(* 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[ 50075, 2365] NotebookOptionsPosition[ 46666, 2248] NotebookOutlinePosition[ 47150, 2270] CellTagsIndexPosition[ 47077, 2265] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["A Review of the FunctionApproximations package", "Title", CellChangeTimes->{{3.393071518003044*^9, 3.393071519372959*^9}}], Cell[TextData[{ "Ken Levasseur\nMathematical Sciences\nUMass Lowell\n", ButtonBox["Kenneth_Levasseur@uml.edu", BaseStyle->"Hyperlink", ButtonData:>{ URL["mailto:Kenneth_Levasseur@uml.edu"], None}] }], "Subsubtitle", FontSize->12, CellTags->"myaddress"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ The Approximations package provides a variety of functions for approximating \ functions by rational functions.\ \>", "Text"], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input"], Cell[TextData[{ "Given a function ", StyleBox["f", FontSlant->"Italic"], " defined on an interval ", Cell[BoxData[ FormBox[ RowBox[{"[", RowBox[{"a", ",", "b"}], "]"}], TraditionalForm]]], ", the basic problem this addresses is that of finding a rational function \ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"r", "(", "x", ")"}], " ", "=", " ", FractionBox[ RowBox[{"p", "(", "x", ")"}], RowBox[{"q", "(", "x", ")"}]]}], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox[ RowBox[{"p", "(", "x", ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"q", "(", "x", ")"}], TraditionalForm]]], " are polynomials with some specified maximal degrees, such that ", StyleBox["f", FontSlant->"Italic"], " and ", StyleBox["r", FontSlant->"Italic"], " are as close as possible in some sense. " }], "Text"], Cell[TextData[{ "In this review, we will approximate this function. If you are reading a ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook, you can change the function to one you might prefer." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", SuperscriptBox["E", RowBox[{"-", " ", "x"}]]}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Minimization of Relative Error", "Section"], Cell["\<\ Here is how to approximate our function in the relative minimax sense. First \ we approximate with polynomials of degree 2 or less.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"r20", "=", RowBox[{"MiniMaxApproximation", "[", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", " ", "2", ",", "0"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ "0.`", ",", "0.3008801244668504`", ",", "0.7927866634890158`", ",", "1.`"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"0.2985571780240445`", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"0.9236364260034658`", " ", "x"}], "+", "0.9948523892989384`"}], ",", "0.005147610701061615`"}], "}"}]}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The output has several parts. The first part is the list of locations of \ relative maxima in the error.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"r20", "[", RowBox[{"[", "1", "]"}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ "0.`", ",", "0.3008801244668504`", ",", "0.7927866634890158`", ",", "1.`"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The second part of the output has two parts. The first is the approximating \ function itself.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"r20", "[", RowBox[{"[", RowBox[{"2", ",", "1"}], "]"}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"0.2985571780240445`", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"0.9236364260034658`", " ", "x"}], "+", "0.9948523892989384`"}], TraditionalForm]], "Output"] }, Open ]], Cell["The second part of the second part is the maximum error ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"r20", "[", RowBox[{"[", RowBox[{"2", ",", "2"}], "]"}], "]"}]], "Input"], Cell[BoxData[ FormBox["0.005147610701061615`", TraditionalForm]], "Output"] }, Open ]], Cell["Here is a plot of the error function", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "-", RowBox[{"r20", "[", RowBox[{"[", RowBox[{"2", ",", "1"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input"], 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