cell[4301, 129, 474, 11, 20, "Print"],
Cell[4778, 142, 570, 14, 40, "Print"],
Cell[5351, 158, 664, 16, 41, "Print"],
Cell[6018, 176, 766, 19, 41, "Print"],
Cell[6787, 197, 864, 21, 41, "Print"]
}, Open  ]],
Cell[7666, 221, 469, 12, 58, "Output"]
}, Open  ]],
Cell[8150, 236, 172, 4, 37, "Input"],
Cell[CellGroupData[{
Cell[8347, 244, 203, 4, 37, "Input"],
Cell[CellGroupData[{
Cell[8575, 252, 359, 10, 22, "Print"],
Cell[8937, 264, 469, 13, 40, "Print"],
Cell[9409, 279, 469, 13, 40, "Print"],
Cell[9881, 294, 177, 3, 20, "Print"]
}, Open  ]],
Cell[10073, 300, 213, 6, 57, "Output"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)