Example_code.nb

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or, if you want to load the source code, point the PacletManager at the \
cloned repository:\
\>", "Text",ExpressionUUID->"28f45967-93e7-4046-8b32-cf276cc7b22c"],

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Cell["Verify with a differential quotient:", \
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Calculate the derivatives of a 2-argument function and compare with the \
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Define a list of dual numbers. You can take the inner product like with \
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Cell["Utilities for working with dual numbers", \
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             GridBoxItemSize->{
              "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}],
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                    TemplateBox[{
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                    RGBColor[0.9686274509803922, 0.4666666666666667, 0.]}, 
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 StyleBox[\\\"a\\\", \\\"TI\\\"]], \\\"TI\\\"]\\), \
\\!\\(\\*StyleBox[SubscriptBox[\\\"f\\\", StyleBox[\\\"b\\\", \\\"TI\\\"]], \
\\\"TI\\\"]\\)}, Dual[\\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\), \
\\!\\(\\*StyleBox[\\\"b\\\", \\\"TI\\\"]\\)]] returns \
Dual[\\!\\(\\*StyleBox[SubscriptBox[\\\"f\\\", StyleBox[\\\"a\\\", \
\\\"TI\\\"]], \\\"TI\\\"]\\)[\\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\)], \
\\!\\(\\*StyleBox[SubscriptBox[\\\"f\\\", StyleBox[\\\"b\\\", \\\"TI\\\"]], \
\\\"TI\\\"]\\)[\\!\\(\\*StyleBox[\\\"b\\\", \
\\\"TI\\\"]\\)]].\\nDualApply[{\\!\\(\\*StyleBox[SubscriptBox[\\\"f\\\", \
StyleBox[\\\"All\\\", \\\"TI\\\"]], \\\"TI\\\"]\\)}, \
Dual[\\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\), \
\\!\\(\\*StyleBox[\\\"b\\\", \\\"TI\\\"]\\)]] returns \
Dual[\\!\\(\\*StyleBox[SubscriptBox[\\\"f\\\", StyleBox[\\\"All\\\", \\\"TI\\\
\"]], \\\"TI\\\"]\\)[\\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\), \
\\!\\(\\*StyleBox[\\\"b\\\", \\\"TI\\\"]\\)][[1]], \
\\!\\(\\*StyleBox[SubscriptBox[\\\"f\\\", StyleBox[\\\"All\\\", \\\"TI\\\"]], \
\\\"TI\\\"]\\)[\\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\), \
\\!\\(\\*StyleBox[\\\"b\\\", \\\"TI\\\"]\\)][[2]]]. \
\\!\\(\\*StyleBox[SubscriptBox[\\\"f\\\", StyleBox[\\\"All\\\", \\\"TI\\\"]], \
\\\"TI\\\"]\\) should return a List of length \
2.\\nDualApply[\\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\), \
Dual[\\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\), \
\\!\\(\\*StyleBox[\\\"b\\\", \\\"TI\\\"]\\)]] returns Dual[\\!\\(\\*StyleBox[\
\\\"f\\\", \\\"TI\\\"]\\)[\\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\)], \
\\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\)[\\!\\(\\*StyleBox[\\\"b\\\", \
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\\!\\(\\*StyleBox[\\\"b\\\", \\\"TI\\\"]\\)], \\!\\(\\*StyleBox[\\\"lvlSpec\\\
\", \\\"TI\\\"]\\)] maps the function(s) to deeper levels of \
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\\\"TI\\\"]\\).\\nDualApply[\\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\)] is \
the operator form of DualApply.\\nDualApply[\\!\\(\\*StyleBox[\\\"f\\\", \
\\\"TI\\\"]\\), \\!\\(\\*StyleBox[\\\"x\\\", \\\"TI\\\"]\\)] will use ToDual[\
\\!\\(\\*StyleBox[\\\"x\\\", \\\"TI\\\"]\\), 0] to cast standard quantities \
\\!\\(\\*StyleBox[\\\"x\\\", \\\"TI\\\"]\\) to duals.\"\>", 
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             StripOnInput->False,
             LineSpacing->{1.5, 1.5, 3.}],
            FrameMargins->{{10, 10}, {8, 10}}],
           BaseStyle->"InformationUsageSubtitleBackground",
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           BaseStyle->"InformationUsageSubtitleBackground",
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             ButtonFunction:>FEPrivate`Set[
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             Evaluator->Automatic,
             FrameMargins->{{9, 0}, {0, 0}},
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             ImageSize->Full,
             Method->"Preemptive"],
            
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              "MouseExited" :> 
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              Method -> "Preemptive", PassEventsDown -> Automatic, 
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        AutoDelete->False,
        FrameStyle->Directive[
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          Thickness[Tiny]],
        GridBoxAlignment->{"Columns" -> {Left, Right}, "Rows" -> {{Center}}},
        GridBoxDividers->{
         "Columns" -> {{None}}, "Rows" -> {False, {True}, False}},
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      FrameMargins->0,
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    BaseStyle->"InformationGridFrame",
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  InformationData[<|
   "ObjectType" -> "Symbol", "Usage" -> 
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\"TI\"]\), \!\(\*StyleBox[SubscriptBox[\"f\", StyleBox[\"b\", \"TI\"]], \
\"TI\"]\)}, Dual[\!\(\*StyleBox[\"a\", \"TI\"]\), \!\(\*StyleBox[\"b\", \
\"TI\"]\)]] returns Dual[\!\(\*StyleBox[SubscriptBox[\"f\", StyleBox[\"a\", \
\"TI\"]], \"TI\"]\)[\!\(\*StyleBox[\"a\", \"TI\"]\)], \
\!\(\*StyleBox[SubscriptBox[\"f\", StyleBox[\"b\", \"TI\"]], \
\"TI\"]\)[\!\(\*StyleBox[\"b\", \"TI\"]\)]].\n\
DualApply[{\!\(\*StyleBox[SubscriptBox[\"f\", StyleBox[\"All\", \"TI\"]], \
\"TI\"]\)}, Dual[\!\(\*StyleBox[\"a\", \"TI\"]\), \!\(\*StyleBox[\"b\", \
\"TI\"]\)]] returns Dual[\!\(\*StyleBox[SubscriptBox[\"f\", StyleBox[\"All\", \
\"TI\"]], \"TI\"]\)[\!\(\*StyleBox[\"a\", \"TI\"]\), \!\(\*StyleBox[\"b\", \
\"TI\"]\)][[1]], \!\(\*StyleBox[SubscriptBox[\"f\", StyleBox[\"All\", \
\"TI\"]], \"TI\"]\)[\!\(\*StyleBox[\"a\", \"TI\"]\), \!\(\*StyleBox[\"b\", \
\"TI\"]\)][[2]]]. \!\(\*StyleBox[SubscriptBox[\"f\", StyleBox[\"All\", \
\"TI\"]], \"TI\"]\) should return a List of length 2.\n\
DualApply[\!\(\*StyleBox[\"f\", \"TI\"]\), Dual[\!\(\*StyleBox[\"a\", \"TI\"]\
\), \!\(\*StyleBox[\"b\", \"TI\"]\)]] returns Dual[\!\(\*StyleBox[\"f\", \"TI\
\"]\)[\!\(\*StyleBox[\"a\", \"TI\"]\)], \!\(\*StyleBox[\"f\", \
\"TI\"]\)[\!\(\*StyleBox[\"b\", \"TI\"]\)]].\n\
DualApply[\!\(\*StyleBox[\"fspec\", \"TI\"]\), Dual[\!\(\*StyleBox[\"a\", \
\"TI\"]\), \!\(\*StyleBox[\"b\", \"TI\"]\)], \!\(\*StyleBox[\"lvlSpec\", \"TI\
\"]\)] maps the function(s) to deeper levels of \!\(\*StyleBox[\"a\", \"TI\"]\
\) and \!\(\*StyleBox[\"b\", \"TI\"]\).\nDualApply[\!\(\*StyleBox[\"f\", \"TI\
\"]\)] is the operator form of DualApply.\nDualApply[\!\(\*StyleBox[\"f\", \
\"TI\"]\), \!\(\*StyleBox[\"x\", \"TI\"]\)] will use \
ToDual[\!\(\*StyleBox[\"x\", \"TI\"]\), 0] to cast standard quantities \
\!\(\*StyleBox[\"x\", \"TI\"]\) to duals.", 
    "Documentation" -> <|"Local" -> "paclet:DualNumbers/ref/DualApply"|>, 
    "OwnValues" -> None, "UpValues" -> None, "DownValues" -> 
    Information`InformationValueForm[
     DownValues, DualNumbers`DualApply, {DualNumbers`DualApply[{
          Pattern[DualNumbers`Private`funa, 
           Blank[]], 
          Pattern[DualNumbers`Private`funb, 
           Blank[]]}, 
         DualNumbers`Dual[
          Pattern[DualNumbers`Private`a, 
           Blank[]], 
          Pattern[DualNumbers`Private`b, 
           Blank[]]]] :> DualNumbers`Dual[
         DualNumbers`Private`funa[DualNumbers`Private`a], 
         DualNumbers`Private`funb[DualNumbers`Private`b]], 
       DualNumbers`DualApply[{
          Pattern[DualNumbers`Private`funa, 
           Blank[]], 
          Pattern[DualNumbers`Private`funb, 
           Blank[]]}, 
         PatternTest[
          DualNumbers`Dual[
           Pattern[DualNumbers`Private`a, 
            Blank[]], 
           Pattern[DualNumbers`Private`b, 
            Blank[]]], DualNumbers`DualArrayQ], 
         Pattern[DualNumbers`Private`spec, 
          Blank[]]] :> DualNumbers`Dual[
         Map[
         DualNumbers`Private`funa, DualNumbers`Private`a, 
          DualNumbers`Private`spec], 
         Map[
         DualNumbers`Private`funb, DualNumbers`Private`b, 
          DualNumbers`Private`spec]], DualNumbers`DualApply[
         Pattern[DualNumbers`Private`fun, 
          Except[
           Blank[List]]], 
         DualNumbers`Dual[
          Pattern[DualNumbers`Private`a, 
           Blank[]], 
          Pattern[DualNumbers`Private`b, 
           Blank[]]]] :> DualNumbers`Dual[
         DualNumbers`Private`fun[DualNumbers`Private`a], 
         DualNumbers`Private`fun[DualNumbers`Private`b]], 
       DualNumbers`DualApply[
         Pattern[DualNumbers`Private`fun, 
          Except[
           Blank[List]]], 
         PatternTest[
          DualNumbers`Dual[
           Pattern[DualNumbers`Private`a, 
            Blank[]], 
           Pattern[DualNumbers`Private`b, 
            Blank[]]], DualNumbers`DualArrayQ], 
         Pattern[DualNumbers`Private`spec, 
          Blank[]]] :> DualNumbers`Dual[
         Map[
         DualNumbers`Private`fun, DualNumbers`Private`a, 
          DualNumbers`Private`spec], 
         Map[
         DualNumbers`Private`fun, DualNumbers`Private`b, 
          DualNumbers`Private`spec]], DualNumbers`DualApply[{
          Pattern[DualNumbers`Private`funAll, 
           Blank[]]}, 
         DualNumbers`Dual[
          Pattern[DualNumbers`Private`a, 
           Blank[]], 
          Pattern[DualNumbers`Private`b, 
           Blank[]]]] :> 
       With[{DualNumbers`Private`try = 
          DualNumbers`Private`funAll[
           DualNumbers`Private`a, DualNumbers`Private`b]}, 
         Condition[
          Apply[DualNumbers`Dual, DualNumbers`Private`try], 
          Replace[
           MatchQ[DualNumbers`Private`try, {
             Blank[], 
             Blank[]}], False :> (Message[
              MessageName[DualNumbers`DualApply, "resultlength"], 
              Short[DualNumbers`Private`funAll]]; False)]]], 
       DualNumbers`DualApply[{
          Pattern[DualNumbers`Private`funAll, 
           Blank[]]}, 
         DualNumbers`Dual[
          Pattern[DualNumbers`Private`a, 
           Blank[]], 
          Pattern[DualNumbers`Private`b, 
           Blank[]]], 0] :> 
       DualNumbers`DualApply[{DualNumbers`Private`funAll}, 
         DualNumbers`Dual[DualNumbers`Private`a, DualNumbers`Private`b]], 
       DualNumbers`DualApply[{
          Pattern[DualNumbers`Private`funAll, 
           Blank[]]}, 
         PatternTest[
          DualNumbers`Dual[
           Pattern[DualNumbers`Private`a, 
            Blank[]], 
           Pattern[DualNumbers`Private`b, 
            Blank[]]], DualNumbers`DualArrayQ], 
         PatternTest[
          Pattern[DualNumbers`Private`n, 
           Blank[Integer]], Positive]] :> 
       With[{DualNumbers`Private`try = 
          MapThread[
           DualNumbers`Private`funAll, {
            DualNumbers`Private`a, DualNumbers`Private`b}, 
            DualNumbers`Private`n]}, 
         Condition[
          DualNumbers`PackDualArray[
           Apply[DualNumbers`Dual, 
            Transpose[DualNumbers`Private`try, 
             TwoWayRule[1, DualNumbers`Private`n + 1]]]], 
          Replace[
           MatchQ[
            Dimensions[DualNumbers`Private`try], {
             Repeated[
              Blank[], {DualNumbers`Private`n}], 2, 
             BlankNullSequence[]}], False :> (Message[
              MessageName[DualNumbers`DualApply, "resultlength"], 
              Short[DualNumbers`Private`funAll]]; False)]]], 
       DualNumbers`DualApply[
         Pattern[DualNumbers`Private`f, 
          Blank[]], 
         Pattern[DualNumbers`Private`other, 
          Except[
           Blank[DualNumbers`Dual]]], 
         Pattern[DualNumbers`Private`rest, 
          BlankNullSequence[]]] :> 
       DualNumbers`DualApply[DualNumbers`Private`f, 
         DualNumbers`ToDual[DualNumbers`Private`other, 0], 
         DualNumbers`Private`rest], Condition[
         DualNumbers`DualApply[
          Pattern[DualNumbers`Private`f, 
           Blank[]], 
          PatternTest[
           Blank[DualNumbers`Dual], DualNumbers`DualScalarQ], 
          Pattern[DualNumbers`Private`spec, 
           Blank[]]], Message[
           MessageName[DualNumbers`DualApply, "arraySpec"], 
           Short[DualNumbers`Private`spec]]; False] :> Undefined, Condition[
         DualNumbers`DualApply[{
           Pattern[DualNumbers`Private`f, 
            Blank[]]}, 
          Blank[DualNumbers`Dual], 
          Pattern[DualNumbers`Private`spec, 
           Except[
            PatternTest[
             Blank[Integer], Positive]]]], Message[
           MessageName[DualNumbers`DualApply, "lvlSpec"], 
           Short[DualNumbers`Private`spec], 
           Short[{DualNumbers`Private`f}]]; False] :> Undefined}], 
    "SubValues" -> 
    Information`InformationValueForm[
     SubValues, DualNumbers`DualApply, {DualNumbers`DualApply[
         Pattern[DualNumbers`Private`fun, 
          Blank[]]][
         Pattern[DualNumbers`Private`d, 
          Blank[DualNumbers`Dual]]] :> 
       DualNumbers`DualApply[
        DualNumbers`Private`fun, DualNumbers`Private`d]}], "DefaultValues" -> 
    Information`InformationValueForm[
     DefaultValues, DualNumbers`DualApply, {
      SyntaxInformation[DualNumbers`DualApply] -> {"ArgumentsPattern" -> {
           BlankSequence[]}}}], "NValues" -> None, "FormatValues" -> None, 
    "Options" -> None, "Attributes" -> {}, "FullName" -> 
    "DualNumbers`DualApply"|>, 
   False]]], "Output",ExpressionUUID->"1076ef84-90bf-48a8-b930-e3e2158c387d"],

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          ItemBox[
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             BaseStyle -> None],
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          ItemBox[
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              Appearance->None,
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\"]\\), \\!\\(\\*StyleBox[SubscriptBox[\\\"f\\\", StyleBox[\\\"n\\\", \
\\\"TI\\\"]], \\\"TI\\\"]\\)}] specifies derivatives for \\!\\(\\*StyleBox[\\\
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\\!\\(\\*StyleBox[\\\"n\\\", \\\"TI\\\"]\\) \
arguments.\\nAddDualHandling[\\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\), \\!\
\\(\\*StyleBox[\\\"n\\\", \\\"TI\\\"]\\)] uses Derivative to infer \
derivatives of \\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\) for when \
\\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\) is called with \\!\\(\\*StyleBox[\
\\\"n\\\", \\\"TI\\\"]\\) \
arguments.\\nAddDualHandling[\\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\), \
{\\!\\(\\*StyleBox[SubscriptBox[\\\"n\\\", StyleBox[\\\"1\\\", \\\"TR\\\"]], \
\\\"TI\\\"]\\), \\!\\(\\*StyleBox[SubscriptBox[\\\"n\\\", StyleBox[\\\"2\\\", \
\\\"TR\\\"]], \\\"TI\\\"]\\), \\!\\(\\*StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\
\"]\\)}] uses Derivative to infer derivatives of \\!\\(\\*StyleBox[\\\"f\\\", \
\\\"TI\\\"]\\) for when \\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\) is called \
with \\!\\(\\*StyleBox[SubscriptBox[\\\"n\\\", StyleBox[\\\"1\\\", \
\\\"TR\\\"]], \\\"TI\\\"]\\), \\!\\(\\*StyleBox[SubscriptBox[\\\"n\\\", \
StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\"TI\\\"]\\), \[Ellipsis] arguments.\"\>",
              "InformationUsageText",
             StripOnInput->False,
             LineSpacing->{1.5, 1.5, 3.}],
            FrameMargins->{{10, 10}, {8, 10}}],
           BaseStyle->"InformationUsageSubtitleBackground",
           StripOnInput->False], 
          ItemBox["\<\"\"\>",
           BaseStyle->"InformationUsageSubtitleBackground",
           StripOnInput->False]},
         {
          PaneBox[
           TagBox[GridBox[{
              {
               StyleBox[
                TagBox[
                 TooltipBox[
                  StyleBox["\<\" Definitions\"\>", "InformationRowLabel",
                   StripOnInput->False],
                  "\"Definitions\"",
                  TooltipStyle->"TextStyling"],
                 Annotation[#, "Definitions", "Tooltip"]& ], "DialogStyle",
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