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even bother with, but really, it is \ not. Mark Sofroniou made a nice ", ButtonBox["Wolfram Blog", BaseStyle->"Hyperlink", ButtonData->{ URL["http://blog.wolfram.com/"], None}], " post on \"", ButtonBox["Arithmetic Is Hard--To Get Right", BaseStyle->"Hyperlink", ButtonData->{ URL["http://blog.wolfram.com/2007/09/arithmetic_is_hardto_get_right.html?\ lid=perma"], None}], "\" recently. " }], "Text", CellChangeTimes->{{3.400602031859375*^9, 3.400602077859375*^9}, { 3.400602126453125*^9, 3.400602132140625*^9}, {3.4006021668125*^9, 3.400602166828125*^9}, {3.400602215375*^9, 3.40060224759375*^9}, { 3.400602689453125*^9, 3.400602690171875*^9}, 3.40060344615625*^9}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " supports arithmetic for four different types of explicit numbers, Integer, \ Rational, Real, and Complex. Exact representations of numbers like ", Cell[BoxData[ SqrtBox["2"]], CellChangeTimes->{3.40060274396875*^9}], "could be considered a fifth type, but those fall more under the purview of \ symbolic computation and numerical evaluation than arithmetic. " }], "Text", CellChangeTimes->{{3.400602254125*^9, 3.400602262453125*^9}, { 3.400602424375*^9, 3.400602424515625*^9}, {3.400602503015625*^9, 3.40060255603125*^9}, {3.40060262046875*^9, 3.400602644609375*^9}, { 3.400602681375*^9, 3.400602955859375*^9}, {3.400603093984375*^9, 3.400603104078125*^9}}], Cell[TextData[{ "Internally, the four types are actually broken into subtypes -- the most \ apparent of these subtypes are machine reals and integers. ", StyleBox["Mathematica", FontSlant->"Italic"], " uses machine hardware numbers when those are sufficient to represent the \ desired quantity to take advantage of the blinding speed of arithmetic on \ modern processors. 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This may seem like an obvious \ thing to you as ", StyleBox["Mathematica", FontSlant->"Italic"], " users, but not all systems support such a consistent model. The key \ things behind the coercion model are" }], "Text", CellChangeTimes->{{3.400603509265625*^9, 3.400603667140625*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Exact in ", StyleBox["\[DoubleRightArrow]", FontSize->16], " exact out" }], "Item", CellChangeTimes->{{3.40060367271875*^9, 3.4006037121875*^9}}], Cell["\<\ The precision of the result is based on the precision of the inputs -- \ coercion is done to achieve the best precision justifiable from the inputs.\ \>", "Item", CellChangeTimes->{{3.40060367271875*^9, 3.4006037121875*^9}, { 3.4006037860625*^9, 3.4006038485*^9}}], Cell["\<\ Precision of results is computed using significance arithmetic.\ \>", "Subitem", CellChangeTimes->{{3.40060367271875*^9, 3.4006037121875*^9}, { 3.4006037860625*^9, 3.4006038485*^9}, {3.4006040836875*^9, 3.400604096984375*^9}}], Cell["\<\ Machine numbers are considered to have less precision than any other numbers\ \>", "Item", CellChangeTimes->{{3.40060367271875*^9, 3.4006037121875*^9}, { 3.400603852015625*^9, 3.400603912984375*^9}, 3.400604285859375*^9}], Cell["\<\ When machine numbers are terms in an arithmetic operation, all the inputs \ will be coerced to machine numbers if possible.\ \>", "Subitem", CellChangeTimes->{{3.40060367271875*^9, 3.4006037121875*^9}, { 3.4006037860625*^9, 3.4006038485*^9}, {3.4006040836875*^9, 3.400604096984375*^9}, {3.40060427909375*^9, 3.40060428171875*^9}}] }, Open ]], Cell[TextData[{ "This coercion model applies generally to computations in ", StyleBox["Mathematica", FontSlant->"Italic"], ", not just arithmetic." }], "Text", CellChangeTimes->{{3.400604352890625*^9, 3.400604389125*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}], ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]], StyleBox["Contents", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "Significance arithmetic keeps track of justified precision using a first \ order approximation. It is not rigorous like interval arithmetic (also \ supported in ", StyleBox["Mathematica", FontSlant->"Italic"], " with Interval objects) but it is fast and, with reasonably high precision \ accurate as well. The ", StyleBox["Precision", "MR"], " command in ", StyleBox["Mathematica", FontSlant->"Italic"], " returns a measure of the relative size of the estimated error interval and \ the Accuracy command gives a measure of the absolute size of the error \ interval." }], "Text", CellChangeTimes->{{3.4006044123125*^9, 3.40060447353125*^9}, { 3.400604545546875*^9, 3.4006045486875*^9}, {3.4006046109375*^9, 3.400604734671875*^9}, {3.40060667746875*^9, 3.400606753625*^9}}], Cell["\<\ This gives the precision and accuracy for a number given with 30 decimal \ places.\ \>", "MathCaption", CellChangeTimes->{{3.400606765625*^9, 3.400606792359375*^9}, 3.400840229234375*^9}], Cell[BoxData[ RowBox[{ RowBox[{"x", " ", "=", " ", "0.999999999999999999999999999999"}], ";"}]], "Input", CellChangeTimes->{{3.40060680565625*^9, 3.4006068274375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Precision", "[", "x", "]"}], ",", " ", RowBox[{"Accuracy", "[", "x", "]"}], ",", " ", RowBox[{"Precision", "[", RowBox[{"32", " ", "x"}], "]"}], ",", " ", RowBox[{"Accuracy", "[", RowBox[{"32", " ", "x"}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.400606821484375*^9, 3.40060682325*^9}, { 3.4006068784375*^9, 3.400606890078125*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"30.`", ",", "30.`", ",", "30.`", ",", "28.494850021680094`"}], "}"}]], "Output", CellChangeTimes->{3.4018118728125*^9}] }, Open ]], Cell[TextData[{ "For the number x above, the error interval is [ \ 0.9999999999999999999999999999985, 0.9999999999999999999999999999995], having \ length ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalDelta]x", " ", "=", " ", SuperscriptBox["10", RowBox[{"-", "30"}]]}], TraditionalForm]]], ". 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The 0 in the output indicates that the error interval \ contains zero.\ \>", "MathCaption", CellChangeTimes->{{3.400604792796875*^9, 3.400604875375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"1", " ", "-", "x"}]], "Input", CellChangeTimes->{{3.400604745046875*^9, 3.400604767265625*^9}, 3.400607231578125*^9}], Cell[BoxData["0``30."], "Output", CellChangeTimes->{3.401811877265625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Precision", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.40060482765625*^9, 3.400604829453125*^9}}], Cell[BoxData["0.`"], "Output", CellChangeTimes->{3.40181187740625*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}], ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]], StyleBox["Contents", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ Most computations do not lose a lot of precision. In fact, some even gain \ precision.\ \>", "Text", CellChangeTimes->{{3.400604885578125*^9, 3.40060493134375*^9}}], Cell[TextData[{ "Repeatedly iterating the function ", Cell[BoxData[ FormBox[ RowBox[{"1", " ", "-", " ", RowBox[{"x", "/", "2"}]}], TraditionalForm]]], " gives a result with higher precision than the input." }], "MathCaption", CellChangeTimes->{{3.4006050023125*^9, 3.40060506859375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Nest", "[", RowBox[{ RowBox[{ RowBox[{"1", " ", "-", " ", RowBox[{"#", "/", "2"}]}], "&"}], ",", "0.999999999999999999999999999999", ",", " ", "1000"}], "]"}]], "Input", CellChangeTimes->{{3.400604563671875*^9, 3.400604596125*^9}, { 3.40060493903125*^9, 3.40060494415625*^9}, 3.40060516421875*^9, 3.40060618090625*^9}], Cell[BoxData["0.\ 666666666666666666666666666666666666666666666666666666666666666666666666666666\ 666666666666666666666666666666666666666666666666666666666666666666666666666666\ 666666666666666666666666666666666666666666666666666666666666666666666666666666\ 666666666666666666666666666666666666666666666666666666666666666666697775453950\ 1072959663363181573672459587148508`330.8539044049256"], "Output", CellChangeTimes->{3.401811882484375*^9}] }, Open ]], Cell[TextData[{ "For the example ", Cell[BoxData[ FormBox[ RowBox[{"1", " ", "-", " ", RowBox[{"x", "/", "2"}]}], TraditionalForm]]], ", you can easily see why there is a precision gain:\nSuppose that there is \ initially an error \[CapitalDelta]x (in this case about ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "30"}]], TraditionalForm]]], "). Then\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"y", " ", "+", " ", "\[CapitalDelta]y"}], " ", "=", " ", RowBox[{ RowBox[{"1", " ", "-", " ", RowBox[{ RowBox[{"(", RowBox[{"x", " ", "+", " ", "\[CapitalDelta]x"}], ")"}], "/", "2"}]}], " ", "=", " ", RowBox[{"1", " ", "-", " ", RowBox[{"x", "/", "2"}], " ", "-", " ", RowBox[{"\[CapitalDelta]x", "/", "2"}]}]}]}], TraditionalForm]]], "\n\nso the error in the result is about half that of the input. Its \ slightly greater than that because of the arithmetic of subtracting from 1 \ (since in ", StyleBox["Mathematica", FontSlant->"Italic"], " numbers are internally represented in binary, division by 2 introduces no \ error). 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Cell["\<\ In general, if you can use listability for operations on large lists, your \ computations will be much faster.\ \>", "Text", CellChangeTimes->{{3.4006145325*^9, 3.40061455840625*^9}, {3.400614602625*^9, 3.400614658796875*^9}}], Cell["\<\ Compare adding a large nested random array to itself using listability to \ using indexing.\ \>", "MathCaption", CellChangeTimes->{{3.40061480153125*^9, 3.400614851078125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"a", " ", "=", " ", RowBox[{"RandomReal", "[", RowBox[{"1", ",", RowBox[{"{", RowBox[{"100", ",", " ", "100", ",", " ", "100"}], "}"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.400614669515625*^9, 3.400614690640625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Timing", "[", RowBox[{ RowBox[{"aa1", " ", "=", " ", RowBox[{"a", " ", "+", " ", "a"}]}], ";"}], "]"}]], "Input", CellChangeTimes->{{3.400614704984375*^9, 3.400614712453125*^9}, { 3.400614768421875*^9, 3.400614769578125*^9}}], Cell[BoxData[ RowBox[{"{", 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Generally when an expression has the NumericQ \ attribute it will evaluate to a number inside of N. At a user level, this \ seems simple and natural, but there is a huge amount of machinery behind it. \ " }], "Text", CellChangeTimes->{{3.40060742396875*^9, 3.400607568484375*^9}}], Cell["\<\ Evaluate an integral to get a numeric expression representing the exact \ result. Then find the numerical value with N.\ \>", "MathCaption", CellChangeTimes->{{3.40060762153125*^9, 3.400607668140625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"in", " ", "=", " ", RowBox[{"Integrate", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"Exp", "[", "s", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"s", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.40060757215625*^9, 3.40060759778125*^9}, { 3.40060767946875*^9, 3.4006076860625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"-", RowBox[{"SinIntegral", "[", FractionBox["1", "\[ExponentialE]"], "]"}]}], "+", RowBox[{"SinIntegral", "[", "\[ExponentialE]", "]"}]}]], "Output", CellChangeTimes->{3.401811908078125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"NumericQ", "[", "in", "]"}], ",", " ", RowBox[{"N", "[", "in", "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.4006076896875*^9, 3.4006076994375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"True", ",", "1.455915572116364`"}], "}"}]], "Output", CellChangeTimes->{3.40181190846875*^9}] }, Open ]], Cell[TextData[{ "This example can be done because we have good numerical algorithms to \ compute the special sine integral function. ", StyleBox["Mathematica", FontSlant->"Italic"], " supports a many special functions all of which require several algorithms \ for computing the results in different ranges of numbers." }], "Text", CellChangeTimes->{{3.400607721265625*^9, 3.40060781546875*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}], ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]], StyleBox["Contents", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ The default for N is to use machine precision, using hardware numbers as much \ as possible. However, N can work adaptively to any precision, so you can get \ the result of the integral as precise as you'd like.\ \>", "Text", CellChangeTimes->{{3.4006078240625*^9, 3.400607865546875*^9}, { 3.400607926640625*^9, 3.400607982328125*^9}}], Cell["This gives the result of the integral to 100 digits.", "MathCaption", CellChangeTimes->{{3.40060799046875*^9, 3.40060800503125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"in", ",", "100"}], "]"}]], "Input", CellChangeTimes->{{3.400608009078125*^9, 3.400608011859375*^9}}], Cell[BoxData["1.\ 455915572116364038693979762323661101822670964425088970117118425476096775005050\ 5937579978041296905579178760401643953`100."], "Output", CellChangeTimes->{3.401811912046875*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}], ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]], StyleBox["Contents", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ Note that when functions, special or otherwise are evaluated, their \ conditioning is taken into account. This way , you get a precision out \ justified by the arguments in. The same thing happens for the hyperbolic \ function coth as for the example I showed before:\ \>", "Text", CellChangeTimes->{{3.40060822675*^9, 3.400608322921875*^9}, { 3.400781514875*^9, 3.40078152003125*^9}}], Cell[TextData[{ "This iterates the hyperbolic function ", Cell[BoxData[ FormBox[ RowBox[{"coth", "(", "x", ")"}], TraditionalForm]]], " 1000 times." }], "MathCaption", CellChangeTimes->{{3.400608327859375*^9, 3.400608373078125*^9}, { 3.40060861165625*^9, 3.400608617234375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Nest", "[", RowBox[{"Coth", ",", " ", "1.11111111111111111111", ",", "1000"}], "]"}]], "Input", CellChangeTimes->{{3.400608375625*^9, 3.400608404828125*^9}, { 3.40060860253125*^9, 3.40060860303125*^9}}], Cell[BoxData["1.\ 199678640257733833916369848641141944261458788418607208915477783918124725223847\ 479999086992146509379885905508415299288536097217804089692766839797586421045590\ 105855796472417575312855320890626919169865733244813105228707591474863036642744\ 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computing a definite \ integral for a function or finding the solution to a system of ordinary \ differential equations given a right hand side function. Typically these \ methods work by sampling the function at points determined by the algorithm \ and error is estimated based on an approximation the procedure uses. \ \>", "Text", CellChangeTimes->{{3.400616840140625*^9, 3.40061715090625*^9}}], Cell["\<\ Though the different sorts of tasks typically done are quite different, we \ have used several design principles to make the interfaces to these functions \ be as similar as possible.\ \>", "Text", CellChangeTimes->{{3.400617205734375*^9, 3.4006172206875*^9}, { 3.4006173066875*^9, 3.4006173345625*^9}, {3.400617374078125*^9, 3.40061739453125*^9}}], Cell[CellGroupData[{ Cell["\<\ Functions are task or problem based rather than algorithm based (e.g. \ FindRoot, NDSolve instead of Newton, VODE).\ \>", "Item", CellChangeTimes->{3.400617414046875*^9}], Cell[CellGroupData[{ Cell["\<\ Particular algorithms and methods can usually be selected and/or tuned with \ Method options.\ \>", "Subitem", CellChangeTimes->{{3.400617414046875*^9, 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This is indeed a very \ quick overview of numerical computation in ", StyleBox["Mathematica", FontSlant->"Italic"], ", but should be enough to give you a fell of how the system was thought out \ as a whole and different tasks can be combined to get otherwise difficult \ results. \nSome of the important points I have illustrated are:" }], "Text", CellChangeTimes->{{3.40078257959375*^9, 3.40078266090625*^9}, { 3.400782697609375*^9, 3.4007827051875*^9}, {3.40078277825*^9, 3.4007829575625*^9}, 3.400840370484375*^9}], Cell[CellGroupData[{ Cell[TextData[{ " In operations based on input values or data, ", StyleBox["Mathematica", FontSlant->"Italic"], " automatically returns a result based on the precision of the inputs." }], "Item", CellChangeTimes->{{3.400782979921875*^9, 3.400782984203125*^9}}], Cell["\<\ Functions are task or problem based rather than algorithm based.\ \>", "Item"], Cell["\<\ Functions try to pick a good algorithm based on the problem given.\ \>", "Item", 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