(* Content-type: application/vnd.wolfram.mathematica *)

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

(* CreatedBy='Mathematica 10.3' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       158,          7]
NotebookDataLength[     20735,        514]
NotebookOptionsPosition[     19551,        469]
NotebookOutlinePosition[     19888,        484]
CellTagsIndexPosition[     19845,        481]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell[BoxData[
 RowBox[{
  RowBox[{"runge", "[", "x_", "]"}], " ", "=", " ", 
  RowBox[{"1", " ", "/", " ", 
   RowBox[{"(", 
    RowBox[{"1", " ", "+", " ", 
     RowBox[{"25", " ", 
      RowBox[{"x", "^", "2"}]}]}], ")"}]}]}]], "Input",
 CellChangeTimes->{{3.6855202639388638`*^9, 3.6855202996308117`*^9}}],

Cell[BoxData[
 FractionBox["1", 
  RowBox[{"1", "+", 
   RowBox[{"25", " ", 
    SuperscriptBox["x", "2"]}]}]]], "Output",
 CellChangeTimes->{{3.6855202971715717`*^9, 3.685520299998887*^9}, 
   3.685529753291757*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"rungeData11", " ", "=", " ", 
  RowBox[{"Table", "[", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", ",", " ", 
      RowBox[{"runge", "[", "x", "]"}]}], "}"}], ",", "\[IndentingNewLine]", 
    RowBox[{"{", 
     RowBox[{"x", ",", " ", 
      RowBox[{"-", "1"}], ",", " ", 
      RowBox[{"+", "1"}], ",", " ", "0.2"}], "}"}]}], "\[IndentingNewLine]", 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.685520443605824*^9, 3.6855204506374273`*^9}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"-", "1.`"}], ",", "0.038461538461538464`"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"-", "0.8`"}], ",", "0.05882352941176469`"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"-", "0.6`"}], ",", "0.1`"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"-", "0.3999999999999999`"}], ",", "0.20000000000000007`"}], 
    "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"-", "0.19999999999999996`"}], ",", "0.5000000000000001`"}], 
    "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0.`", ",", "1.`"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0.20000000000000018`", ",", "0.49999999999999956`"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0.40000000000000013`", ",", "0.1999999999999999`"}], "}"}], ",", 
   
   RowBox[{"{", 
    RowBox[{"0.6000000000000001`", ",", "0.09999999999999998`"}], "}"}], ",", 
   
   RowBox[{"{", 
    RowBox[{"0.8`", ",", "0.05882352941176469`"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"1.`", ",", "0.038461538461538464`"}], "}"}]}], "}"}]], "Output",
 CellChangeTimes->{{3.685520443884881*^9, 3.6855204591362753`*^9}, 
   3.685529753470796*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"rungeApprox11", " ", "=", " ", 
  RowBox[{
   RowBox[{"InterpolatingPolynomial", "[", 
    RowBox[{"rungeData11", ",", " ", "x"}], "]"}], " ", "//", " ", 
   "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.6855204739097023`*^9, 3.6855204741090593`*^9}, {
  3.685529748047958*^9, 3.685529757823937*^9}}],

Cell[BoxData[
 RowBox[{"1.`", "\[VeryThinSpace]", "+", 
  RowBox[{"1.3322676295501878`*^-15", " ", "x"}], "-", 
  RowBox[{"16.855203619909503`", " ", 
   SuperscriptBox["x", "2"]}], "-", 
  RowBox[{"3.019806626980426`*^-14", " ", 
   SuperscriptBox["x", "3"]}], "+", 
  RowBox[{"123.35972850678728`", " ", 
   SuperscriptBox["x", "4"]}], "+", 
  RowBox[{"1.4210854715202004`*^-13", " ", 
   SuperscriptBox["x", "5"]}], "-", 
  RowBox[{"381.43382352941154`", " ", 
   SuperscriptBox["x", "6"]}], "-", 
  RowBox[{"2.2737367544323206`*^-13", " ", 
   SuperscriptBox["x", "7"]}], "+", 
  RowBox[{"494.9095022624431`", " ", 
   SuperscriptBox["x", "8"]}], "+", 
  RowBox[{"1.1368683772161603`*^-13", " ", 
   SuperscriptBox["x", "9"]}], "-", 
  RowBox[{"220.94174208144779`", " ", 
   SuperscriptBox["x", "10"]}]}]], "Output",
 CellChangeTimes->{3.685530508856052*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"rungePlot", " ", "=", " ", 
  RowBox[{"Show", "[", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{"Plot", "[", 
     RowBox[{
      RowBox[{"runge", "[", "x", "]"}], ",", " ", 
      RowBox[{"{", 
       RowBox[{"x", ",", " ", 
        RowBox[{"-", "1"}], ",", " ", 
        RowBox[{"+", "1"}]}], "}"}]}], "]"}], ",", "\[IndentingNewLine]", 
    RowBox[{"ListPlot", "[", 
     RowBox[{"rungeData11", ",", " ", 
      RowBox[{"PlotStyle", " ", "\[Rule]", " ", "Red"}]}], "]"}], ",", 
    "\[IndentingNewLine]", 
    RowBox[{"Plot", "[", 
     RowBox[{"rungeApprox11", ",", " ", 
      RowBox[{"{", 
       RowBox[{"x", ",", " ", 
        RowBox[{"-", "1"}], ",", " ", 
        RowBox[{"+", "1"}]}], "}"}], ",", " ", 
      RowBox[{"PlotStyle", " ", "\[Rule]", " ", "Red"}]}], "]"}]}], 
   "\[IndentingNewLine]", "]"}]}]], "Input",
 CellChangeTimes->{{3.6855203042714443`*^9, 3.685520320302579*^9}, {
  3.685520372318378*^9, 3.68552039602993*^9}, {3.685520460981647*^9, 
  3.6855204931812057`*^9}}],

Cell[BoxData[
 RowBox[{
  StyleBox[
   RowBox[{"InterpolatingPolynomial", "::", "list"}], "MessageName"], 
  RowBox[{
  ":", " "}], "\<\"List expected at position \[NoBreak]\\!\\(\\*RowBox[{\\\"1\
\\\"}]\\)\[NoBreak] in \[NoBreak]\\!\\(\\*RowBox[{\\\"InterpolatingPolynomial\
\\\", \\\"[\\\", RowBox[{\\\"rungeData\\\", \\\",\\\", RowBox[{\\\"-\\\", \
\\\"0.9999591428571428`\\\"}]}], \\\"]\\\"}]\\)\[NoBreak]. \
\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \
ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/General/list\\\", \
ButtonNote -> \\\"InterpolatingPolynomial::list\\\"]\\)\"\>"}]], "Message", \
"MSG",
 CellChangeTimes->{3.6855297537585983`*^9}],

Cell[BoxData[
 RowBox[{
  StyleBox[
   RowBox[{"InterpolatingPolynomial", "::", "list"}], "MessageName"], 
  RowBox[{
  ":", " "}], "\<\"List expected at position \[NoBreak]\\!\\(\\*RowBox[{\\\"1\
\\\"}]\\)\[NoBreak] in \[NoBreak]\\!\\(\\*RowBox[{\\\"InterpolatingPolynomial\
\\\", \\\"[\\\", RowBox[{\\\"rungeData\\\", \\\",\\\", RowBox[{\\\"-\\\", \
\\\"0.9591428163265305`\\\"}]}], \\\"]\\\"}]\\)\[NoBreak]. \
\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \
ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/General/list\\\", \
ButtonNote -> \\\"InterpolatingPolynomial::list\\\"]\\)\"\>"}]], "Message", \
"MSG",
 CellChangeTimes->{3.685529753818915*^9}],

Cell[BoxData[
 RowBox[{
  StyleBox[
   RowBox[{"InterpolatingPolynomial", "::", "list"}], "MessageName"], 
  RowBox[{
  ":", " "}], "\<\"List expected at position \[NoBreak]\\!\\(\\*RowBox[{\\\"1\
\\\"}]\\)\[NoBreak] in \[NoBreak]\\!\\(\\*RowBox[{\\\"InterpolatingPolynomial\
\\\", \\\"[\\\", RowBox[{\\\"rungeData\\\", \\\",\\\", RowBox[{\\\"-\\\", \
\\\"0.9183264897959182`\\\"}]}], \\\"]\\\"}]\\)\[NoBreak]. \
\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \
ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/General/list\\\", \
ButtonNote -> \\\"InterpolatingPolynomial::list\\\"]\\)\"\>"}]], "Message", \
"MSG",
 CellChangeTimes->{3.685529753874078*^9}],

Cell[BoxData[
 RowBox[{
  StyleBox[
   RowBox[{"General", "::", "stop"}], "MessageName"], 
  RowBox[{
  ":", " "}], "\<\"Further output of \
\[NoBreak]\\!\\(\\*StyleBox[RowBox[{\\\"InterpolatingPolynomial\\\", \\\"::\\\
\", \\\"list\\\"}], \\\"MessageName\\\"]\\)\[NoBreak] will be suppressed \
during this calculation. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \
ButtonStyle->\\\"Link\\\", ButtonFrame->None, \
ButtonData:>\\\"paclet:ref/message/General/stop\\\", ButtonNote -> \
\\\"General::stop\\\"]\\)\"\>"}]], "Message", "MSG",
 CellChangeTimes->{3.685529753881908*^9}],

Cell[BoxData[
 GraphicsBox[{{{}, {}, 
    {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[
     1.], LineBox[CompressedData["
1:eJw1mHk01F/YwJEkZUklW4WsKdrs8fySrCmJEipkrUQqImuWKFtIdmmGSEmy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      "]]}}, {{}, {{}, 
     {RGBColor[1, 0, 0], PointSize[0.012833333333333334`], AbsoluteThickness[
      1.6], 
      PointBox[{{-1., 0.038461538461538464`}, {-0.8, 
       0.05882352941176469}, {-0.6, 0.1}, {-0.3999999999999999, 
       0.20000000000000007`}, {-0.19999999999999996`, 0.5000000000000001}, {
       0., 1.}, {0.20000000000000018`, 0.49999999999999956`}, {
       0.40000000000000013`, 0.1999999999999999}, {0.6000000000000001, 
       0.09999999999999998}, {0.8, 0.05882352941176469}, {1., 
       0.038461538461538464`}}]}, {}}, {}}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->{True, True},
  AxesLabel->{None, None},
  AxesOrigin->{0, 0},
  DisplayFunction->Identity,
  Frame->{{False, False}, {False, False}},
  FrameLabel->{{None, None}, {None, None}},
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  Method->{
   "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "ScalingFunctions" -> None},
  PlotRange->{{-1, 1}, {0., 0.9999994079562567}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{
  3.6855203207892113`*^9, {3.685520384995545*^9, 3.685520396339651*^9}, 
   3.685520461663477*^9, 3.6855205001903887`*^9, 3.6855297538885403`*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"rungeInt", " ", "=", " ", 
  RowBox[{"Integrate", "[", 
   RowBox[{
    RowBox[{"runge", "[", "x", "]"}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"x", ",", " ", 
      RowBox[{"-", "1"}], ",", " ", 
      RowBox[{"+", "1"}]}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.6855204254572487`*^9, 3.685520430517417*^9}, {
  3.685520516108758*^9, 3.685520560228278*^9}}],

Cell[BoxData[
 FractionBox[
  RowBox[{"2", " ", 
   RowBox[{"ArcTan", "[", "5", "]"}]}], "5"]], "Output",
 CellChangeTimes->{{3.685520518937872*^9, 3.6855205284217854`*^9}, 
   3.685520568516642*^9, 3.685529754132867*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"rungeIntApprox11", " ", "=", " ", 
  RowBox[{"Integrate", "[", 
   RowBox[{"rungeApprox11", ",", " ", 
    RowBox[{"{", 
     RowBox[{"x", ",", " ", 
      RowBox[{"-", "1"}], ",", " ", 
      RowBox[{"+", "1"}]}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.685520529848077*^9, 3.685520566340179*^9}}],

Cell[BoxData[
 RowBox[{
  StyleBox[
   RowBox[{"InterpolatingPolynomial", "::", "list"}], "MessageName"], 
  RowBox[{
  ":", " "}], "\<\"List expected at position \[NoBreak]\\!\\(\\*RowBox[{\\\"1\
\\\"}]\\)\[NoBreak] in \[NoBreak]\\!\\(\\*RowBox[{\\\"InterpolatingPolynomial\
\\\", \\\"[\\\", RowBox[{\\\"rungeData\\\", \\\",\\\", \\\"x\\\"}], \
\\\"]\\\"}]\\)\[NoBreak]. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \
ButtonStyle->\\\"Link\\\", ButtonFrame->None, \
ButtonData:>\\\"paclet:ref/message/General/list\\\", ButtonNote -> \
\\\"InterpolatingPolynomial::list\\\"]\\)\"\>"}]], "Message", "MSG",
 CellChangeTimes->{3.68552975425229*^9}],

Cell[BoxData[
 RowBox[{
  SubsuperscriptBox["\[Integral]", 
   RowBox[{"-", "1"}], "1"], 
  RowBox[{
   RowBox[{"InterpolatingPolynomial", "[", 
    RowBox[{"rungeData", ",", "x"}], "]"}], 
   RowBox[{"\[DifferentialD]", "x"}]}]}]], "Output",
 CellChangeTimes->{3.685520537852358*^9, 3.685520569087985*^9, 
  3.685529754312867*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(", 
   RowBox[{"rungeIntApprox11", " ", "-", " ", "rungeInt"}], ")"}], " ", "/", 
  " ", "rungeInt"}]], "Input",
 CellChangeTimes->{{3.6855205445248613`*^9, 3.68552058114006*^9}}],

Cell[BoxData[
 FractionBox[
  RowBox[{"5", " ", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"-", 
      FractionBox[
       RowBox[{"2", " ", 
        RowBox[{"ArcTan", "[", "5", "]"}]}], "5"]}], "+", 
     RowBox[{
      SubsuperscriptBox["\[Integral]", 
       RowBox[{"-", "1"}], "1"], 
      RowBox[{
       RowBox[{"InterpolatingPolynomial", "[", 
        RowBox[{"rungeData", ",", "x"}], "]"}], 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}], ")"}]}], 
  RowBox[{"2", " ", 
   RowBox[{"ArcTan", "[", "5", "]"}]}]]], "Output",
 CellChangeTimes->{3.6855205817013617`*^9, 3.685529754579514*^9}]
}, Open  ]]
},
WindowSize->{1054, 1179},
WindowMargins->{{0, Automatic}, {Automatic, 19}},
FrontEndVersion->"10.3 for Linux x86 (64-bit) (October 9, 2015)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[580, 22, 308, 8, 32, "Input"],
Cell[891, 32, 216, 6, 51, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1144, 43, 490, 12, 99, "Input"],
Cell[1637, 57, 1203, 35, 55, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2877, 97, 326, 7, 32, "Input"],
Cell[3206, 106, 863, 21, 58, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[4106, 132, 1021, 25, 121, "Input"],
Cell[5130, 159, 689, 13, 23, "Message"],
Cell[5822, 174, 687, 13, 23, "Message"],
Cell[6512, 189, 687, 13, 23, "Message"],
Cell[7202, 204, 578, 12, 23, "Message"],
Cell[7783, 218, 8890, 158, 238, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[16710, 381, 397, 10, 32, "Input"],
Cell[17110, 393, 221, 5, 49, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[17368, 403, 332, 8, 32, "Input"],
Cell[17703, 413, 645, 12, 23, "Message"],
Cell[18351, 427, 331, 9, 49, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[18719, 441, 215, 5, 32, "Input"],
Cell[18937, 448, 598, 18, 63, "Output"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)