(* 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[ 413054, 10499] NotebookOptionsPosition[ 392157, 9854] NotebookOutlinePosition[ 395008, 9928] CellTagsIndexPosition[ 394928, 9923] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", Magnification->1., CellTags->"SlideShowHeader"], Cell["\<\ Real Root Isolation for Tame Elementary Functions\ \>", "Title", CellChangeTimes->{{3.424433385624853*^9, 3.424433389043407*^9}, 3.4564711960483856`*^9}], Cell["Adam Strzebonski", "Subtitle", CellChangeTimes->{{3.424433449329602*^9, 3.424433454945587*^9}}], Cell["\<\ Wolfram Research Inc. 100 Trade Centre Dr. Champaign, IL 61820 U.S.A.\ \>", "Subsubtitle", CellChangeTimes->{{3.424433498014389*^9, 3.424433601117033*^9}}], Cell["\<\ adams@wolfram.com http://members.wolfram.com/adams\ \>", "Subsubtitle", CellChangeTimes->{{3.424433498014389*^9, 3.424433605925003*^9}, { 3.424493823192981*^9, 3.4244938330471506`*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", Magnification->1., CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Abstract", "Section", CellChangeTimes->{{3.424433639503929*^9, 3.424433641402786*^9}}], Cell[TextData[{ "We present a real root isolation procedure for univariate elementary \ functions. The procedure finds the domain and the zero set of a function ", StyleBox["f", FontSlant->"Italic"], " in an arbitrary, possibly unbounded, interval as long as ", StyleBox["f", FontSlant->"Italic"], " is represented by a tame expression. An elementary expression is tame if \ the arguments of its trigonometric subexpressions are bounded. We discuss \ implementation of the procedure and give empirical results. The procedure \ requires the ability to determine signs of elementary functions at simple \ roots of other elementary functions. The currently known method to do this \ [Richardson 1997, 2009] depends on Schanuel\[CloseCurlyQuote]s conjecture." }], "Text", CellChangeTimes->{{3.424433660313063*^9, 3.424433713142749*^9}, { 3.4244413441131153`*^9, 3.424441345054469*^9}, {3.424442097396283*^9, 3.424442099669552*^9}, 3.425645278548315*^9, {3.4564712280143504`*^9, 3.45647126766136*^9}, {3.4574445267114224`*^9, 3.4574445301663904`*^9}, { 3.457444572036597*^9, 3.4574445934573984`*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, 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hyperbolic functions are given in the table below.\n\ \[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"D", "(", RowBox[{"f", "+", "g"}], ")"}], "=", RowBox[{ RowBox[{"D", "(", RowBox[{"f", " ", "g"}], ")"}], "=", RowBox[{ RowBox[{"D", "(", "f", ")"}], "\[Intersection]", RowBox[{"D", "(", "g", ")"}]}]}]}], TraditionalForm]]], ",\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"D", "(", RowBox[{"f", "(", "g", ")"}], ")"}], "=", RowBox[{ SuperscriptBox["g", RowBox[{"-", "1"}]], "(", RowBox[{"D", "(", "f", ")"}], ")"}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.42444254300704*^9, 3.424442954699024*^9}, { 3.4564713621572385`*^9, 3.456471443263864*^9}}], Cell[BoxData[GridBox[{ { RowBox[{"f", RowBox[{"(", "x", ")"}]}], RowBox[{"D", RowBox[{"(", "f", ")"}]}]}, { RowBox[{ RowBox[{"sin", RowBox[{"(", "x", ")"}]}], ",", RowBox[{"cos", RowBox[{"(", "x", ")"}]}]}], "\[DoubleStruckCapitalR]"}, { RowBox[{ RowBox[{"tan", " ", RowBox[{"(", "x", ")"}]}], ",", RowBox[{"sec", " 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From now on, \ when we refer to the domain, point values, zero sets and roots of an \ elementary expression, we will mean the domain, point values and roots of the \ corresponding elementary function.\ \>", "Text", CellChangeTimes->{{3.424449073217021*^9, 3.4244491307197056`*^9}, { 3.4564727519857135`*^9, 3.45647278784728*^9}}], Cell[TextData[{ "Two elementary expressions are ", StyleBox["equivalent", FontSlant->"Italic"], " if they represent the same elementary function. An elementary function \ (expression) ", Cell[BoxData[ FormBox["g", TraditionalForm]]], " ", StyleBox["extends", FontSlant->"Italic"], " an elementary function (expression) ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " if ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"D", "(", "f", ")"}], "\[SubsetEqual]", RowBox[{"D", "(", "g", ")"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[ForAll]", RowBox[{"x", "\[Element]", RowBox[{"D", "(", "f", ")"}]}]], " ", RowBox[{"f", "(", "x", ")"}]}], "=", RowBox[{"g", "(", "x", ")"}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.4244491430975037`*^9, 3.4244492791631565`*^9}, { 3.456476215636198*^9, 3.4564762386593037`*^9}}], Cell[TextData[{ "For an elementary expression ", Cell[BoxData[ FormBox["f", TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"f", "'"}], TraditionalForm]]], " denotes the expression obtained from ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " by applying the rules of differentiation with respect to ", Cell[BoxData[ FormBox["x", TraditionalForm]]], ". The function represented by ", Cell[BoxData[ FormBox[ RowBox[{"f", "'"}], TraditionalForm]]], " extends the derivative of the function represented by ", Cell[BoxData[ FormBox["f", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.4244494557871294`*^9, 3.4244495212011905`*^9}, { 3.456476259669515*^9, 3.456476261191704*^9}}], Cell[TextData[{ "The following notation will be used for the domain and the zero set of the \ restriction of an elementary function ", StyleBox["f", FontSlant->"Italic"], " to an interval ", StyleBox["(a, b)", FontSlant->"Italic"], ".\n", StyleBox["D(f, a, b) := D(f) \:2229 (a, b)", FontSlant->"Italic"], "\n", StyleBox["Z(f, a, b) := {x \[Element] (a, b) : f(x) = 0}", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.4244491430975037`*^9, 3.4244492791631565`*^9}, { 3.456472942139141*^9, 3.456472957971907*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Tame elementary expressions and functions\ \>", "Section", CellChangeTimes->{{3.424448737564376*^9, 3.424448743753275*^9}, { 3.456472492092005*^9, 3.456472493594165*^9}, {3.456473479812277*^9, 3.4564734883445454`*^9}}], Cell[TextData[{ StyleBox["Definition", FontWeight->"Bold"], ": Let ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"a", "<", "b"}], "\[Element]", RowBox[{"\[DoubleStruckCapitalR]", "\[Union]", RowBox[{"{", RowBox[{ RowBox[{"\[Minus]", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". An elementary expression ", StyleBox["f", FontSlant->"Italic"], " is tame over the interval ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " if for every subexpression of ", StyleBox["f", FontSlant->"Italic"], " of the form ", Cell[BoxData[ FormBox[ RowBox[{"g", "(", RowBox[{"h", "(", "x", ")"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], ", where ", StyleBox["g", FontSlant->"Italic"], " is a trigonometric function, there exist ", Cell[BoxData[ FormBox[ RowBox[{"c", ",", RowBox[{"d", "\[Element]", "\[DoubleStruckCapitalR]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " such that ", Cell[BoxData[ FormBox[ RowBox[{"c", "<", RowBox[{"h", "(", "u", ")"}], "<", "d"}], TraditionalForm]], FormatType->"TraditionalForm"], " for all ", Cell[BoxData[ FormBox[ RowBox[{"u", "\[Element]", RowBox[{"D", "(", RowBox[{"h", ",", "a", ",", "b"}], ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.4244488931180515`*^9, 3.4244490544600496`*^9}, { 3.4564725097774353`*^9, 3.456472511249552*^9}, {3.4564725779855137`*^9, 3.456472591364752*^9}, {3.456472622789939*^9, 3.456472667073616*^9}, { 3.456473508403389*^9, 3.4564735387570353`*^9}, {3.456473737332573*^9, 3.4564737403669357`*^9}, {3.457444818470952*^9, 3.457444897845086*^9}}], Cell[TextData[{ "An elementary function ", Cell[BoxData[ FormBox[ RowBox[{"f", ":", RowBox[{ RowBox[{"D", RowBox[{"(", RowBox[{"f", ",", "a", ",", "b"}], ")"}]}], "\[RightArrow]", "\[DoubleStruckCapitalR]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " is tame over ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " if it is the function represented by an elementary expression tame over ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.424449073217021*^9, 3.4244491307197056`*^9}, { 3.4564727519857135`*^9, 3.45647278784728*^9}, {3.4564735949778767`*^9, 3.456473615176922*^9}, {3.457444914959696*^9, 3.4574449411573668`*^9}}], Cell[TextData[{ StyleBox["Example", FontWeight->"Bold"], ": Expression ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"sin", "(", RowBox[{"exp", "(", RowBox[{"-", SuperscriptBox["x", "2"]}], ")"}], ")"}], "-", "x"}], TraditionalForm]]], " is a tame elementary expression over \[DoubleStruckCapitalR]." }], "Text", CellChangeTimes->{{3.4244491430975037`*^9, 3.4244492791631565`*^9}, { 3.456472942139141*^9, 3.456472957971907*^9}, {3.4564736438882065`*^9, 3.45647369023485*^9}, {3.4574449532347326`*^9, 3.457444956449355*^9}, { 3.4579916030784416`*^9, 3.45799169882612*^9}}], Cell[TextData[{ StyleBox["Example", FontWeight->"Bold"], ": Expression ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"sin", "(", "x", ")"}], "2"], "+", SuperscriptBox[ RowBox[{"cos", "(", "x", ")"}], "2"]}], TraditionalForm]]], " represents an elementary function tame over \[DoubleStruckCapitalR], but \ it is not a tame elementary expression over \[DoubleStruckCapitalR]. The \ algorithm given in this paper requires tame expressions, not just expressions \ representing tame functions." }], "Text", CellChangeTimes->{{3.4244491430975037`*^9, 3.4244492791631565`*^9}, { 3.456472942139141*^9, 3.456472957971907*^9}, {3.4564736438882065`*^9, 3.45647369023485*^9}, {3.4574449532347326`*^9, 3.457444956449355*^9}}], Cell[TextData[{ StyleBox["Claim", FontWeight->"Bold"], ": The domain and the zero set of a tame elementary function consist of a \ finite number of open, possibly unbounded, intervals and a finite number of \ points." }], "Text", CellChangeTimes->{{3.4244494557871294`*^9, 3.4244495212011905`*^9}, { 3.456473732385459*^9, 3.4564737661339874`*^9}}], Cell["\<\ This claim follows from the fact that sets definable using exponential and \ restricted trigonometric functions form an o-minimal structure. The algorithm \ I will describe in this talk computes domains and zero sets of tame \ elementary expressions. The proof of correctness of the algorithm does not \ use the o-minimality and hence it provides a new proof of the claim.\ \>", "Text", CellChangeTimes->{{3.4564738028367634`*^9, 3.4564738576956463`*^9}, { 3.4564764234750557`*^9, 3.4564764287426305`*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Definition of roots", "Section", CellChangeTimes->{{3.424444782096693*^9, 3.4244447906089325`*^9}}], Cell[TextData[{ "Let ", Cell[BoxData[ FormBox[ RowBox[{"f", ":", RowBox[{"\[DoubleStruckCapitalR]", "\[SupersetEqual]", RowBox[{ RowBox[{"D", "(", "f", ")"}], "\[RightArrow]", "\[DoubleStruckCapitalR]"}]}]}], TraditionalForm]]], " and let ", Cell[BoxData[ FormBox[ RowBox[{"a", "\[Element]", RowBox[{"D", "(", "f", ")"}]}], TraditionalForm]]], ". ", Cell[BoxData[ FormBox["a", TraditionalForm]]], " is a ", StyleBox["root", FontSlant->"Italic"], " of ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " iff \n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "a", ")"}], "=", "0"}], TraditionalForm]]], ",\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Exists]", RowBox[{"\[Epsilon]", ">", "0"}]], RowBox[{ SubscriptBox["\[ForAll]", RowBox[{ RowBox[{ RowBox[{"a", "-", "\[Epsilon]"}], "<", "x", "<", "a"}], "\[Or]", RowBox[{"a", "<", "x", "<", RowBox[{"a", "+", "\[Epsilon]"}]}]}]], " ", RowBox[{"x", "\[Element]", RowBox[{"D", "(", "f", ")"}]}]}]}], " ", "\[And]", " ", RowBox[{"x", "\[NotEqual]", "0", " "}]}], TraditionalForm]]], ".\n" }], "Text", CellChangeTimes->{{3.4244448861162653`*^9, 3.4244452864318914`*^9}, { 3.424445316475091*^9, 3.424445317376387*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Root constants", "Section", CellChangeTimes->{{3.4244495575835056`*^9, 3.4244495608181567`*^9}}], Cell[TextData[{ StyleBox["Definition", FontWeight->"Bold"], ": An elementary root constant with coefficients in a computable field K \ \[SubsetEqual] \[DoubleStruckCapitalR] is an expression ", Cell[BoxData[ FormBox[ RowBox[{"Root", "(", RowBox[{"f", ",", "a", ",", "b"}], ")"}], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " is a tame elementary expression with coefficients in K, ", Cell[BoxData[ FormBox[ RowBox[{"a", "<", "b"}], TraditionalForm]]], " are rational numbers, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"[", RowBox[{"a", ",", "b"}], "]"}], "\[SubsetEqual]", RowBox[{"D", "(", "f", ")"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", " ", RowBox[{"(", "a", ")"}], " ", "f", " ", RowBox[{"(", "b", ")"}]}], "<", "0"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"f", "'"}], TraditionalForm]]], " has a constant nonzero sign on ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], ". An elementary root constant ", Cell[BoxData[ FormBox[ RowBox[{"Root", " ", RowBox[{"(", RowBox[{"f", ",", " ", "a", ",", " ", "b"}], ")"}]}], TraditionalForm]]], " represents the only root of ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " in ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], ". Form now on we will identify elementary root expressions with the roots \ they represent." }], "Text", CellChangeTimes->{{3.4244495868155394`*^9, 3.424449780243675*^9}, { 3.4244498153741903`*^9, 3.4244498294243937`*^9}, {3.456476493996461*^9, 3.456476637332568*^9}, {3.456490050389597*^9, 3.456490053133542*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " representation:" }], "Text", CellChangeTimes->{{3.4244498593173776`*^9, 3.424449870172987*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Sin", "[", SuperscriptBox["E", RowBox[{"-", SuperscriptBox["x", "2"]}]], "]"}], "-", "x"}], "\[Equal]", "0"}], ",", "x", ",", "Reals"}], "]"}]], "Input", CellChangeTimes->{{3.4244499187428274`*^9, 3.42444998445732*^9}, { 3.4564766654630175`*^9, 3.456476704659379*^9}}], Cell[BoxData[ RowBox[{"x", "\[Equal]", RowBox[{"Root", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["#1", "2"]}]], "]"}]}], "+", "#1"}], "&"}], ",", "0.62570617795779608338755743995111372801`20.60205986823331"}], "}"}], "]"}]}]], "Output", CellChangeTimes->{3.427016473323704*^9, 3.456476707453397*^9, 3.4564767478314576`*^9, 3.4574450448865213`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"r", "=", RowBox[{"%", "[", RowBox[{"[", "2", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.42445003177536*^9, 3.424450058013088*^9}, { 3.456476745438016*^9, 3.4564767455982466`*^9}}], Cell[BoxData[ RowBox[{"Root", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["#1", "2"]}]], "]"}]}], "+", "#1"}], "&"}], ",", "0.62570617795779608338755743995111372801`20.60205986823331"}], "}"}], "]"}]], "Output", CellChangeTimes->{ 3.427016476127736*^9, {3.4564767296453075`*^9, 3.4564767496540785`*^9}, 3.457445053789323*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Precision", "[", "r", "]"}]], "Input", CellChangeTimes->{{3.424450039065843*^9, 3.424450067176264*^9}}], Cell[BoxData["\[Infinity]"], "Output", CellChangeTimes->{3.4270164797529488`*^9, 3.456476752057534*^9, 3.4574450563630238`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"r", ",", "100"}], "]"}]], "Input", CellChangeTimes->{{3.424450053486579*^9, 3.4244500723937664`*^9}}], Cell[BoxData["0.\ 625706177957796083389121704952639087666352300890720999941494498926619016688919\ 563321046137870915531795331320568038`100."], "Output", CellChangeTimes->{3.427016481705757*^9, 3.456476754280731*^9, 3.4574450582457314`*^9}] }, Open ]], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Root isolation", "Section", CellChangeTimes->{{3.424450960080197*^9, 3.4244509849259233`*^9}, { 3.424451034517232*^9, 3.4244510369707603`*^9}, {3.456477808696907*^9, 3.456477810449427*^9}, {3.4564827586445827`*^9, 3.4564827613985424`*^9}}], Cell[TextData[{ "I will present a procedure which given an elementary expression ", StyleBox["f", FontSlant->"Italic"], " and a possibly infinite interval ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], StyleBox[" ", FontSlant->"Italic"], "finds out whether ", StyleBox["f", FontSlant->"Italic"], " is tame over ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " and if yes compute its domain and zero set. ", Cell[BoxData[ FormBox[ RowBox[{"D", "(", RowBox[{"f", ",", "a", ",", "b"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"Z", "(", RowBox[{"f", ",", "a", ",", "b"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " are finite unions of intervals and points expressed using elements of ", StyleBox["K", FontSlant->"Italic"], " and elementary root constants." }], "Text", CellChangeTimes->{{3.4244505727933054`*^9, 3.4244506040182047`*^9}, { 3.456482789408819*^9, 3.4564828564251842`*^9}, {3.457445069381744*^9, 3.4574450935064335`*^9}}], Cell[TextData[{ "The assumption that ", StyleBox["K", FontSlant->"Italic"], " is a computable field implies that there is an algorithm for computation \ of signs of elements of ", StyleBox["K", FontSlant->"Italic"], ". However, the root isolation procedure requires the ability to determine \ signs of constants obtained by evaluation of tame elementary functions at \ arbitrary elementary root constants. At present, there is a known method to \ do this only when K \[SubsetEqual] ", Cell[BoxData[ FormBox[ SubscriptBox["E", "\[DoubleStruckCapitalR]"], TraditionalForm]], FormatType->"TraditionalForm"], ", where ", Cell[BoxData[ FormBox[ SubscriptBox["E", "\[DoubleStruckCapitalR]"], TraditionalForm]], FormatType->"TraditionalForm"], " is the field of real elementary numbers." }], "Text", CellChangeTimes->{{3.4244507164699025`*^9, 3.424450755265688*^9}, { 3.424450789434821*^9, 3.4244508076710434`*^9}, {3.4244508430519185`*^9, 3.4244508782825775`*^9}, {3.4564828742908735`*^9, 3.456482930882248*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Elementary numbers", "Section", CellChangeTimes->{{3.424450960080197*^9, 3.4244509849259233`*^9}, { 3.424451034517232*^9, 3.4244510369707603`*^9}, {3.456477808696907*^9, 3.456477830818717*^9}, 3.4565052561644273`*^9}], Cell[TextData[{ StyleBox["Definition", FontWeight->"Bold"], ": An elementary number is a coordinate of a point \[Alpha] \[Element] ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalC]", "n"], TraditionalForm]], FormatType->"TraditionalForm"], " such that ", Cell[BoxData[ FormBox[ SubscriptBox["f", "1"], TraditionalForm]], FormatType->"TraditionalForm"], "(\[Alpha]) = . . . = ", Cell[BoxData[ FormBox[ SubscriptBox["f", "n"], TraditionalForm]], FormatType->"TraditionalForm"], "(\[Alpha]) = 0, where \n\[Bullet] for each ", StyleBox["1\[LessEqual]i \[LessEqual]n", FontSlant->"Italic"], " either ", Cell[BoxData[ FormBox[ SubscriptBox["f", "i"], TraditionalForm]], FormatType->"TraditionalForm"], " \[Element] ", Cell[BoxData[ FormBox[ RowBox[{"\[DoubleStruckCapitalQ]", "[", RowBox[{ SubscriptBox["x", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["x", "n"]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "] or ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["f", "i"], "=", RowBox[{ SubscriptBox["x", "p"], "-", RowBox[{"exp", "(", SubscriptBox["x", "q"], ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " for some ", StyleBox["1\[LessEqual]p,q \[LessEqual]n", FontSlant->"Italic"], " \n\[Bullet] the Jacobian ", Cell[BoxData[ FormBox[ RowBox[{"Jac", "(", RowBox[{ SubscriptBox["f", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["f", "n"]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ") is nonzero at \[Alpha].\nLet ", StyleBox["E", FontSlant->"Italic"], " denote the set of all elementary numbers. The set of real elementary \ numbers is defined as", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox["E", "\[DoubleStruckCapitalR]"], TraditionalForm]], FormatType->"TraditionalForm", FontSlant->"Italic"], StyleBox[" := E \:2229 \[DoubleStruckCapitalR]", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.4244505727933054`*^9, 3.4244506040182047`*^9}, { 3.456477833873109*^9, 3.4564779430000257`*^9}, {3.456478013241027*^9, 3.4564782305334787`*^9}}], Cell[TextData[{ StyleBox["Proposition", FontWeight->"Bold"], ": \n\[Bullet] ", Cell[BoxData[ FormBox[ SubscriptBox["E", "\[DoubleStruckCapitalR]"], TraditionalForm]], FontSlant->"Italic"], " is a computable field (assuming Schanuel\[CloseCurlyQuote]s conjecture).\n\ \[Bullet] If ", StyleBox["f", FontSlant->"Italic"], " is an elementary expression with coefficients in ", Cell[BoxData[ FormBox[ SubscriptBox["E", "\[DoubleStruckCapitalR]"], TraditionalForm]], FontSlant->"Italic"], " and ", Cell[BoxData[ FormBox[ RowBox[{"x", "\[Element]", RowBox[{ SubscriptBox["E", "\[DoubleStruckCapitalR]"], "\[Intersection]", RowBox[{"D", "(", "f", ")"}]}]}], TraditionalForm]], FontSlant->"Italic"], " then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "\[Element]", SubscriptBox["E", "\[DoubleStruckCapitalR]"]}], TraditionalForm]], FontSlant->"Italic"], ".\n\[Bullet] If ", StyleBox["r", FontSlant->"Italic"], " is an elementary root constant with coefficients in ", Cell[BoxData[ FormBox[ SubscriptBox["E", "\[DoubleStruckCapitalR]"], TraditionalForm]], FontSlant->"Italic"], ", then ", Cell[BoxData[ FormBox[ RowBox[{"r", "\[Element]", SubscriptBox["E", "\[DoubleStruckCapitalR]"]}], TraditionalForm]], FontSlant->"Italic"], ".\nThe last two items are effective, that is one can compute the functions ", Cell[BoxData[ FormBox[ SubscriptBox["f", "i"], TraditionalForm]]], " as in the definition of elementary numbers. " }], "Text", CellChangeTimes->{{3.4244507164699025`*^9, 3.424450755265688*^9}, { 3.424450789434821*^9, 3.4244508076710434`*^9}, {3.4244508430519185`*^9, 3.4244508782825775`*^9}, {3.4564783142338333`*^9, 3.456478395861208*^9}, 3.4564784760364943`*^9, {3.4564785351414833`*^9, 3.4564785982322035`*^9}, { 3.456478651348581*^9, 3.456478744632717*^9}, {3.4564788043585987`*^9, 3.4564789573586016`*^9}, {3.4564822256581855`*^9, 3.4564822395982304`*^9}, {3.45648232105536*^9, 3.456482418896048*^9}, { 3.456494725842568*^9, 3.456494738640971*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Reduction to Exp, Log, Tan and ArcTan\ \>", "Section", CellChangeTimes->{{3.4244528530421424`*^9, 3.4244528619649725`*^9}, { 3.424453178580243*^9, 3.4244531793513517`*^9}, {3.456483762598197*^9, 3.456483780684203*^9}}], Cell["\<\ The problem of finding the domain and the zero set of an arbitrary tame \ elementary function can be reduced to the problem of finding the domain and \ the zero set of a tame elementary function constructed only with exp, log, \ tan and arctan.\ \>", "Text", CellChangeTimes->{{3.4244532506739087`*^9, 3.424453607236621*^9}, { 3.4564838139520397`*^9, 3.4564838277318544`*^9}, {3.456485818784848*^9, 3.4564858198263454`*^9}}], Cell[TextData[{ StyleBox["Definition", FontWeight->"Bold"], ": The set of ELT functions is the smallest set of partial functions \ \[DoubleStruckCapitalR] \[RightArrow] \[DoubleStruckCapitalR] containing the \ constant functions, the identity function, exp, log, tan and arctan, closed \ under addition, multiplication and composition of functions." }], "Text", CellChangeTimes->{{3.456495020686533*^9, 3.45649509125801*^9}, { 3.4564970460388465`*^9, 3.456497082481248*^9}}], Cell[TextData[{ StyleBox["Fact", FontWeight->"Bold"], ": Let ", StyleBox["f", FontSlant->"Italic"], " be a trigonometric, hyperbolic, inverse trigonometric or inverse \ hyperbolic function. There exist an ELT function ", Cell[BoxData[ FormBox[ SubscriptBox["f", "ELT"], TraditionalForm]]], " and an at most countable set ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"EX", "(", "f", ")"}], "\[Subset]", "\[DoubleStruckCapitalR]"}], TraditionalForm]]], ", such that\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"D", "(", SubscriptBox["f", "ELT"], ")"}], "=", RowBox[{ RowBox[{"D", "(", "f", ")"}], "\\", RowBox[{"EX", "(", "f", ")"}]}]}], TraditionalForm]]], ",\n\[Bullet] ", Cell[BoxData[ FormBox[ SubscriptBox["\[ForAll]", RowBox[{"u", "\[Element]", RowBox[{"D", "(", SubscriptBox["f", "ELT"], ")"}]}]], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["f", "ELT"], "(", "u", ")"}], "=", RowBox[{"f", "(", "u", ")"}]}], TraditionalForm]]], ",\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{"D", "(", SubscriptBox["f", "ELT"], ")"}], TraditionalForm]]], " is an open set,\n\[Bullet] for any ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"a", "<", "b"}], "\[Element]", "\[DoubleStruckCapitalR]"}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{"D", "(", RowBox[{ SubscriptBox["f", "ELT"], ",", "a", ",", "b"}], ")"}], TraditionalForm]]], " is a union of a finite number of disjoint open intervals and ", StyleBox["EX(f) \:2229 (a, b) ", FontSlant->"Italic"], "is a finite set of points.\nExplicit formulas for ", Cell[BoxData[ FormBox[ SubscriptBox["f", "ELT"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"EX", "(", "f", ")"}], TraditionalForm]]], " are given in the table below." }], "Text", CellChangeTimes->{{3.456495020686533*^9, 3.4564950271157775`*^9}, 3.4564950575595536`*^9, {3.4564950948331504`*^9, 3.4564951224929233`*^9}, { 3.4564965901132574`*^9, 3.4564966177730303`*^9}, {3.4564966608349504`*^9, 3.4564968792490144`*^9}, {3.4564969108845043`*^9, 3.4564969430507565`*^9}, {3.4564969749165773`*^9, 3.456496980354397*^9}, { 3.457445591362315*^9, 3.457445598492568*^9}}], Cell[TextData[{ 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"ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}]]], "\n" }], "Text", CellChangeTimes->{{3.4244536604331136`*^9, 3.424453671639227*^9}, { 3.4244537076309805`*^9, 3.4244539529236946`*^9}, {3.4244541399225855`*^9, 3.424454247206853*^9}, 3.4564838820699887`*^9, {3.4564839444997587`*^9, 3.4564839637874928`*^9}, {3.4564840027134657`*^9, 3.456484307041067*^9}, { 3.4564843501230164`*^9, 3.4564844390108304`*^9}, {3.4564844941601315`*^9, 3.4564846086647806`*^9}, {3.456484648672309*^9, 3.456484778308717*^9}, { 3.456484898161056*^9, 3.4564849959717007`*^9}, {3.4564850498091154`*^9, 3.4564850724116163`*^9}, {3.4564852321513104`*^9, 3.456485328109291*^9}, { 3.456485370390088*^9, 3.456485479717293*^9}}], Cell[TextData[{ StyleBox["Definition", FontWeight->"Bold"], ": An ELT expression is an elementary expression which does not contain \ trigonometric functions other than tan, does not contain inverse \ trigonometric functions other than arctan and does not contain hyperbolic or \ inverse hyperbolic functions. An ELT root constant is an elementary root \ constant ", Cell[BoxData[ FormBox[ RowBox[{"Root", "(", RowBox[{"f", ",", "a", ",", "b"}], ")"}], TraditionalForm]]], ", where ", StyleBox["f", FontSlant->"Italic"], " is an ELT expression." }], "Text", CellChangeTimes->{{3.456495020686533*^9, 3.45649509125801*^9}, { 3.4564970460388465`*^9, 3.4564971326634064`*^9}}], Cell[TextData[{ StyleBox["Claim", FontWeight->"Bold"], ": If ", StyleBox["f", FontSlant->"Italic"], " is an ELT expression, then ", Cell[BoxData[ FormBox[ RowBox[{"D", "(", "f", ")"}], TraditionalForm]]], " is open and ", StyleBox["f", FontSlant->"Italic"], " is tame and analytic in every interval contained in ", Cell[BoxData[ FormBox[ RowBox[{"D", "(", "f", ")"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.424454327552384*^9, 3.4244546597200174`*^9}, { 3.4244547042640686`*^9, 3.4244550451041727`*^9}, {3.4564971494575553`*^9, 3.45649717866956*^9}, 3.4574458860861073`*^9}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["False derivatives", "Section", CellChangeTimes->{{3.4244528530421424`*^9, 3.4244528619649725`*^9}, { 3.424453178580243*^9, 3.4244531793513517`*^9}, {3.456505277344883*^9, 3.4565052819314785`*^9}, 3.456640775443925*^9}], Cell[TextData[{ StyleBox["Definition", FontWeight->"Bold"], ": Let ", Cell[BoxData[ FormBox[ RowBox[{"f", ":", RowBox[{ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], "\[RightArrow]", "\[DoubleStruckCapitalR]"}]}], TraditionalForm]]], " be a differentiable function. A continuous function ", Cell[BoxData[ FormBox[ RowBox[{"g", ":", RowBox[{ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], "\[RightArrow]", "\[DoubleStruckCapitalR]"}]}], TraditionalForm]]], " is a false derivative of ", StyleBox["f", FontSlant->"Italic"], " in ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " if for each ", Cell[BoxData[ FormBox[ RowBox[{"r", "\[Element]", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm]]], " such that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "r", ")"}], "=", "0"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"sign", RowBox[{"(", "f"}]}], "\[CloseCurlyQuote]"}], RowBox[{"(", "r", ")"}]}], ")"}], "=", RowBox[{"sign", "(", RowBox[{"g", "(", "r", ")"}], ")"}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.4244532506739087`*^9, 3.424453607236621*^9}, { 3.4565052974537983`*^9, 3.45650540015147*^9}}], Cell[TextData[{ StyleBox["Fact", FontWeight->"Bold"], ": If ", StyleBox["g", FontSlant->"Italic"], " is a false derivative of ", StyleBox["f", FontSlant->"Italic"], " in ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", "u", ")"}], "\[NotEqual]", "0"}], TraditionalForm]]], " for all ", Cell[BoxData[ FormBox[ RowBox[{"u", "\[Element]", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm]]], ", then there is at most one ", Cell[BoxData[ FormBox[ RowBox[{"r", "\[Element]", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm]]], " such that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "r", ")"}], "=", "0"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.4244536604331136`*^9, 3.424453671639227*^9}, { 3.4244537076309805`*^9, 3.4244539529236946`*^9}, {3.4244541399225855`*^9, 3.424454247206853*^9}, {3.4565054227940288`*^9, 3.456505504090928*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Simple recursive real root isolation\ \>", "Section", CellChangeTimes->{{3.424491579476675*^9, 3.4244915847342353`*^9}, { 3.45650638340532*^9, 3.4565063932895327`*^9}}], Cell[TextData[{ StyleBox["Algorithm idea", FontWeight->"Bold"], ": Let ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"a", "<", "b"}], "\[Element]", RowBox[{"R", "\[Union]", RowBox[{"{", RowBox[{ RowBox[{"\[Minus]", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}]}], TraditionalForm]]], " and let ", StyleBox["f", FontSlant->"Italic"], " be an analytic function in ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], ". Suppose that \n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"h", " ", RowBox[{"f", "'"}]}], "=", RowBox[{ RowBox[{"q", " ", "f"}], "+", "g"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", \n\[Bullet] we can recursively find ", Cell[BoxData[ FormBox[ RowBox[{"Z", "(", RowBox[{"g", ",", "a", ",", "b"}], ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"Z", "(", RowBox[{"h", ",", "a", ",", "b"}], ")"}], TraditionalForm]]], ",\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Z", "(", RowBox[{"g", ",", "a", ",", "b"}], ")"}], "=", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm]]], " or ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Z", "(", RowBox[{"g", ",", "a", ",", "b"}], ")"}], "=", RowBox[{"{", RowBox[{ SubscriptBox["r", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["r", "n"]}], "}"}]}], TraditionalForm]]], ",\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Z", "(", RowBox[{"h", ",", "a", ",", "b"}], ")"}], "=", RowBox[{"{", RowBox[{ SubscriptBox["s", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["s", "m"]}], "}"}]}], TraditionalForm]]], ".\nThen we can find ", Cell[BoxData[ FormBox[ RowBox[{"Z", "(", RowBox[{"f", ",", "a", ",", "b"}], ")"}], TraditionalForm]]], " as follows:" }], "Text", CellChangeTimes->{{3.424491808025312*^9, 3.4244922703000307`*^9}, 3.424492760554982*^9, {3.4565063562162237`*^9, 3.4565063794496317`*^9}, { 3.4565064338578672`*^9, 3.456506702313888*^9}, {3.456506738115368*^9, 3.4565067407591696`*^9}, {3.456507021723176*^9, 3.4565070527277584`*^9}, 3.456507084543507*^9, {3.4565075861247454`*^9, 3.4565078075331154`*^9}, { 3.4565078400899296`*^9, 3.456507845027029*^9}, {3.456507891904435*^9, 3.4565081433860483`*^9}, {3.456508176874202*^9, 3.456508372545563*^9}, { 3.456508414976576*^9, 3.456508584370152*^9}, {3.457446639118915*^9, 3.4574466451375694`*^9}, {3.4574467011080513`*^9, 3.457446741265795*^9}, { 3.4574467831760592`*^9, 3.4574467833062468`*^9}, {3.4574468453654833`*^9, 3.457447011604523*^9}, {3.4574506356756816`*^9, 3.4574506479433217`*^9}, { 3.457450691756322*^9, 3.4574506992370787`*^9}, {3.457450848171235*^9, 3.457450856523245*^9}, {3.4574509195839214`*^9, 3.457451025646432*^9}, { 3.4574512922497888`*^9, 3.4574513392173247`*^9}, {3.457451888437064*^9, 3.457451904329917*^9}, {3.4574519542416863`*^9, 3.4574520788708944`*^9}, { 3.4574521430131264`*^9, 3.457452152887325*^9}, {3.457452191552923*^9, 3.457452241044088*^9}, {3.4574522785880737`*^9, 3.457452327748763*^9}, { 3.457452464565496*^9, 3.4574525080179777`*^9}, {3.4574525784692817`*^9, 3.457452602233453*^9}, {3.45745263625237*^9, 3.457453011421837*^9}}], Cell[TextData[{ StyleBox["Case 1", FontWeight->"Bold"], ": ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Z", "(", RowBox[{"g", ",", "a", ",", "b"}], ")"}], "=", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm]]], ".\n", StyleBox["f", FontSlant->"Italic"], " satisfies the differential equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "'"}], "=", RowBox[{ FractionBox["q", "h"], "f"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " in ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], "\\", RowBox[{"{", RowBox[{ SubscriptBox["s", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["s", "m"]}], "}"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". 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We can decide between the two possibilities by computing the sign of ", StyleBox["f", FontSlant->"Italic"], " at any ", Cell[BoxData[ FormBox[ RowBox[{"c", "\[Element]", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.424491808025312*^9, 3.4244922703000307`*^9}, 3.424492760554982*^9, {3.4565063562162237`*^9, 3.4565063794496317`*^9}, { 3.4565064338578672`*^9, 3.456506702313888*^9}, {3.456506738115368*^9, 3.4565067407591696`*^9}, {3.456507021723176*^9, 3.4565070527277584`*^9}, 3.456507084543507*^9, {3.4565075861247454`*^9, 3.4565078075331154`*^9}, { 3.4565078400899296`*^9, 3.456507845027029*^9}, {3.456507891904435*^9, 3.4565081433860483`*^9}, {3.456508176874202*^9, 3.456508372545563*^9}, { 3.456508414976576*^9, 3.456508584370152*^9}, {3.457446639118915*^9, 3.4574466451375694`*^9}, {3.4574467011080513`*^9, 3.457446741265795*^9}, { 3.4574467831760592`*^9, 3.4574467833062468`*^9}, {3.4574468453654833`*^9, 3.457447011604523*^9}, {3.4574506356756816`*^9, 3.4574506479433217`*^9}, { 3.457450691756322*^9, 3.4574506992370787`*^9}, {3.457450848171235*^9, 3.457450856523245*^9}, {3.4574509195839214`*^9, 3.457451025646432*^9}, { 3.4574512922497888`*^9, 3.4574513392173247`*^9}, {3.457451888437064*^9, 3.457451904329917*^9}, {3.4574519542416863`*^9, 3.4574520788708944`*^9}, { 3.4574521430131264`*^9, 3.457452152887325*^9}, {3.457452191552923*^9, 3.457452241044088*^9}, {3.4574522785880737`*^9, 3.457452327748763*^9}, { 3.457452464565496*^9, 3.4574525080179777`*^9}, {3.4574525784692817`*^9, 3.457452602233453*^9}, {3.45745263625237*^9, 3.457452978304216*^9}, { 3.457453016278821*^9, 3.457453047964382*^9}}], Cell[TextData[{ StyleBox["Case 2", FontWeight->"Bold"], ": If ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Z", "(", RowBox[{"g", ",", "a", ",", "b"}], ")"}], "=", RowBox[{"{", RowBox[{ SubscriptBox["r", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["r", "n"]}], "}"}]}], TraditionalForm]]], ". \nLet ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["t", "0"], ":=", RowBox[{ RowBox[{"a", "<", SubscriptBox["t", "1"], "<", "\[Ellipsis]", "<", SubscriptBox["t", "k"], "<", SubscriptBox["r", RowBox[{"k", "+", "1"}]]}], ":=", "b"}]}], TraditionalForm]]], " be such that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["t", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["t", "k"]}], "}"}], "=", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["r", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["r", "n"]}], "}"}], "\[Union]", RowBox[{"{", RowBox[{ SubscriptBox["s", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["s", "m"]}], "}"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". For ", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessEqual]", "i", "\[LessEqual]", "k"}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{"g", "\[NotEqual]", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["g", TraditionalForm]], FormatType->"TraditionalForm"], " or ", Cell[BoxData[ FormBox[ RowBox[{"-", "g"}], TraditionalForm]], FormatType->"TraditionalForm"], " is a false derivative of ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], " in ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["t", "i"], ",", SubscriptBox["t", RowBox[{"i", "+", "1"}]]}], ")"}], TraditionalForm]]], ". Hence ", Cell[BoxData[ FormBox[ RowBox[{"Z", "(", RowBox[{"f", ",", "a", ",", "b"}], ")"}], TraditionalForm]]], " consists of:\n a) ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["t", "i"], ":", RowBox[{"1", "\[LessEqual]", "i", "\[LessEqual]", RowBox[{"k", "\[Wedge]", RowBox[{"f", "(", SubscriptBox["t", "i"], ")"}]}]}]}], "=", "0"}], "}"}], TraditionalForm]]], "\n b) ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["u", "i"], ":", " ", RowBox[{"0", "\[LessEqual]", "i", "\[LessEqual]", RowBox[{"n", "\[Wedge]", RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", RowBox[{ SubscriptBox["t", "i"], "+"}]}]], RowBox[{"sign", "(", RowBox[{"f", "(", "x", ")"}], ")"}]}]}], "\[NotEqual]", RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", SuperscriptBox[ SubscriptBox["t", RowBox[{"i", "+", "1"}]], "-"]}]], RowBox[{ RowBox[{"sign", "(", RowBox[{"f", "(", "x", ")"}], ")"}], "\[Wedge]", RowBox[{"{", SubscriptBox["u", "i"], "}"}]}]}]}]}], "=", RowBox[{"Z", "(", RowBox[{"f", ",", SubscriptBox["t", "i"], ",", SubscriptBox["t", RowBox[{"i", "+", "1"}]]}], ")"}]}], "}"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.424491808025312*^9, 3.4244922703000307`*^9}, 3.424492760554982*^9, {3.4565063562162237`*^9, 3.4565063794496317`*^9}, { 3.4565064338578672`*^9, 3.456506702313888*^9}, {3.456506738115368*^9, 3.4565067407591696`*^9}, {3.456507021723176*^9, 3.4565070527277584`*^9}, 3.456507084543507*^9, {3.4565075861247454`*^9, 3.4565078075331154`*^9}, { 3.4565078400899296`*^9, 3.456507845027029*^9}, {3.456507891904435*^9, 3.4565081433860483`*^9}, {3.456508176874202*^9, 3.456508372545563*^9}, { 3.456508414976576*^9, 3.456508584370152*^9}, {3.457446639118915*^9, 3.4574466451375694`*^9}, {3.4574467011080513`*^9, 3.457446741265795*^9}, { 3.4574467831760592`*^9, 3.4574467833062468`*^9}, {3.4574468453654833`*^9, 3.457447011604523*^9}, {3.4574506356756816`*^9, 3.4574506479433217`*^9}, { 3.457450691756322*^9, 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"The algorithm can be used for ", StyleBox["K", FontSlant->"Italic"], " not contained in ", Cell[BoxData[ FormBox[ SubscriptBox["E", "\[DoubleStruckCapitalR]"], TraditionalForm]]], ", however it may fail if a call to the sign determination heuristic it uses \ fails." }], "Text", CellChangeTimes->{{3.4244505727933054`*^9, 3.4244506040182047`*^9}, { 3.456482789408819*^9, 3.4564828564251842`*^9}, {3.4564835495017796`*^9, 3.4564835931745777`*^9}}], Cell["\<\ The algorithm could be extended to compute domains and zero sets for an \ arbitrary class of Pfaffian functions, provided there are algorithms \ available for numeric evaluation and computation of limits and Pfaffian \ chains for the functions in the class, as well as for zero testing of \ constants constructed with functions from the class and their simple roots.\ \>", "Text", CellChangeTimes->{{3.4244507164699025`*^9, 3.424450755265688*^9}, { 3.424450789434821*^9, 3.4244508076710434`*^9}, {3.4244508430519185`*^9, 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Comparison with the exp-log root isolation algorithm from ISSAC 2008\ \>", "Section", CellChangeTimes->{{3.424450960080197*^9, 3.4244509849259233`*^9}, { 3.424451034517232*^9, 3.4244510369707603`*^9}, {3.456477808696907*^9, 3.456477810449427*^9}, {3.4564827586445827`*^9, 3.4564827613985424`*^9}, { 3.456483475194931*^9, 3.456483485199317*^9}, {3.456746633433714*^9, 3.456746637569661*^9}, {3.4567466905057793`*^9, 3.4567466984071407`*^9}}], Cell["\<\ The set of real exp-log functions is contained in the set of tame elementary \ functions, so the algorithm presented here can be used to find domains and \ zero sets of exp-log functions.\ \>", "Text", CellChangeTimes->{{3.4244505727933054`*^9, 3.4244506040182047`*^9}, { 3.456482789408819*^9, 3.4564828564251842`*^9}, {3.4564835495017796`*^9, 3.4564835931745777`*^9}, {3.4567462586948657`*^9, 3.456746333542491*^9}}], Cell["\<\ I have compared the performance of the two algorithms on a set of examples \ not involving trigonometric functions (the exp-log root isolation algorithm \ can be extended to handle inverse trigonometric functions). The results of \ the comparison are rather inconclusive. 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