(* 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[ 455177, 10505] NotebookOptionsPosition[ 428741, 9658] NotebookOutlinePosition[ 432264, 9757] CellTagsIndexPosition[ 431661, 9740] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Funktionen und Auswertung", "Title"], Cell[CellGroupData[{ Cell["Einf\[UDoubleDot]hrung", "Section"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Set", "//", "Attributes"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"HoldFirst", ",", "Protected", ",", "SequenceHold"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SetDelayed", "//", "Attributes"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"HoldAll", ",", "Protected", ",", "SequenceHold"}], "}"}]], "Output"] }, Open ]], Cell["\<\ Ein und dieselbe Notation auf der rechten Seite kann f\[UDoubleDot]r \ unterschiedliche Objekte stehen. f1 ist ein Ausdruck mit zwei Symbolvariablen f2 ist eine Funktion mit formalem Parameter x, in der eine weitere \ Symbolvariable vorkommt. f3 ist eine zweistelllige Funktion. f4 ist eine Schar einstelliger Funktionen mit Scharparameter t und \ Funktionsparameter x.\ \>", "SmallText"], Cell[BoxData[{ RowBox[{"f1", ":=", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"t", " ", "x"}], "+", "1"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f2", "[", "x_", "]"}], ":=", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"t", " ", "x"}], "+", "1"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f3", "[", RowBox[{"x_", ",", "t_"}], "]"}], ":=", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"t", " ", "x"}], "+", "1"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"f4", "[", "t_", "]"}], "[", "x_", "]"}], ":=", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"t", " ", "x"}], "+", "1"}]}]}], "Input"], Cell["\<\ Das erste Listenelement ist der Ausdruck f1. Das zweite Listenelement ist der Funktionswert von f2 an der (symbolischen) \ Stelle x f\[UDoubleDot]r den speziellen Wert t\[Equal]3. Das dritte Listenelement ist der Funktionswert von f3 an der Stelle \ (x,t)=(2,x). Das erste Argument (f\[UDoubleDot]r x) ist eine Zahl, das zweite \ (f\[UDoubleDot]r t) das Symbol x. Das vierte Listenelement ist der Funktionswert der \"allgemeinen\" Funktion \ f4[t] der Schar (der Unterschied ist wie der zwischen f2 und f2[x] im zweiten \ Fall) an der Stelle x\[Equal]3.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{"f1", ",", RowBox[{ RowBox[{"f2", "[", "x", "]"}], "/.", RowBox[{"t", "\[Rule]", "3"}]}], ",", RowBox[{"f3", "[", RowBox[{"2", ",", "x"}], "]"}], ",", RowBox[{ RowBox[{"f4", "[", "t", "]"}], "[", "3", "]"}]}], "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"1", "+", RowBox[{"t", " ", "x"}], "+", SuperscriptBox["x", "2"]}], ",", RowBox[{"1", "+", RowBox[{"3", " ", "x"}], "+", SuperscriptBox["x", "2"]}], ",", RowBox[{"5", "+", RowBox[{"2", " ", "x"}]}], ",", RowBox[{"10", "+", RowBox[{"3", " ", "t"}]}]}], "}"}]], "Output"] }, Open ]], Cell["\<\ Und hier werden schlie\[SZ]lich f\[UDoubleDot]nf der Funktionen aus der Schar \ f4 grafisch dargestellt. Ganz genau: Dargestellt werden nicht die Funktionen \ f4[t], sondern die Ausdr\[UDoubleDot]cke f4[t][x].\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"u", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"f4", "[", "t", "]"}], "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "1", ",", "5"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"v", "=", RowBox[{"Plot", "[", RowBox[{"u", ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], " ", ",", RowBox[{"Ticks", "\[Rule]", RowBox[{"{", RowBox[{"Join", "[", RowBox[{ RowBox[{"Range", "[", RowBox[{ RowBox[{"-", "1"}], ",", "1", ",", ".5"}], "]"}], ",", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"i", ",", "\"\<\>\""}], "}"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "1"}], ",", "1", ",", ".1"}], "}"}]}], "]"}]}], "]"}], "}"}]}]}], "]"}]}]}], "Input"], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 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Hier soll nicht untersucht werden, ob dies f\[UDoubleDot]r \ alle (x,y) gilt, sondern es wird das ", StyleBox["Problem", FontSlant->"Italic"], " formuliert, die L\[ODoubleDot]sungspaare (x,y) zu finden. 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In den \ 1960er Jahren ist im Zusammenhang mit dem 10. 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Beachten Sie aber, dass nat\[UDoubleDot]rlich auch hier vor dem \ Vergleich die Argumente von ", StyleBox["SameQ", FontWeight->"Bold"], " ausgewertet werden." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"x", "+", "y"}], "===", RowBox[{"y", "+", "x"}]}]], "Input"], Cell[BoxData["True"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"x", "+", "x"}], "===", RowBox[{"2", "x"}]}]], "Input"], Cell[BoxData["True"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", "x"}], "===", RowBox[{"x", RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}]}]}]], "Input"], Cell[BoxData["False"], "Output"] }, Open ]], Cell[TextData[{ "=== (", StyleBox["SameQ", FontWeight->"Bold"], ") und \[Equal] (", StyleBox["Equal", FontWeight->"Bold"], ") reagieren in diesem Beispiel unterschiedlich. " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", "x"}], "===", RowBox[{"x", RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}]}]}], "//", "Expand"}]], "Input"], Cell[BoxData["False"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", "x"}], "==", RowBox[{"x", RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}]}]}], "//", "Expand"}]], "Input"], Cell[BoxData["True"], "Output"] }, Open ]], Cell["\<\ Auch in Steuerstrukturen kann es geschehen, dass Boolesche \ Ausdr\[UDoubleDot]cke nicht vollst\[ADoubleDot]ndig ausgewertet werden k\ \[ODoubleDot]nnen. Verzweigungen werden dann in symbolischer Form als Funktionsausdruck zur\ \[UDoubleDot]ckgegeben.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"u", "=", RowBox[{"If", "[", RowBox[{ RowBox[{"x", "\[Equal]", "y"}], ",", "1", ",", "2"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"If", "[", RowBox[{ RowBox[{"x", "\[Equal]", "y"}], ",", "1", ",", "2"}], "]"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"u", "/.", RowBox[{"x", "\[Rule]", "y"}]}]], "Input"], Cell[BoxData["1"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"u", "/.", RowBox[{"x", "\[Rule]", RowBox[{"y", "+", "1"}]}]}]], "Input"], Cell[BoxData[ RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"1", "+", "y"}], "\[Equal]", "y"}], ",", "1", ",", "2"}], "]"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"%", "//", "Simplify"}]], "Input"], Cell[BoxData["2"], "Output"] }, Open ]], Cell[TextData[{ "Schleifenk\[ODoubleDot]rper werden dagegen nur dann betreten, wenn die \ Testbedingung explizit zu ", StyleBox["True", FontWeight->"Bold"], " auswertet." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", "n", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"i", "=", "0"}], ";", RowBox[{"While", "[", RowBox[{ RowBox[{"i", "\[LessEqual]", "n"}], ",", RowBox[{ RowBox[{"i", "++"}], ";", RowBox[{"Print", "[", "i", "]"}]}]}], "]"}], ";", " ", "i"}]}], "Input"], Cell[BoxData["0"], "Output"] }, Open ]], Cell[TextData[{ "Dies kann zu falschen Ergebnissen f\[UDoubleDot]hren, wenn eine Aussage als \ Testbedingung zwar den Wahrheitswert ", StyleBox["True", FontWeight->"Bold"], " hat, dies aber von ", StyleBox["Mathematica", FontSlant->"Italic"], " nicht erkannt wird." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ SqrtBox[ RowBox[{"2", "+", SqrtBox["3"]}]], "+", SqrtBox[ RowBox[{"2", "-", SqrtBox["3"]}]]}], "\[Equal]", SqrtBox["6"]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ SqrtBox[ RowBox[{"2", "-", SqrtBox["3"]}]], "+", SqrtBox[ RowBox[{"2", "+", SqrtBox["3"]}]]}], "\[Equal]", SqrtBox["6"]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"i", "=", "5"}], ";", RowBox[{"While", "[", RowBox[{ RowBox[{ SqrtBox["i"], "\[LessEqual]", RowBox[{ SqrtBox[ RowBox[{"2", "+", SqrtBox["3"]}]], "+", SqrtBox[ RowBox[{"2", "-", SqrtBox["3"]}]]}]}], ",", RowBox[{"i", "++"}]}], "]"}], ";", " ", "i"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"N", "::", "\<\"meprec\"\>"}], RowBox[{ ":", " "}], "\<\"Internal precision limit $MaxExtraPrecision = \ \\!\\(50.`\\) reached while evaluating \\!\\(\\@6 - \\@\\(2 - \\@3\\) - \ \\@\\(2 + \\@3\\)\\).\"\>"}]], "Message", "MSG"], Cell[BoxData["6"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s", "=", RowBox[{ RowBox[{ SqrtBox[ RowBox[{"2", "+", SqrtBox["3"]}]], "+", SqrtBox[ RowBox[{"2", "-", SqrtBox["3"]}]]}], "\[LessEqual]", SqrtBox["6"]}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"N", "::", "\<\"meprec\"\>"}], RowBox[{ ":", " "}], "\<\"Internal precision limit $MaxExtraPrecision = \ \\!\\(50.`\\) reached while evaluating \\!\\(\\(\\(-\\@6\\)\\) + \\@\\(2 - \ \\@3\\) + \\@\\(2 + \\@3\\)\\).\"\>"}]], "Message", "MSG"], Cell[BoxData[ RowBox[{ RowBox[{ SqrtBox[ RowBox[{"2", "-", SqrtBox["3"]}]], "+", SqrtBox[ RowBox[{"2", "+", SqrtBox["3"]}]]}], "\[LessEqual]", SqrtBox["6"]}]], "Output"] }, Open ]], Cell[TextData[{ "Interessanterweise wird als Ergebnis der numerischen N\[ADoubleDot]herung \ ", StyleBox["True", FontWeight->"Bold"], " zur\[UDoubleDot]ckgegeben, obwohl dies auf Grund von \ Rundungsungenauigkeiten numerisch niemals genau festgestellt werden kann. " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s", "//", "N"}]], "Input"], Cell[BoxData["True"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], "-", RowBox[{"s", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "//", "N"}]], "Input"], Cell[BoxData["4.440892098500626`*^-16"], "Output"] }, Open ]], Cell[TextData[{ "Mehr noch, wir bekommen damit zwei Ausdr\[UDoubleDot]cke, von denen ", StyleBox["Mathematica", FontSlant->"Italic"], " behauptet, dass sie numerisch gleich seien, ihre Differenz aber nicht \ numerisch gleich Null. 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