(*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 0, 0] NotebookDataLength[ 390777, 8523] NotebookOptionsPosition[ 361269, 8006] NotebookOutlinePosition[ 362491, 8040] CellTagsIndexPosition[ 362267, 8032] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["M\[EAcute]todos computacionais / 2019", "Title", CellChangeTimes->{{3.762026892275013*^9, 3.7620269012290277`*^9}, { 3.7620269647693653`*^9, 3.762026964881485*^9}, {3.762027090556736*^9, 3.762027091422873*^9}},ExpressionUUID->"f4434330-f7fd-4afb-9cf1-\ 244f47b9bf7a"], Cell["\<\ Davi C. 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I thought I would share the list of things that I look for first when \ trying to optimize ", StyleBox["Mathematica", FontSlant->"Italic"], " code." }], "Text", CellChangeTimes->{{3.527775826528042*^9, 3.5277759214260087`*^9}, { 3.530359517814147*^9, 3.5303595180325475`*^9}, {3.5303610411990232`*^9, 3.530361063850263*^9}, {3.530513926894823*^9, 3.5305139540055337`*^9}, { 3.531742933542495*^9, 3.531742945052101*^9}, {3.5317479944336233`*^9, 3.531747994435823*^9}},ExpressionUUID->"ed3b39d8-5989-4d1a-93c6-\ fd716b59eac0"], Cell[CellGroupData[{ Cell["\<\ 1. Use floating-point numbers if you can, and use them early.\ \>", "Section", CellChangeTimes->{{3.5184325766916385`*^9, 3.5184326012040896`*^9}, { 3.5277764737771792`*^9, 3.5277764806411915`*^9}, 3.5305143045665865`*^9, { 3.531672793467746*^9, 3.5316727974939337`*^9}, {3.53219463574444*^9, 3.5321946358290854`*^9}},ExpressionUUID->"26186c71-c331-4458-a136-\ c292b4b80778"], Cell[TextData[{ "Of the most common issues that I see when I review slow code is that the \ programmer has inadvertently asked ", StyleBox["Mathematica", FontSlant->"Italic"], " to do things more carefully than needed. Unnecessary use of exact \ arithmetic is the most common case. \n\nIn most numerical software, there is \ no such thing as exact arithmetic. 1/3 is the same thing as 0.33333333333333. \ That difference can be pretty important when you hit nasty, numerically \ unstable problems, but in the majority of tasks, floating-point numbers are \ good enough and, importantly, much faster. In ", StyleBox["Mathematica", FontSlant->"Italic"], " any number with a decimal point and less than 16 digits of input is \ automatically treated as a machine float, so always use the decimal point if \ you want speed ahead of accuracy (e.g. enter a third as 1./3.). Here is a \ simple example where working with floating-point numbers is nearly 40 times \ faster than doing the computation exactly and then converting the result to a \ decimal afterward. And in this case it gets the same result." }], "Text", CellChangeTimes->{{3.518432605860555*^9, 3.5184327617791452`*^9}, { 3.5184328061585827`*^9, 3.5184328256455317`*^9}, {3.51843663211314*^9, 3.5184366985567837`*^9}, {3.5184370154674716`*^9, 3.5184371794318666`*^9}, {3.518441461183013*^9, 3.518441509319826*^9}, { 3.5277759468540535`*^9, 3.5277759632184825`*^9}, {3.5277760053697567`*^9, 3.52777602446419*^9}, {3.5277761397015924`*^9, 3.5277761506216116`*^9}, { 3.5277762280757475`*^9, 3.527776234612159*^9}, {3.5277762752970304`*^9, 3.527776444277527*^9}, {3.5277765078008385`*^9, 3.5277767141902018`*^9}, { 3.5277767473246593`*^9, 3.5277767559202747`*^9}, {3.530340566284503*^9, 3.5303405944425526`*^9}, {3.5303595675470343`*^9, 3.530359568015035*^9}, { 3.5305139923153644`*^9, 3.5305141087000017`*^9}, {3.530524374523055*^9, 3.5305243925048532`*^9}, {3.5305245044970512`*^9, 3.530524505007102*^9}, { 3.531672819056211*^9, 3.531672901240128*^9}, {3.531673093175789*^9, 3.5316731107531424`*^9}, {3.5317429612986794`*^9, 3.5317429630512753`*^9}, 3.531747573816592*^9},ExpressionUUID->"38acd910-8d1a-4d53-be4c-\ 800013ec5332"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"N", "[", RowBox[{"Det", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{"1", "+", RowBox[{"Abs", "[", RowBox[{"i", "-", "j"}], "]"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "150"}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "1", ",", "150"}], "}"}]}], "]"}], "]"}], "]"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.5243691652914495`*^9, 3.5243692146967363`*^9}, { 3.5243692650224247`*^9, 3.5243692686572313`*^9}, {3.524369299077285*^9, 3.524369377935423*^9}, {3.524369408464677*^9, 3.5243695142172623`*^9}, { 3.524369545042917*^9, 3.524369557164138*^9}, {3.524369599377812*^9, 3.5243696320910697`*^9}, {3.530359617857123*^9, 3.53035962195993*^9}, { 3.5305141305271845`*^9, 3.53051413718285*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"9277e830-3da2-4176-98c1-29b544ba4835"], Cell[BoxData[ RowBox[{"{", RowBox[{"3.9469012`8.047801248639436", ",", "9.303106865686802`*^-21"}], "}"}]], "Output", CellChangeTimes->{3.5243696362094765`*^9, 3.5277766298554535`*^9, 3.5306113630871897`*^9}, CellLabel->"Out[1]=",ExpressionUUID->"6190f83f-4c5d-4777-b93d-d448b5f709a5"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Det", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{"1.", "+", RowBox[{"Abs", "[", RowBox[{"i", "-", "j"}], "]"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"i", ",", "1.", ",", "150."}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "1.", ",", "150."}], "}"}]}], "]"}], "]"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.5243692251175547`*^9, 3.5243692328707685`*^9}, { 3.52436933604935*^9, 3.5243693672182045`*^9}, {3.5243694006646633`*^9, 3.524369400992264*^9}, {3.5243695744957685`*^9, 3.5243695943234034`*^9}, { 3.530514140168148*^9, 3.5305141427684083`*^9}}, CellLabel->"In[2]:=",ExpressionUUID->"ee20ad1c-545b-494d-b178-16ab357dcb65"], Cell[BoxData[ RowBox[{"{", RowBox[{"0.078002`6.343650731799636", ",", "9.303106865686791`*^-21"}], "}"}]], "Output", CellChangeTimes->{{3.5243693440209637`*^9, 3.5243693678110056`*^9}, 3.5243694018346653`*^9, {3.5243695784425755`*^9, 3.5243695949474044`*^9}, 3.524369636240677*^9, 3.527776630027054*^9, 3.5306113670808926`*^9}, CellLabel->"Out[2]=",ExpressionUUID->"785f7e4d-bb64-4ef3-a430-77b1640dc7d6"] }, Open ]], Cell[TextData[{ "The same is true for symbolic computation. If you don\[CloseCurlyQuote]t \ care about the symbolic answer and are not worried about stability, then \ substitute numerical values as soon as you can. 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This takes away \ some of the flexibility of the ", StyleBox["Mathematica", FontSlant->"Italic"], " language, but freed from having to worry about \[OpenCurlyDoubleQuote]What \ if the argument was symbolic?\[CloseCurlyDoubleQuote] and the like, ", StyleBox["Mathematica", FontSlant->"Italic"], " can optimize the program and create a byte code to run on its own virtual \ machine. Not everything can be compiled, and very simple code might not \ benefit, but complex low-level numerical code can get a really big speedup.\n\ \nHere is an example:" }], "Text", CellChangeTimes->{{3.5184328813040967`*^9, 3.518433004069372*^9}, { 3.5184330371266775`*^9, 3.5184330725202165`*^9}, {3.518437191017025*^9, 3.5184372158425074`*^9}, {3.5277769857398787`*^9, 3.527777023616745*^9}, { 3.5277770569852037`*^9, 3.527777062070812*^9}, {3.5303597152958937`*^9, 3.5303597434695435`*^9}, {3.5305143610972385`*^9, 3.530514395800709*^9}, { 3.530523283221281*^9, 3.530523283223281*^9}, 3.530524415265129*^9, { 3.5306020390928507`*^9, 3.53060204798874*^9}, {3.530612934801506*^9, 3.530612943600132*^9}, {3.531674098234971*^9, 3.531674107936288*^9}, { 3.531674196407691*^9, 3.531674238835453*^9}, {3.53174763126728*^9, 3.53174763354841*^9}},ExpressionUUID->"2e0b123f-ba56-42d4-abc3-\ b3f57c2d6e11"], Cell[BoxData[ RowBox[{ RowBox[{"arg", "=", RowBox[{"Range", "[", " ", RowBox[{ RowBox[{"-", "50."}], 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FontColor->GrayLevel[0]], " some hints about the parallelizable nature of the code, getting an even \ better result." }], "Text", CellChangeTimes->{{3.527777120477315*^9, 3.527777171614205*^9}, { 3.5305144892520533`*^9, 3.530514490196148*^9}, {3.5316743837132587`*^9, 3.531674400120215*^9}, {3.5321897668806562`*^9, 3.532189768527027*^9}},ExpressionUUID->"287c1224-a0c1-4055-8e4f-\ 9ddb18985093"], Cell[BoxData[ RowBox[{ RowBox[{"cfn2", "=", RowBox[{"Compile", "[", RowBox[{ RowBox[{"{", "x", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"sum", "=", "1.0"}], ",", RowBox[{"inc", "=", "1.0"}]}], "}"}], ",", RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"inc", "=", RowBox[{"inc", "*", RowBox[{"x", "/", "i"}]}]}], ";", RowBox[{"sum", "=", RowBox[{"sum", "+", "inc"}]}]}], ",", RowBox[{"{", RowBox[{"i", ",", "10000"}], "}"}]}], "]"}], ";", "sum"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"RuntimeAttributes", "\[Rule]", RowBox[{"{", 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Cell[TextData[{ "On my dual-core machine I get a result 150 times faster than the original; \ the benefit would be even greater with more cores.\n\nBe aware though that \ many ", StyleBox["Mathematica", FontSlant->"Italic"], " functions like ", StyleBox[ButtonBox["Table", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Table.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Table.html"], "FunctionLink"], ", ", StyleBox[ButtonBox["Plot", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Plot.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Plot.html"], "FunctionLink"], ", ", Cell[BoxData[ FormBox[ StyleBox[ ButtonBox["NIntegrate", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/NIntegrate.html"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/NIntegrate.html"], "FunctionLink"], TraditionalForm]],ExpressionUUID-> "1a65418a-72b7-4d1f-8387-583a39754fb3"], ", and so on automatically compile their arguments, so you won\ \[CloseCurlyQuote]t see any improvement when passing them compiled versions \ of your code." }], "Text", CellChangeTimes->{{3.527777212236676*^9, 3.527777227087902*^9}, { 3.5303597859952183`*^9, 3.5303598085684576`*^9}, {3.5305145169098186`*^9, 3.530514517252853*^9}, {3.5306021942008357`*^9, 3.530602195375836*^9}, { 3.5306024188518357`*^9, 3.530602525372836*^9}, {3.5306122110233383`*^9, 3.5306122111325407`*^9}, {3.5316744746702957`*^9, 3.531674558539873*^9}, { 3.5316746028262177`*^9, 3.531674638336638*^9}, {3.531743017968853*^9, 3.531743027400189*^9}},ExpressionUUID->"98a97090-126c-4f31-8749-\ f2d1c6f42cfa"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "2.5. ...and use ", StyleBox["Compile", "Program", FontFamily->"Courier"], " to generate C code." }], "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5277751787213044`*^9, 3.52777518171651*^9}, {3.5303593967735343`*^9, 3.530359397085535*^9}, 3.531674663799675*^9, {3.532194643604498*^9, 3.532194643860396*^9}},ExpressionUUID->"b710fc8b-2444-4769-a5d2-\ 5fa09b6448bd"], Cell[TextData[{ "Furthermore, if your code is compilable, then you can also use the option \ ", StyleBox[ButtonBox["CompilationTarget", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/CompilationTarget.html"]\ , None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/CompilationTarget.html"], "FunctionLink"], StyleBox[ButtonBox["->\[CloseCurlyDoubleQuote]", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/CompilationTarget.html"]\ , None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/CompilationTarget.html"], FontColor->GrayLevel[0]], StyleBox[ButtonBox["C", BaseStyle->"Hyperlink", ButtonData->{ 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Use built-in functions.", "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, {3.51843359260822*^9, 3.5184336015751166`*^9}, 3.5184339650794635`*^9, {3.5184339985448093`*^9, 3.518434010359991*^9}, 3.518437409746896*^9, {3.5277749472450976`*^9, 3.5277749545459104`*^9}, { 3.52787108199835*^9, 3.52787108658235*^9}, {3.5303460271309757`*^9, 3.530346028035777*^9}, {3.5303582888429885`*^9, 3.530358288998989*^9}, 3.5303594077871537`*^9, 3.53052482310856*^9, 3.531675532835513*^9, { 3.532194646860272*^9, 3.532194647140368*^9}},ExpressionUUID->"172416fc-9ae3-4ad8-b1fb-\ 51aeef7720e6"], Cell[TextData[{ StyleBox["Mathematica ", FontSlant->"Italic"], "has a lot of functions. More than the average person would care to sit down \ and learn in one go. So it is not surprising that I often see code where \ someone has implemented some operation without having realized that ", StyleBox["Mathematica", FontSlant->"Italic"], " already knows how to do it. Not only is it a waste of time re-implementing \ work that is already done, but our guys are paid to worry about what the best \ algorithms are for different kinds of input and how to implement them \ efficiently, so most built-in functions are really fast.\n\nIf you find \ something close-but-not-quite-right, then check the options and optional \ arguments; often they generalize functions to cover many specialized uses or \ abstracted applications.\n\nHere is such an example. If I have a list of a \ million 2\[Times]2 matrices that I want to turn into a list of a million flat \ lists of 4 elements, the conceptually easiest way might be to ", StyleBox[ButtonBox["Map", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Map.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Map.html"], "FunctionLink"], " the basic ", StyleBox[ButtonBox["Flatten", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Flatten.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Flatten.html"], "FunctionLink"], " operation down the list of them. 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Use Wolfram ", StyleBox["Workbench.", FontSlant->"Italic"] }], "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, {3.51843359260822*^9, 3.5184336015751166`*^9}, 3.5184339650794635`*^9, {3.5184339985448093`*^9, 3.518434010359991*^9}, { 3.5184344107140226`*^9, 3.5184344162735786`*^9}, {3.518434642964245*^9, 3.5184346525882072`*^9}, 3.5184356995218906`*^9, 3.5303594030603456`*^9, { 3.5305233681152563`*^9, 3.5305233716189566`*^9}, {3.53052482626756*^9, 3.5305248263955603`*^9}, 3.531675879039959*^9, {3.532194649940309*^9, 3.532194650260353*^9}},ExpressionUUID->"b794ddc0-578b-4583-b81b-\ 8a92441feb36"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can be quite forgiving of some kinds of programming mistakes\[LongDash]it \ will proceed happily in symbolic mode if you forget to initialize a variable \ at the right point and doesn\[CloseCurlyQuote]t care about recursion or \ unexpected data types. That\[CloseCurlyQuote]s great when you just need to \ get a quick answer, but it will also let you get away with less than optimal \ solutions without realizing it. \n\n", StyleBox[ButtonBox["Workbench", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.wolfram.com/products/workbench/"], None}, ButtonNote->"http://www.wolfram.com/products/workbench/"], FontSlant->"Italic"], " helps in several ways. First it lets you debug and organize large code \ projects better, and having clean, organized code should make it easier to \ write good code. But the key feature in this context is the profiler that \ lets you see which lines of code used up the time, and how many times they \ were called.\n\nTake this example, a truly horrible way (computationally \ speaking) to implement ", StyleBox[ButtonBox["Fibonacci", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Fibonacci.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Fibonacci.html"], "FunctionLink"], " numbers. If you didn\[CloseCurlyQuote]t think about the consequences of \ the double recursion, you might be surprised by the 22 seconds it takes to \ evaluate ", StyleBox["fib[35]", "FunctionLink", FontColor->GrayLevel[0]], " (about the same time it takes the built-in function to calculate all \ 208,987,639 digits of ", StyleBox["Fibonacci[", "Program", FontSize->13], StyleBox["1000000000]", "Program", FontSize->13, FontColor->GrayLevel[0]], " [see tip 3])." }], "Text", CellChangeTimes->{{3.5184346553404827`*^9, 3.5184349880267477`*^9}, { 3.5243700356293783`*^9, 3.524370138386759*^9}, {3.527777512897004*^9, 3.5277776468076396`*^9}, {3.5277784575798283`*^9, 3.527778477189063*^9}, { 3.52787098652035*^9, 3.5278710025743504`*^9}, {3.530340802125717*^9, 3.530340832420971*^9}, {3.53035879086847*^9, 3.530358991609623*^9}, { 3.530359448050824*^9, 3.530359448378425*^9}, {3.530359870531767*^9, 3.530359913401642*^9}, 3.5303600660479097`*^9, {3.530523396650962*^9, 3.530523426082847*^9}, {3.5305234605467386`*^9, 3.530523465538737*^9}, { 3.530524424203023*^9, 3.530524435531155*^9}, 3.53052483691356*^9, 3.530611519157198*^9, {3.531675890293474*^9, 3.531675899367023*^9}, 3.531675993480646*^9, {3.53167604454745*^9, 3.531676044768195*^9}, { 3.531676671297133*^9, 3.531676683688246*^9}, {3.531748051119424*^9, 3.531748099238312*^9}, 3.5321897024672832`*^9, {3.532189800415896*^9, 3.5321898047183332`*^9}},ExpressionUUID->"83fac6b5-ae6a-4641-8d6f-\ 5d60142e3be9"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"fib", "[", "n_", "]"}], ":=", RowBox[{ RowBox[{"fib", "[", "n", "]"}], "=", " ", RowBox[{ RowBox[{"fib", "[", RowBox[{"n", "-", "1"}], "]"}], "+", RowBox[{"fib", "[", RowBox[{"n", "-", "2"}], "]"}]}]}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"fib", "[", "1", "]"}], "=", "1"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"fib", "[", "2", "]"}], "=", "1"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"fib", "[", "35", "]"}], ";"}], "//", "AbsoluteTiming"}]}], "Input",\ CellChangeTimes->{{3.52777924924942*^9, 3.527779265240448*^9}, { 3.527779371102234*^9, 3.527779385345059*^9}, 3.5277795004928617`*^9, { 3.5277795984620333`*^9, 3.5277796119716573`*^9}, {3.530519723846079*^9, 3.530519730908785*^9}, {3.771198353255514*^9, 3.771198359329719*^9}}, CellLabel->"In[5]:=",ExpressionUUID->"db4c3810-cd90-4909-ab8c-dd004f282fda"], Cell[BoxData[ RowBox[{"{", RowBox[{"0.000216`", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{3.5303600560326924`*^9, 3.5306115116846066`*^9, 3.7711983149541807`*^9, 3.771198363636553*^9}, CellLabel->"Out[8]=",ExpressionUUID->"a43e0d60-7cb9-4a8d-bc9d-d7d340beed5e"] }, Open ]], Cell[TextData[{ "Running the code in the profiler reveals the reason. The main rule is \ invoked 9,227,464 times, and the ", StyleBox["fib[1]", "Program"], " value is requested 18,454,929 times. \n\nBeing told what your code really \ does, rather than what you thought it would do, can be a real eye-opener." }], "Text", CellChangeTimes->{{3.5303589994720364`*^9, 3.5303590234960785`*^9}, { 3.530359095880206*^9, 3.530359122899453*^9}, {3.5305146837519493`*^9, 3.530514700326949*^9}, {3.5305218810197153`*^9, 3.530521881179747*^9}, { 3.5316768827508507`*^9, 3.531676887716514*^9}},ExpressionUUID->"e245515f-9596-459a-8e00-\ 70c133afbab4"] }, Open ]], Cell[CellGroupData[{ Cell["5. Remember values that you will need in the future.", "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, {3.51843359260822*^9, 3.5184336015751166`*^9}, 3.5184339650794635`*^9, {3.5184339985448093`*^9, 3.518434010359991*^9}, { 3.5184344107140226`*^9, 3.5184344162735786`*^9}, {3.518434642964245*^9, 3.5184346525882072`*^9}, 3.5184352176447077`*^9, {3.5184352961035523`*^9, 3.5184353023281746`*^9}, 3.5184357016731052`*^9, 3.518437411243045*^9, 3.530359412513962*^9, {3.5316769500052443`*^9, 3.531676952561431*^9}, { 3.5321946543323174`*^9, 3.5321946546200123`*^9}},ExpressionUUID->"c79d4664-468c-4f2c-87c6-\ cf7d69f31bce"], Cell[TextData[{ "This is good programming advice in any language. The ", StyleBox["Mathematica", FontSlant->"Italic"], " construct that you will want to know is this: " }], "Text", CellChangeTimes->{{3.5184353165765996`*^9, 3.518435349513893*^9}, { 3.5243729675803022`*^9, 3.5243729806531253`*^9}, {3.524373633030671*^9, 3.5243736686455336`*^9}, {3.530340912980512*^9, 3.5303409342121496`*^9}, { 3.5316769368044147`*^9, 3.531676937842472*^9}},ExpressionUUID->"6d9351c1-1937-4992-9575-\ 9a820b07b440"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{"f", "[", "x", "]"}], "=", RowBox[{"(*", RowBox[{"What", " ", "the", " ", "function", " ", "does"}], "*)"}]}]}]], "Input", CellChangeTimes->{{3.5184353522781696`*^9, 3.5184353964935904`*^9}, { 3.5303409417937627`*^9, 3.530340943572166*^9}, 3.532189855752719*^9},ExpressionUUID->"2987726e-3092-4a8b-8b63-\ 53a34e201067"], Cell[TextData[{ "It saves the result of calling ", StyleBox["f", "Program"], " on any value, so that if it is called again on the same value, ", StyleBox["Mathematica", FontSlant->"Italic"], " will not need to work it out again. You are trading speed for memory here, \ so it isn\[CloseCurlyQuote]t appropriate if your function is likely to be \ called for a huge number of values, but rarely the same ones twice. But if \ the possible input set is constrained, this can really help. Here it is \ rescuing the program that I used to illustrate tip 3. Change the first rule \ to this:" }], "Text", CellChangeTimes->{{3.5184353165765996`*^9, 3.518435472615202*^9}, { 3.527778935029668*^9, 3.5277790242024245`*^9}, {3.527779096524152*^9, 3.527779115852586*^9}, {3.527779325128953*^9, 3.5277793547690053`*^9}, { 3.527779435251547*^9, 3.527779479088624*^9}, {3.5277796736083655`*^9, 3.5277796827967815`*^9}, {3.52787105300035*^9, 3.52787106428935*^9}, { 3.530341107512854*^9, 3.53034118495339*^9}, {3.530341220973853*^9, 3.530341258413919*^9}, {3.5303412922659783`*^9, 3.530341330361245*^9}, { 3.53035917785835*^9, 3.530359236639253*^9}, {3.530359459329644*^9, 3.5303594594700446`*^9}, {3.530521961495807*^9, 3.530521972128933*^9}, 3.5305244507226744`*^9, {3.531677475540741*^9, 3.531677517727303*^9}, { 3.531743071628285*^9, 3.531743072704618*^9}},ExpressionUUID->"fbfe45b5-66cd-4060-9bdf-\ bec5e5f5a5a2"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"fib", "[", "n_", "]"}], ":=", RowBox[{ RowBox[{"fib", "[", "n", "]"}], "=", RowBox[{ RowBox[{"fib", "[", RowBox[{"n", "-", "1"}], "]"}], "+", RowBox[{"fib", "[", RowBox[{"n", "-", "2"}], "]"}]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.5277788789163694`*^9, 3.5277788808039722`*^9}, { 3.5277789223780456`*^9, 3.527778922440446*^9}, 3.527779032626439*^9, { 3.527779511444081*^9, 3.5277795152660875`*^9}, {3.5303603029831266`*^9, 3.530360306976733*^9}},ExpressionUUID->"1251ac82-1b25-4b83-bfad-\ 0904fd3b26c1"], Cell[TextData[{ "And it becomes immeasurably fast, since ", StyleBox["fib[35]", "Program"], " now only requires the main rule to be evaluated 33 times. Looking up \ previous results prevents the need to repeatedly recurse down to ", StyleBox["fib[1]", "Program"], "." }], "Text", CellChangeTimes->{{3.530359245640469*^9, 3.530359336697829*^9}, { 3.5303603139499454`*^9, 3.530360315431948*^9}, {3.5305219992911673`*^9, 3.530522001779416*^9}, 3.5305244533859406`*^9, {3.531677544672453*^9, 3.531677546398597*^9}},ExpressionUUID->"edc82d87-630c-4d62-807a-\ 17ffd1b56fd8"] }, Open ]], Cell[CellGroupData[{ Cell["6. Parallelize.", "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, 3.5184357047054086`*^9, 3.5303594140271645`*^9, 3.53167756333573*^9, {3.532194657980077*^9, 3.532194658340001*^9}},ExpressionUUID->"6e9ba304-6e28-4815-a78f-\ bcf969b5fa75"], Cell[TextData[{ "An increasing number of ", StyleBox["Mathematica", FontSlant->"Italic"], " operations will automatically parallelize over local cores (most linear \ algebra, image processing, and statistics), and, as we have seen, so does ", StyleBox["Compile", "Program", FontSize->13], " if manually requested. But for other operations, or if you want to \ parallelize over remote hardware, you can use the built-in parallel \ programming constructs.\n\nThere is a collection of tools for this, but for \ very independent tasks, you can get quite a long way with just ", StyleBox[ButtonBox["ParallelTable", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/ParallelTable.html"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/ParallelTable.html"], "FunctionLink"], ", ", StyleBox[ButtonBox["ParallelMap", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/ParallelMap.html"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/ParallelMap.html"], "FunctionLink"], ", and ", StyleBox[ButtonBox["ParallelTry", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/ParallelTry.html"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/ParallelTry.html"], "FunctionLink"], ". Each of these automatically takes care of communication, worker \ management, and collection of results. There is some overhead for sending the \ task and retrieving the result, so there is a trade-off of time gained versus \ time lost. Your ", StyleBox["Mathematica", FontSlant->"Italic"], " comes with four compute kernels, and you can scale up with ", Cell[BoxData[ FormBox[ ButtonBox[ StyleBox[ RowBox[{"grid", StyleBox[ AdjustmentBox["Mathematica", BoxMargins->{{-0.175, 0}, {0, 0}}], FontSlant->"Italic"]}]], BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.wolfram.com/gridmathematica/"], None}, ButtonNote->"http://www.wolfram.com/gridmathematica/"], TextForm]], ExpressionUUID->"bb491aaf-9296-48f4-8ca2-61696469b092"], " if you have access to additional CPU power. Here, ", StyleBox["ParallelTable", "Program", FontSize->13], " gives me double the performance, since it is running on my dual-core \ machine. More CPUs would give a better speedup." }], "Text", CellChangeTimes->{{3.518433176332597*^9, 3.5184331922131844`*^9}, { 3.518433226046568*^9, 3.5184334944344034`*^9}, {3.524373024239602*^9, 3.524373192454697*^9}, {3.52787122979035*^9, 3.52787126131735*^9}, 3.530341425584813*^9, {3.530360348628806*^9, 3.5303604619786053`*^9}, { 3.5303604934292607`*^9, 3.5303605518669634`*^9}, {3.530522023148553*^9, 3.530522094615699*^9}, {3.530523569557536*^9, 3.530523646549932*^9}, { 3.5305244691415157`*^9, 3.530524476598262*^9}, {3.530524528304432*^9, 3.5305245362752285`*^9}, {3.531677619814473*^9, 3.531677646530725*^9}, 3.531677676580605*^9, 3.5316777107945967`*^9, {3.531678816447279*^9, 3.531678876434552*^9}, {3.531748162780654*^9, 3.5317481711386147`*^9}, { 3.531748201910081*^9, 3.531748266921167*^9}},ExpressionUUID->"2ba55bf2-21fe-418d-8557-\ c8b75aadfcdf"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"PrimeQ", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"10", "^", "1000"}], ",", RowBox[{ RowBox[{"10", "^", "1000"}], "+", "5000"}]}], "}"}]}], "]"}], ";"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.5243732121575317`*^9, 3.524373343275762*^9}, { 3.5243734449099407`*^9, 3.5243735703965607`*^9}}, CellLabel->"In[24]:=",ExpressionUUID->"e329fe44-b889-407d-9bf7-c62377c7758c"], Cell[BoxData[ RowBox[{"{", RowBox[{"8.8298264`8.397497158651891", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{{3.5243732967252803`*^9, 3.524373343868563*^9}, { 3.524373447140744*^9, 3.524373549102524*^9}, 3.5243735809265795`*^9, 3.5306115423081913`*^9}, CellLabel->"Out[24]=",ExpressionUUID->"3c8e4b0f-b37b-4c7d-9d79-6b49cd60c217"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"ParallelTable", "[", RowBox[{ RowBox[{"PrimeQ", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"10", "^", "1000"}], ",", RowBox[{ RowBox[{"10", "^", "1000"}], "+", "5000"}]}], "}"}]}], "]"}], ";"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.524373555576535*^9, 3.524373577010973*^9}}, CellLabel->"In[25]:=",ExpressionUUID->"42b2e24c-7fe3-41cc-b4f3-f8af575a0382"], Cell[BoxData[ RowBox[{"{", RowBox[{"4.992128`8.149830705783524", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{{3.5243735603501434`*^9, 3.52437358682339*^9}, { 3.5306115497495823`*^9, 3.5306115770190816`*^9}}, CellLabel->"Out[25]=",ExpressionUUID->"51a5668a-0a9c-4fe3-b973-e5212d35df46"] }, Open ]], Cell[TextData[{ "Anything that ", StyleBox["Mathematica", FontSlant->"Italic"], " can do, it can also do in parallel. For example, you could send a set of \ parallel tasks to remote hardware, each of which compiles and runs in C or on \ a GPU." }], "Text", CellChangeTimes->{{3.5278712720849*^9, 3.527871382783969*^9}, { 3.5317430969336863`*^9, 3.5317430978538513`*^9}},ExpressionUUID->"1f1f216b-ce50-45b1-95db-\ f7693ef2c81c"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "6.5. Think about ", StyleBox["CUDALink", FontSlant->"Italic"], " and ", StyleBox["OpenCLLink.", FontSlant->"Italic"] }], "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, {3.51843359260822*^9, 3.5184336015751166`*^9}, 3.5184339650794635`*^9, {3.5184339985448093`*^9, 3.518434010359991*^9}, { 3.5184344107140226`*^9, 3.5184344162735786`*^9}, 3.5184357087058086`*^9, { 3.5184376952224407`*^9, 3.518437715879506*^9}, {3.5277750279908395`*^9, 3.52777502831844*^9}, 3.530357255416173*^9, 3.530359419845975*^9, 3.5317484703545103`*^9, 3.532194661676175*^9},ExpressionUUID->"30d63032-6c95-4efa-b59b-\ 70e247d9ac9c"], Cell[TextData[{ "If you have GPU hardware, there are some really fast things you can do with \ massive parallelization. Unless one of the built-in CUDA-optimized functions \ happens to do what you want, you will need to do a little work, but the ", StyleBox[ButtonBox["CUDALink", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/CUDALink/guide/CUDALink.\ html"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/CUDALink/guide/CUDALink.html"], FontSlant->"Italic"], " and ", StyleBox[ButtonBox["OpenCLLink", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/OpenCLLink/tutorial/\ Overview.html"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/OpenCLLink/tutorial/Overview.\ html"], FontSlant->"Italic"], " tools automate a lot of the messy details for you." }], "Text", CellChangeTimes->{{3.5184377188638043`*^9, 3.51843783222814*^9}, { 3.530357269159797*^9, 3.530357291857837*^9}, {3.5303605979514446`*^9, 3.5303606162502766`*^9}, {3.5305221452357597`*^9, 3.5305221600902452`*^9}, {3.530523711812379*^9, 3.530523756069953*^9}, 3.5316790559816103`*^9, {3.532189952347513*^9, 3.5321899524900312`*^9}, 3.532194702674841*^9},ExpressionUUID->"73fe3437-3587-45ca-ae2b-\ a20fefecef80"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "7. Use ", StyleBox["Sow", "Program"], " and ", StyleBox["Reap", "Program"], " to accumulate large amounts of data (not ", StyleBox["AppendTo", "Program"], ")." }], "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, {3.51843359260822*^9, 3.5184336015751166`*^9}, 3.5184339650794635`*^9, 3.5184357067776155`*^9, {3.5243737976139603`*^9, 3.524373800967966*^9}, {3.527774963952727*^9, 3.5277749888347707`*^9}, 3.530359421171977*^9, 3.531679133220194*^9, {3.532194664467731*^9, 3.532194664803981*^9}},ExpressionUUID->"f1a2a3e9-6c13-426a-bf0e-\ a5fcdc6ed94f"], Cell[TextData[{ "Because of the flexibility of ", StyleBox["Mathematica", FontSlant->"Italic"], " data structures, ", StyleBox[ButtonBox["AppendTo", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/AppendTo.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/AppendTo.html"], "FunctionLink"], " can\[CloseCurlyQuote]t assume that you will be appending a number, because \ you might equally append a document or a sound or an image. As a result, ", StyleBox["AppendTo", "Program", FontSize->13], " must create a fresh copy of all of the data, restructured to accommodate \ the appended information. This makes it progressively slower as the data \ accumulates. (And the construct ", StyleBox["data=Append[data,value] ", "Program"], "is the same as ", StyleBox["AppendTo", "Program"], ".)\n\nInstead use ", StyleBox[ButtonBox["Sow", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Sow.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Sow.html"], "FunctionLink"], " and ", StyleBox[ButtonBox["Reap", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Reap.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Reap.html"], "FunctionLink"], ". ", StyleBox["Sow", "Program", FontSize->13], " throws out the values that you want to accumulate, and ", StyleBox["Reap", "Program", FontSize->13], " collects them and builds a data object once at the end. The following are \ equivalent:" }], "Text", CellChangeTimes->{{3.5184336114471035`*^9, 3.518433776014559*^9}, { 3.518433815240481*^9, 3.5184339594619017`*^9}, {3.5243738166303935`*^9, 3.524373898546138*^9}, {3.5277797254160566`*^9, 3.527779816005416*^9}, { 3.5278714106257524`*^9, 3.527871464643154*^9}, {3.5303606431291237`*^9, 3.530360659821153*^9}, {3.530360694952415*^9, 3.530360701754027*^9}, { 3.5305237793076286`*^9, 3.530523813981161*^9}, {3.5305244834689484`*^9, 3.5305244930369053`*^9}, 3.531679158857614*^9, 3.531679205842345*^9, { 3.5316794108987722`*^9, 3.531679411456296*^9}, {3.5317431373249817`*^9, 3.531743138060718*^9}, {3.531743446629601*^9, 3.531743446874591*^9}, { 3.53174848476062*^9, 3.5317484949278803`*^9}},ExpressionUUID->"f60ca44e-dfd8-44e1-8b68-\ 45f9f6b68ff5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"data", "=", RowBox[{"{", "}"}]}], ";", RowBox[{"Do", "[", RowBox[{ RowBox[{"AppendTo", "[", RowBox[{"data", ",", RowBox[{"RandomReal", "[", "x", "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "40000"}], "}"}]}], "]"}], ";"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.524373834570425*^9, 3.524373855006461*^9}, { 3.5243739066581516`*^9, 3.5243739529434333`*^9}, {3.52777982410183*^9, 3.5277798557854853`*^9}, {3.5303607149984503`*^9, 3.530360719912459*^9}}, CellLabel->"In[26]:=",ExpressionUUID->"bb8660e4-9a70-4af1-baba-9ed9edf14c95"], Cell[BoxData[ RowBox[{"{", RowBox[{"5.8813508`8.221022077669412", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{ 3.5243739542538357`*^9, {3.5277798433366632`*^9, 3.5277798621190968`*^9}, 3.5306116085006886`*^9}, CellLabel->"Out[26]=",ExpressionUUID->"a7621763-5cb5-4400-a38d-ed39fde6397d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"data", "=", RowBox[{ RowBox[{"Reap", "[", RowBox[{"Do", "[", RowBox[{ RowBox[{"Sow", "[", RowBox[{"RandomReal", "[", "x", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "40000"}], "}"}]}], "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], ";"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.5243739588090434`*^9, 3.524373991584701*^9}, { 3.524374037667182*^9, 3.5243740382755833`*^9}, 3.527779831168642*^9, { 3.527779867282706*^9, 3.527779868577508*^9}, {3.530360723001264*^9, 3.530360725965269*^9}}, CellLabel->"In[27]:=",ExpressionUUID->"4bcbe873-4003-48f5-9072-de6c9200ccd2"], Cell[BoxData[ RowBox[{"{", RowBox[{"0.1092028`6.489778767477871", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{3.5243739923803024`*^9, 3.5243740431115913`*^9, 3.5277798698567104`*^9, 3.5306116097799215`*^9}, CellLabel->"Out[27]=",ExpressionUUID->"64124e99-6f2a-4146-8316-c2d385ac18ff"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "8. Use ", StyleBox["Block", "Program"], " or ", StyleBox["With", "Program"], " rather than ", StyleBox["Module.", "Program"] }], "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, {3.51843359260822*^9, 3.5184336015751166`*^9}, 3.5184339650794635`*^9, 3.5184357067776155`*^9, {3.518437281637086*^9, 3.5184372932532473`*^9}, 3.518437693239242*^9, {3.527872582324911*^9, 3.527872614625141*^9}, 3.5303594262887864`*^9, 3.5316794316481857`*^9, { 3.532194667787909*^9, 3.5321946680677223`*^9}},ExpressionUUID->"f6f2d7ac-a0fd-4f74-bc90-\ e83275846225"], Cell[TextData[{ StyleBox[ButtonBox["Block", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Block.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Block.html"], "FunctionLink"], ", ", StyleBox[ButtonBox["With", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/With.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/With.html"], "FunctionLink"], StyleBox[ButtonBox[",", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/With.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/With.html"], FontColor->GrayLevel[0]], " and ", StyleBox[ButtonBox["Module", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Module.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Module.html"], "FunctionLink"], " are all localization constructs with slightly different properties. In my \ experience, ", StyleBox["Block", "Program"], " and ", StyleBox[ButtonBox["Module", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Module.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Module.html"], "Program", FontColor->GrayLevel[0]], " are interchangeable in at least 95% of code that I write, but ", StyleBox[ButtonBox["Block", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Block.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Block.html"], "Program", FontColor->GrayLevel[0]], " is usually faster, and in some cases ", StyleBox[ButtonBox["With", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/With.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/With.html"], "Program", FontColor->GrayLevel[0]], " (effectively ", StyleBox[ButtonBox["Block", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Block.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Block.html"], "Program", FontColor->GrayLevel[0]], " with the variables in a read-only state) is faster still." }], "Text", CellChangeTimes->{{3.518437300237946*^9, 3.518437357447666*^9}, { 3.5243806647037907`*^9, 3.524380666279393*^9}, {3.5278726173154097`*^9, 3.527872785452222*^9}, {3.5278728782995057`*^9, 3.5278728802617016`*^9}, { 3.530523825668992*^9, 3.5305239185667014`*^9}, 3.5305244961802197`*^9, 3.53167951187016*^9, {3.53167957045483*^9, 3.531679620881929*^9}},ExpressionUUID->"b892fb27-0fcf-4e2a-a01c-\ 02373a1953fe"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"x", "=", "2."}], "}"}], ",", RowBox[{"1", "/", "x"}]}], "]"}], ",", RowBox[{"{", "1000000", "}"}]}], "]"}], ";"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.5243805877010555`*^9, 3.5243806423645515`*^9}, { 3.5278720387095547`*^9, 3.5278720569773817`*^9}, {3.530523923293229*^9, 3.530523926092949*^9}}, CellLabel->"In[28]:=",ExpressionUUID->"b8c65504-4942-4a8b-8bfc-6eb826da8dab"], Cell[BoxData[ RowBox[{"{", RowBox[{"4.1497064`8.069562364094677", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{{3.524380588761857*^9, 3.5243806471225595`*^9}, { 3.5278720460772915`*^9, 3.527872062311915*^9}, 3.527872809805657*^9, 3.530611616924905*^9}, CellLabel->"Out[28]=",ExpressionUUID->"b3f0c017-65f2-409f-9769-8387ad0fcdb6"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{"Block", "[", RowBox[{ RowBox[{"{", RowBox[{"x", "=", "2."}], "}"}], ",", RowBox[{"1", "/", "x"}]}], "]"}], ",", RowBox[{"{", "1000000", "}"}]}], "]"}], ";"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.5278725688375626`*^9, 3.5278725695176306`*^9}, { 3.5305239275488033`*^9, 3.5305239307644815`*^9}}, CellLabel->"In[29]:=",ExpressionUUID->"1606768e-bac8-4f38-92dd-71d56acab6f3"], Cell[BoxData[ RowBox[{"{", RowBox[{"1.4664376`7.617808581063314", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{3.5278725714598246`*^9, 3.527872802708947*^9, 3.530611620294591*^9}, CellLabel->"Out[29]=",ExpressionUUID->"dbfc979a-45be-4ad0-98ff-75b34a89f1b0"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["9. Go easy on pattern matching.", "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, {3.51843359260822*^9, 3.5184336015751166`*^9}, 3.5184339650794635`*^9, {3.5184339985448093`*^9, 3.518434010359991*^9}, { 3.5184344107140226`*^9, 3.5184344162735786`*^9}, 3.5184357087058086`*^9, { 3.5184376952224407`*^9, 3.5184376952624445`*^9}, 3.530359431389995*^9, 3.5316796462594767`*^9, {3.532194671515539*^9, 3.532194672123567*^9}},ExpressionUUID->"cee14ecd-9ff9-46dc-a9ae-\ 94f6530ee016"], Cell[TextData[{ "Pattern matching is great. It can make complicated tasks easy to program. \ But it isn\[CloseCurlyQuote]t always fast, especially the fuzzier patterns \ like ", StyleBox[ButtonBox["BlankNullSequence", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/BlankNullSequence.html"]\ , None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/BlankNullSequence.html"], "FunctionLink"], " (usually written as \[OpenCurlyDoubleQuote]___\[CloseCurlyDoubleQuote]), \ which can search long and hard through your data for patterns that you, as a \ programmer, might already know will never be there. If execution speed \ matters, use tighter patterns, or none at all.\n\nAs an example, here is a \ rather neat way to implement a ", ButtonBox["bubble sort", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Bubble_sort"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Bubble_sort"], " in a single line of code using patterns:" }], "Text", CellChangeTimes->{{3.518434418817833*^9, 3.5184345766566153`*^9}, { 3.518435731050043*^9, 3.5184357391068487`*^9}, {3.527780038727007*^9, 3.527780042190213*^9}, {3.527780462290351*^9, 3.5277804794503813`*^9}, { 3.5303607680425434`*^9, 3.530360778088961*^9}, {3.530522259777213*^9, 3.5305222949127264`*^9}, {3.530523953719186*^9, 3.530523987481809*^9}, { 3.5305240938919945`*^9, 3.5305240938939953`*^9}, {3.531679653305306*^9, 3.5316796874898033`*^9}, {3.531748537321587*^9, 3.531748544038336*^9}},ExpressionUUID->"e149e072-2194-49ed-9027-\ 4c8dc4dd8711"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"data", "=", 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that I was \ taught when I first learned programming:\ \>", "Text", CellChangeTimes->{{3.5277804894187984`*^9, 3.5277804903548007`*^9}, { 3.527871526839373*^9, 3.527871586082297*^9}, {3.530524104351041*^9, 3.5305241052621317`*^9}, {3.531679712361116*^9, 3.531679718590417*^9}},ExpressionUUID->"6a5191d6-7217-4dc4-9aed-\ f504e80da1f8"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"flag", "=", "True"}], ";", "\[IndentingNewLine]", RowBox[{"While", "[", RowBox[{ RowBox[{"TrueQ", "[", "flag", "]"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"flag", "=", "False"}], ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{"If", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"data", "[", RowBox[{"[", "i", "]"}], "]"}], ">", RowBox[{"data", "[", RowBox[{"[", RowBox[{"i", "+", "1"}], "]"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"temp", "=", RowBox[{"data", "[", RowBox[{"[", "i", "]"}], "]"}]}], ";", "\[IndentingNewLine]", 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Try doing things differently.", "Section", CellChangeTimes->{{3.5184328635503216`*^9, 3.518432872207187*^9}, { 3.5184330767446384`*^9, 3.5184330886728315`*^9}, {3.5184331625812216`*^9, 3.5184331685798216`*^9}, {3.51843359260822*^9, 3.5184336015751166`*^9}, 3.5184339650794635`*^9, {3.5184339985448093`*^9, 3.518434010359991*^9}, { 3.5184344107140226`*^9, 3.5184344162735786`*^9}, {3.518434642964245*^9, 3.5184346525882072`*^9}, {3.5184349968196273`*^9, 3.5184350186038055`*^9}, {3.5184352192598686`*^9, 3.518435228764819*^9}, { 3.5184359876056957`*^9, 3.5184359937653117`*^9}, 3.518437699558874*^9, 3.530359380175105*^9, {3.530359435898403*^9, 3.5303594361792035`*^9}, 3.531679833693572*^9, {3.532194675259603*^9, 3.532194675507477*^9}},ExpressionUUID->"50e652d4-5e7a-43ea-9cfa-\ 47530b22c7f2"], Cell[TextData[{ "One of ", StyleBox["Mathematica", FontSlant->"Italic"], "\[CloseCurlyQuote]s great strengths is that it can tackle the same problem \ in different ways. It allows you to program the way you think, as opposed to \ reconceptualizing the problem for the style of the programming language. \ However, conceptual simplicity is not always the same as computational \ efficiency. Sometimes the easy-to-understand idea does more work than is \ necessary." }], "Text", CellChangeTimes->{{3.527872353249006*^9, 3.5278723547451553`*^9}, 3.5278723880694876`*^9, {3.5303453275701466`*^9, 3.530345371484224*^9}, { 3.5303454151955004`*^9, 3.5303454919388356`*^9}, 3.530360820287035*^9, { 3.5305225650197344`*^9, 3.530522607413973*^9}, {3.5316798415900917`*^9, 3.531679856954521*^9}, 3.531748642644395*^9, {3.532189993309868*^9, 3.532189995145186*^9}},ExpressionUUID->"5b9dd76e-4880-4e06-af92-\ 4f3669262795"], Cell[TextData[{ "But another issue is that because special optimizations and smart \ algorithms are applied automatically in ", StyleBox["Mathematica", FontSlant->"Italic"], ", it is often hard to predict when something clever is going to happen. For \ example, here are two ways of calculating factorial, but the second is over \ 10 times faster. 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You might guess that the ", StyleBox[ButtonBox["Do", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Do.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Do.html"], "FunctionLink"], " loop is slow, or all those ", ButtonBox["assignments", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/Set.html"], None}, ButtonNote->"http://reference.wolfram.com/mathematica/ref/Set.html"], " to ", StyleBox["temp", FontSlant->"Italic"], " take time, or that there is something else \[OpenCurlyDoubleQuote]wrong\ \[CloseCurlyDoubleQuote] with the first implementation, but the real reason \ is probably quite unexpected. 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There are lots of such pieces of hidden magic in ", StyleBox["Mathematica", FontSlant->"Italic"], ", and more get added with each release.\n\nOf course the best way here is \ to use the built-in function (tip 3 again):" }], "Text", CellChangeTimes->{ 3.5303448042662277`*^9, {3.5303449355570583`*^9, 3.530345169027068*^9}, { 3.5303455814059925`*^9, 3.530345590313608*^9}, {3.530359476598875*^9, 3.5303594769108753`*^9}, {3.53036087421633*^9, 3.5303608857447505`*^9}, { 3.5303609168356047`*^9, 3.530360918988409*^9}, {3.530522660678299*^9, 3.530522671361514*^9}, {3.530522705539765*^9, 3.530522710899372*^9}, 3.5305227568711596`*^9, {3.530522797288281*^9, 3.5305228508133335`*^9}, { 3.5305241447360787`*^9, 3.5305241642710323`*^9}, {3.5305242680294065`*^9, 3.530524275408145*^9}, 3.5305248370995603`*^9, 3.531679932804001*^9, { 3.531679962822941*^9, 3.531680070139244*^9}, 3.5321900645443707`*^9},ExpressionUUID->"c66a6136-6edc-4212-b9b6-\ d094a4e756bc"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"AbsoluteTiming", "[", RowBox[{ RowBox[{"65536", "!"}], ";"}], "]"}]], "Input", CellChangeTimes->{{3.530344282302911*^9, 3.530344288776922*^9}}, CellLabel->"In[37]:=",ExpressionUUID->"1e533cf8-d97c-4222-8973-b70d34274031"], Cell[BoxData[ RowBox[{"{", RowBox[{"0.0156004`5.644680727463611", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{3.5303442892137227`*^9, 3.5303443528794346`*^9, 3.530611725269683*^9}, CellLabel->"Out[37]=",ExpressionUUID->"b101afc8-0cc8-4bdf-b570-bb4a04dcf88d"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is capable of superb computational performance, and also superb robustness \ and accuracy, but not always both at the same time. 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diagnostic_file = (Default)\\n refresh = 100 \ (Default)\\n\\n------------------------------------------------------------\\\ nEXPERIMENTAL ALGORITHM:\\n This procedure has not been thoroughly tested \ and may be unstable\\n or buggy. The interface is subject to \ change.\\n------------------------------------------------------------\\n\\n\\\ n\\nGradient evaluation took 1.5e-05 seconds\\n1000 transitions using 10 \ leapfrog steps per transition would take 0.15 seconds.\\nAdjust your \ expectations accordingly!\\n\\n\\nBegin eta adaptation.\\nIteration: 1 / \ 250 [ 0%] (Adaptation)\\nIteration: 50 / 250 [ 20%] \ (Adaptation)\\nIteration: 100 / 250 [ 40%] (Adaptation)\\nIteration: 150 / \ 250 [ 60%] (Adaptation)\\nSuccess! Found best value [eta = 10] earlier than \ expected.\\n\\nBegin stochastic gradient ascent.\\n iter ELBO \ delta_ELBO_mean delta_ELBO_med notes \\n 100 -203.793 \ 1.000 1.000\\n 200 -92.425 1.102 \ 1.205\\n 300 -74.748 0.814 1.000\\n 400 \ -59.214 0.676 1.000\\n 500 -51.906 \ 0.569 0.262\\n 600 -47.135 \ 0.491 0.262\\n 700 -47.779 0.423 \ 0.236\\n 800 -55.654 0.388 0.236\\n 900 \ -47.277 0.364 0.177\\n 1000 -47.187 \ 0.328 0.177\\n 1100 -66.118 0.257 \ 0.177\\n 1200 -53.570 0.160 \ 0.177\\n 1300 -60.720 0.148 0.141\\n 1400 \ -58.112 0.126 0.141\\n 1500 -66.765 \ 0.125 0.130\\n 1600 -48.800 0.151 \ 0.141\\n 1700 -65.511 0.176 \ 0.177\\n 1800 -51.127 0.190 0.234\\n 1900 \ -45.029 0.185 0.234\\n 2000 -45.918 \ 0.187 0.234\\n 2100 -45.157 0.160 \ 0.135\\n 2200 -50.209 0.147 \ 0.130\\n 2300 -47.776 0.140 0.130\\n 2400 \ -43.555 0.145 0.130\\n 2500 -49.649 \ 0.145 0.123\\n 2600 -48.829 0.110 \ 0.101\\n 2700 -44.125 0.095 \ 0.101\\n 2800 -47.077 0.073 0.097\\n 2900 \ -45.373 0.063 0.063\\n 3000 -46.760 \ 0.064 0.063\\n 3100 -46.524 0.063 \ 0.063\\n 3200 -44.714 0.057 \ 0.051\\n 3300 -44.017 0.053 0.040\\n 3400 \ -46.968 0.050 0.040\\n 3500 -46.266 \ 0.039 0.038\\n 3600 -46.776 0.039 \ 0.038\\n 3700 -47.046 0.029 \ 0.030\\n 3800 -44.071 0.029 0.030\\n 3900 \ -47.973 0.033 0.030\\n 4000 -45.675 \ 0.036 0.040\\n 4100 -56.085 0.054 \ 0.050\\n 4200 -44.926 0.074 \ 0.063\\n 4300 -50.123 0.083 0.067\\n 4400 \ -43.795 0.091 0.081\\n 4500 -45.084 \ 0.093 0.081\\n 4600 -44.961 0.092 \ 0.081\\n 4700 -44.461 0.092 \ 0.081\\n 4800 -43.472 0.088 0.081\\n 4900 \ -44.613 0.082 0.050\\n 5000 -44.176 \ 0.078 0.029\\n 5100 -48.106 0.068 \ 0.029\\n 5200 -45.437 0.049 \ 0.029\\n 5300 -46.797 0.041 0.029\\n 5400 \ -48.534 0.031 0.029\\n 5500 -46.544 \ 0.032 0.029\\n 5600 -51.182 0.041 \ 0.036\\n 5700 -43.673 0.057 \ 0.043\\n 5800 -72.604 0.094 0.059\\n 5900 \ -50.377 0.136 0.082\\n 6000 -43.840 \ 0.150 0.091\\n 6100 -46.651 0.148 \ 0.091\\n 6200 -43.322 0.150 \ 0.091\\n 6300 -44.351 0.149 0.091\\n 6400 \ -45.429 0.148 0.091\\n 6500 -44.583 \ 0.145 0.091\\n 6600 -47.587 0.143 \ 0.077\\n 6700 -46.078 0.129 \ 0.063\\n 6800 -44.617 0.092 0.060\\n 6900 \ -43.343 0.051 0.033\\n 7000 -43.572 \ 0.037 0.033\\n 7100 -43.586 0.031 \ 0.029\\n 7200 -44.674 0.025 \ 0.024\\n 7300 -43.995 0.025 0.024\\n 7400 \ -48.740 0.032 0.029\\n 7500 -44.784 \ 0.039 0.033\\n 7600 -45.071 0.033 \ 0.029\\n 7700 -44.939 0.030 \ 0.024\\n 7800 -44.673 0.028 0.015\\n 7900 \ -44.022 0.026 0.015\\n 8000 -45.451 \ 0.029 0.015\\n 8100 -44.070 0.032 \ 0.024\\n 8200 -44.866 0.031 \ 0.018\\n 8300 -46.729 0.034 0.031\\n 8400 \ -46.338 0.025 0.018\\n 8500 -43.761 \ 0.022 0.018\\n 8600 -43.867 0.021 \ 0.018\\n 8700 -46.412 0.027 \ 0.031\\n 8800 -43.567 0.033 0.031\\n 8900 \ -44.834 0.034 0.031\\n 9000 -43.922 \ 0.033 0.031\\n 9100 -47.031 0.036 \ 0.040\\n 9200 -51.890 0.044 \ 0.055\\n 9300 -47.337 0.049 0.059\\n 9400 \ -45.950 0.052 0.059\\n 9500 -44.222 \ 0.050 0.055\\n 9600 -49.575 0.060 \ 0.065\\n 9700 -45.020 0.065 \ 0.066\\n 9800 -44.430 0.060 0.066\\n 9900 \ -43.766 0.058 0.066\\n 10000 -43.845 \ 0.056 0.066\\nInformational Message: The maximum number \ of iterations is reached! 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