(* 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[ 25682, 723] NotebookOptionsPosition[ 24081, 664] NotebookOutlinePosition[ 24535, 682] CellTagsIndexPosition[ 24492, 679] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Math 481A California State University, Northridge Newton's Interpolation Method: Forward Differences\ \>", "Section"], Cell["\[Copyright]2005 BE Shapiro", "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell["NewtonInterpolate", "Section"], Cell["\<\ Input: List of points of the form {{x0,f0}, {x1, f1}, ....} and a point x Output: Interpolated value of x to the highest order possible, given the data To print out the interpolation table, use NewtonInterpolationTable, which is \ otherwise identical to this function but has lots of printing.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"NewtonInterpolate", "[", RowBox[{"points_", ",", " ", "X_"}], "]"}], ":=", " ", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "n", ",", " ", "F", ",", " ", "i", ",", " ", "j", ",", "\[CapitalDelta]F", ",", " ", "P", ",", "t", ",", "h", ",", " ", "x"}], "}"}], ",", "\[IndentingNewLine]", " ", "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ "points", " ", "should", " ", "have", " ", "the", " ", "form", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x0", ",", " ", RowBox[{"f", "[", "x0", "]"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x1", ",", " ", RowBox[{"f", "[", "x1", "]"}]}], "}"}], ",", " ", "..."}], "}"}]}], "*)"}], CellFrame->True, Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", CellFrame->True, Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", CellFrame->True, Background->RGBColor[1, 1, 0]], StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{"NOTE", ":", " ", RowBox[{ "if", " ", "the", " ", "xi", " ", "are", " ", "not", " ", "equally", " ", "spaced"}]}], ",", " ", RowBox[{ RowBox[{ "this", " ", "will", " ", "not", " ", "calculate", " ", "the", " ", "correct", " ", "answer"}], " ", "-", " ", RowBox[{ "ERROR", " ", "CHECKING", " ", "IS", " ", "NOT", " ", "PERFORMED"}]}]}], " ", "*)"}], CellFrame->True, FontSize->18, Background->RGBColor[1, 1, 0]], StyleBox[" ", CellFrame->True, FontSize->18, Background->RGBColor[1, 1, 0]], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", RowBox[{"Length", "[", "points", "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"x", ",", " ", "F"}], "}"}], " ", "=", " ", RowBox[{"Transpose", "[", "points", "]"}]}], ";", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{"The", " ", "variables", " ", "ddtable"}], ",", " ", RowBox[{ RowBox[{ "nextline", " ", "are", " ", "purely", " ", "for", " ", "show"}], " ", "-", " ", RowBox[{ "it", " ", "is", " ", "used", " ", "to", " ", "print", " ", "the", " ", "table"}]}], ",", " ", RowBox[{ "but", " ", "is", " ", "not", " ", "otherwise", " ", "referenced"}]}], " ", "*)"}], Background->RGBColor[1, 1, 0]], StyleBox[" ", Background->GrayLevel[0.666667]], StyleBox["\[IndentingNewLine]", Background->GrayLevel[0.666667]], "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"0", ",", " ", "i"}], "]"}], " ", "represents", " ", SubscriptBox["f", "i"]}], " ", "=", " ", RowBox[{"f", RowBox[{"(", SubscriptBox["x", "i"], ")"}]}]}], "*)"}], Background->RGBColor[1, 1, 0]], StyleBox[" ", Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", " ", RowBox[{"i", "\[LessEqual]", " ", "n"}], ",", " ", RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"0", ",", "i"}], "]"}], "=", RowBox[{"F", "[", RowBox[{"[", "i", "]"}], "]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"i", ",", " ", "j"}], "]"}], " ", "represents", " ", SuperscriptBox["\[CapitalDelta]", "i"], RowBox[{"(", SubscriptBox["x", "j"], ")"}]}], "*)"}], Background->RGBColor[1, 1, 0]], StyleBox[" ", Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", " ", RowBox[{"i", "\[LessEqual]", " ", RowBox[{"n", "-", "1"}]}], ",", RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"For", "[", RowBox[{ RowBox[{"j", "=", "1"}], ",", " ", RowBox[{"j", "\[LessEqual]", " ", RowBox[{"n", "-", "i"}]}], ",", " ", RowBox[{"j", "++"}], ",", "\[IndentingNewLine]", " ", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"i", ",", "j"}], "]"}], "=", " ", RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{ RowBox[{"i", "-", "1"}], ",", " ", RowBox[{"j", "+", "1"}]}], "]"}], "-", RowBox[{"\[CapitalDelta]F", "[", RowBox[{ RowBox[{"i", "-", "1"}], ",", "j"}], "]"}]}]}], ";"}]}], "\[IndentingNewLine]", "]"}], ";"}]}], " ", "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Mathematica", " ", "indices", " ", "start", " ", "at", " ", "1"}], ",", " ", RowBox[{"not", " ", "zero"}]}], " ", "*)"}], Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], RowBox[{"h", "=", RowBox[{ RowBox[{"x", "[", RowBox[{"[", "2", "]"}], "]"}], "-", RowBox[{"x", "[", RowBox[{"[", "1", "]"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"t", "=", RowBox[{ RowBox[{"(", RowBox[{"X", "-", RowBox[{"x", "[", RowBox[{"[", "1", "]"}], "]"}]}], ")"}], "/", "h"}]}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"P", "[", RowBox[{"u_", ",", "j_"}], "]"}], ":=", RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"0", ",", "1"}], "]"}], "+", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "j"], RowBox[{ RowBox[{"Binomial", "[", RowBox[{"u", ",", "i"}], "]"}], RowBox[{"\[CapitalDelta]F", "[", RowBox[{"i", ",", "1"}], "]"}]}]}]}]}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"P", "[", RowBox[{"t", ",", RowBox[{"n", "-", "1"}]}], "]"}], "]"}], ";"}]}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", "]"}]}], ";"}], " "}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["NewtonInterpolationTable", "Section"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"NewtonInterpolationTable", "[", RowBox[{"points_", ",", " ", "X_"}], "]"}], ":=", " ", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "n", ",", " ", "F", ",", " ", "i", ",", " ", "j", ",", "\[CapitalDelta]F", ",", " ", "P", ",", "t", ",", "h", ",", " ", "x", ",", " ", "ddtable", ",", " ", "nextline"}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ "points", " ", "should", " ", "have", " ", "the", " ", "form", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x0", ",", " ", RowBox[{"f", "[", "x0", "]"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x1", ",", " ", RowBox[{"f", "[", "x1", "]"}]}], "}"}], ",", " ", "..."}], "}"}]}], "*)"}], Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", RowBox[{"Length", "[", "points", "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"x", ",", " ", "F"}], "}"}], " ", "=", " ", RowBox[{"Transpose", "[", "points", "]"}]}], ";", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{"The", " ", "variables", " ", "ddtable"}], ",", " ", RowBox[{ RowBox[{ "nextline", " ", "are", " ", "purely", " ", "for", " ", "show"}], " ", "-", " ", RowBox[{ "it", " ", "is", " ", "used", " ", "to", " ", "print", " ", "the", " ", "table"}]}], ",", " ", RowBox[{ "but", " ", "is", " ", "not", " ", "otherwise", " ", "referenced"}]}], " ", "*)"}], Background->RGBColor[1, 1, 0]], StyleBox[" ", Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], RowBox[{"ddtable", " ", "=", " ", RowBox[{"{", RowBox[{"x", ",", " ", "F"}], "}"}]}], ";", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"0", ",", " ", "i"}], "]"}], " ", "represents", " ", SubscriptBox["f", "i"]}], " ", "=", " ", RowBox[{"f", RowBox[{"(", SubscriptBox["x", "i"], ")"}]}]}], "*)"}], Background->RGBColor[1, 1, 0]], StyleBox[" ", Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", " ", RowBox[{"i", "\[LessEqual]", " ", "n"}], ",", " ", RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"0", ",", "i"}], "]"}], "=", RowBox[{"F", "[", RowBox[{"[", "i", "]"}], "]"}]}], ";"}]}], " ", "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"i", ",", " ", "j"}], "]"}], " ", "represents", " ", SuperscriptBox["\[CapitalDelta]", "i"], RowBox[{"(", SubscriptBox["x", "j"], ")"}]}], "*)"}], Background->RGBColor[1, 1, 0]], StyleBox[" ", Background->RGBColor[1, 1, 0]], "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", " ", RowBox[{"i", "\[LessEqual]", " ", RowBox[{"n", "-", "1"}]}], ",", RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"nextline", "=", RowBox[{"{", "}"}]}], ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"j", "=", "1"}], ",", " ", RowBox[{"j", "\[LessEqual]", " ", RowBox[{"n", "-", "i"}]}], ",", " ", RowBox[{"j", "++"}], ",", "\[IndentingNewLine]", " ", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"i", ",", "j"}], "]"}], "=", " ", RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{ RowBox[{"i", "-", "1"}], ",", " ", RowBox[{"j", "+", "1"}]}], "]"}], "-", RowBox[{"\[CapitalDelta]F", "[", RowBox[{ RowBox[{"i", "-", "1"}], ",", "j"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"nextline", " ", "=", " ", RowBox[{"Append", "[", RowBox[{"nextline", ",", " ", RowBox[{"\[CapitalDelta]F", "[", RowBox[{"i", ",", "j"}], "]"}]}], "]"}]}], ";"}]}], " ", "\[IndentingNewLine]", "]"}], ";", " ", "\[IndentingNewLine]", RowBox[{"ddtable", " ", "=", " ", RowBox[{"Append", "[", RowBox[{"ddtable", ",", " ", RowBox[{"PadRight", "[", RowBox[{"nextline", ",", "n", ",", "\"\< \>\""}], "]"}]}], "]"}]}], ";"}]}], " ", "\[IndentingNewLine]", "]"}], ";", StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{ "format", " ", "the", " ", "table", " ", "for", " ", "printing"}], " ", "-", " ", RowBox[{ "this", " ", "part", " ", "is", " ", "not", " ", "needed", " ", "for", " ", "the", " ", "algorithm"}]}], " ", "*)"}], Background->RGBColor[1, 1, 0]], "\[IndentingNewLine]", RowBox[{"ddtable", " ", "=", " ", RowBox[{"Transpose", "[", "ddtable", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"nextline", " ", "=", RowBox[{"Join", "[", RowBox[{ RowBox[{"{", RowBox[{ "\"\<\!\(\*SubscriptBox[\(x\), \(k\)]\)\>\"", ",", "\"\\""}], "}"}], ",", " ", RowBox[{"\"\<\[CapitalDelta]\>\"", "^", RowBox[{"ToString", "/@", RowBox[{"Range", "[", RowBox[{"1", ",", RowBox[{"n", "-", "1"}]}], "]"}]}]}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"ddtable", " ", "=", " ", RowBox[{"Prepend", "[", RowBox[{"ddtable", ",", " ", "nextline"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"ddtable", "//", "TableForm"}], "]"}], ";", StyleBox["\[IndentingNewLine]", Background->RGBColor[1, 1, 0]], "\[IndentingNewLine]", StyleBox[ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Mathematica", " ", "indices", " ", "start", " ", "at", " ", "1"}], ",", " ", RowBox[{"not", " ", "zero"}]}], " ", "*)"}], Background->RGBColor[1, 1, 0]], "\[IndentingNewLine]", RowBox[{"h", "=", RowBox[{ RowBox[{"x", "[", RowBox[{"[", "2", "]"}], "]"}], "-", RowBox[{"x", "[", RowBox[{"[", "1", "]"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"t", "=", RowBox[{ RowBox[{"(", RowBox[{"X", "-", RowBox[{"x", "[", RowBox[{"[", "1", "]"}], "]"}]}], ")"}], "/", "h"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", "h", ",", "\"\< t=\>\"", ",", "t"}], "]"}], ";", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"P", "[", RowBox[{"u_", ",", "j_"}], "]"}], ":=", RowBox[{ RowBox[{"\[CapitalDelta]F", "[", RowBox[{"0", ",", "1"}], "]"}], "+", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "j"], RowBox[{ RowBox[{"Binomial", "[", RowBox[{"u", ",", "i"}], "]"}], RowBox[{"\[CapitalDelta]F", "[", RowBox[{"i", ",", "1"}], "]"}]}]}]}]}], ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", " ", RowBox[{"i", "\[LessEqual]", " ", RowBox[{"n", "-", "1"}]}], ",", RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", "X", ",", "\"\<] to \>\"", ",", "i", ",", RowBox[{"Switch", "[", RowBox[{ "i", ",", "1", ",", "\"\\"", ",", "2", ",", "\"\\"", ",", "3", ",", "\"\\"", ",", "_", ",", " ", "\"\\""}], "]"}], ",", "\"\< order = \>\"", ",", " ", RowBox[{"P", "[", RowBox[{"t", ",", "i"}], "]"}]}], "]"}], ";"}]}], "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"P", "[", RowBox[{"t", ",", RowBox[{"n", "-", "1"}]}], "]"}], "]"}], ";"}]}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", "]"}]}], ";"}], " "}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Examples", "Section"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"xvalues", " ", "=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"1.0", "+", RowBox[{"k", "*", "0.5"}]}], ",", " ", RowBox[{"{", RowBox[{"k", ",", "0", ",", "4"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"1.`", ",", "1.5`", ",", "2.`", ",", "2.5`", ",", "3.`"}], "}"}]], "Output", CellChangeTimes->{3.398601615380411*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"yvalues", " ", "=", " ", RowBox[{"E", "^", "xvalues"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ "2.718281828459045`", ",", "4.4816890703380645`", ",", "7.38905609893065`", ",", "12.182493960703473`", ",", "20.085536923187668`"}], "}"}]], "Output",\ CellChangeTimes->{3.398601618558092*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"points", "=", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{"xvalues", ",", "yvalues"}], "}"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1.`", ",", "2.718281828459045`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.5`", ",", "4.4816890703380645`"}], "}"}], ",", RowBox[{"{", RowBox[{"2.`", ",", "7.38905609893065`"}], "}"}], ",", RowBox[{"{", RowBox[{"2.5`", ",", "12.182493960703473`"}], "}"}], ",", RowBox[{"{", RowBox[{"3.`", ",", "20.085536923187668`"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.398601620917406*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NewtonInterpolate", "[", RowBox[{"points", ",", "1.2"}], "]"}]], "Input"], Cell[BoxData["3.31383745178261`"], "Output", CellChangeTimes->{3.398601623259399*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NewtonInterpolationTable", "[", RowBox[{"points", ",", "1.2"}], "]"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ TagBox[GridBox[{ {"\<\"\\!\\(\\*SubscriptBox[\\(x\\), \\(k\\)]\\)\"\>", \ "\<\"f(\\!\\(\\*SubscriptBox[\\(x\\), \\(k\\)]\\))\"\>", SuperscriptBox["\<\"\[CapitalDelta]\"\>", "\<\"1\"\>"], SuperscriptBox["\<\"\[CapitalDelta]\"\>", "\<\"2\"\>"], SuperscriptBox["\<\"\[CapitalDelta]\"\>", "\<\"3\"\>"], SuperscriptBox["\<\"\[CapitalDelta]\"\>", "\<\"4\"\>"]}, {"1.`", "2.718281828459045`", "1.7634072418790194`", "1.1439597867135665`", "0.7421110464666705`", "0.48142322106446445`"}, {"1.5`", "4.4816890703380645`", "2.907367028592586`", "1.886070833180237`", "1.223534267531135`", "\<\" \"\>"}, {"2.`", "7.38905609893065`", "4.793437861772823`", "3.109605100711372`", "\<\" \"\>", "\<\" \"\>"}, {"2.5`", "12.182493960703473`", "7.903042962484195`", "\<\" \"\>", "\<\" \"\>", "\<\" \"\>"}, {"3.`", "20.085536923187668`", "\<\" \"\>", "\<\" \"\>", "\<\" \"\>", "\<\" \ \"\>"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Function[BoxForm`e$, TableForm[BoxForm`e$]]]], "Print", CellChangeTimes->{3.398601625629938*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"h=\"\>", "\[InvisibleSpace]", "0.5`", "\[InvisibleSpace]", "\<\" t=\"\>", "\[InvisibleSpace]", "0.3999999999999999`"}], SequenceForm["h=", 0.5, " t=", 0.3999999999999999], Editable->False]], "Print", CellChangeTimes->{3.398601625633524*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"P[\"\>", "\[InvisibleSpace]", "1.2`", "\[InvisibleSpace]", "\<\"] to \"\>", "\[InvisibleSpace]", "1", "\[InvisibleSpace]", "\<\"st\"\>", "\[InvisibleSpace]", "\<\" order = \"\>", "\[InvisibleSpace]", "3.4236447252106528`"}], SequenceForm["P[", 1.2, "] to ", 1, "st", " order = ", 3.4236447252106528`], Editable->False]], "Print", CellChangeTimes->{3.398601625637177*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"P[\"\>", "\[InvisibleSpace]", "1.2`", "\[InvisibleSpace]", "\<\"] to \"\>", "\[InvisibleSpace]", "2", "\[InvisibleSpace]", "\<\"d\"\>", "\[InvisibleSpace]", "\<\" order = \"\>", "\[InvisibleSpace]", "3.286369550805025`"}], SequenceForm["P[", 1.2, "] to ", 2, "d", " order = ", 3.286369550805025], Editable->False]], "Print", CellChangeTimes->{3.398601625638682*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"P[\"\>", "\[InvisibleSpace]", "1.2`", "\[InvisibleSpace]", "\<\"] to \"\>", "\[InvisibleSpace]", "3", "\[InvisibleSpace]", "\<\"rd\"\>", "\[InvisibleSpace]", "\<\" order = \"\>", "\[InvisibleSpace]", "3.3338646577788915`"}], SequenceForm["P[", 1.2, "] to ", 3, "rd", " order = ", 3.3338646577788915`], Editable->False]], "Print", CellChangeTimes->{3.398601625640085*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"P[\"\>", "\[InvisibleSpace]", "1.2`", "\[InvisibleSpace]", "\<\"] to \"\>", "\[InvisibleSpace]", "4", "\[InvisibleSpace]", "\<\"th\"\>", "\[InvisibleSpace]", "\<\" order = \"\>", "\[InvisibleSpace]", "3.31383745178261`"}], SequenceForm["P[", 1.2, "] to ", 4, "th", " order = ", 3.31383745178261], Editable->False]], "Print", CellChangeTimes->{3.398601625641438*^9}] }, Open ]], Cell[BoxData["3.31383745178261`"], "Output", CellChangeTimes->{3.3986016256425877`*^9}] }, Open ]] }, Open ]] }, WindowToolbars->"RulerBar", WindowSize->{686, 1037}, WindowMargins->{{Automatic, 490}, {Automatic, 11}}, Magnification->1, FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (June 19, 2007)", StyleDefinitions->FrontEnd`FileName[{"Creative"}, "PastelColor.nb", CharacterEncoding -> "UTF-8"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 127, 4, 112, "Section"], Cell[720, 29, 49, 0, 34, "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell[806, 34, 36, 0, 72, "Section"], Cell[845, 36, 318, 6, 86, "Text"], Cell[1166, 44, 7385, 195, 629, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[8588, 244, 43, 0, 72, "Section"], Cell[8634, 246, 9843, 242, 851, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[18514, 493, 27, 0, 72, "Section"], Cell[CellGroupData[{ Cell[18566, 497, 242, 7, 37, "Input"], Cell[18811, 506, 160, 4, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19008, 515, 94, 2, 37, "Input"], Cell[19105, 519, 241, 6, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19383, 530, 151, 4, 37, "Input"], Cell[19537, 536, 489, 13, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20063, 554, 100, 2, 37, "Input"], Cell[20166, 558, 86, 1, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20289, 564, 107, 2, 37, "Input"], Cell[CellGroupData[{ Cell[20421, 570, 1460, 32, 105, "Print"], Cell[21884, 604, 302, 7, 20, "Print"], Cell[22189, 613, 443, 10, 20, "Print"], Cell[22635, 625, 431, 8, 20, "Print"], Cell[23069, 635, 443, 10, 20, "Print"], Cell[23515, 647, 435, 9, 20, "Print"] }, Open ]], Cell[23965, 659, 88, 1, 37, "Output"] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)