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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 258187, 9030]*) (*NotebookOutlinePosition[ 258844, 9054]*) (* CellTagsIndexPosition[ 258800, 9050]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "How and why?\n", Cell[BoxData[ FormBox[ StyleBox[\(Mathematica\ Techniques\), "Subtitle", FontSize->18, FontWeight->"Plain"], TraditionalForm]]], "\n", StyleBox["By Allan Hayes", FontSize->14] }], "Title"], Cell["\<\ For this issue I look at timing computations, using patterns, \ proving equality, getting the axes to go through the origin (in 2D) and \ precise evaluation.\ \>", "Text"], Cell[CellGroupData[{ Cell["Timing", "Section"], Cell["\<\ I shall be timing computations a lot in this issue; so to reduce \ the clutter I define a utility. \ \>", "Text"], Cell[BoxData[ \(\(Tim\ = \ Function[x, First[Timing[x]], \ HoldAll]; \)\)], "Input"], Cell["which gives me", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Tim[Integrate[Sum[Cos[n\ x], {n, 0, 10}], x]]\)], "Input"], Cell[BoxData[ \(1.26000000000203726`\ Second\)], "Output"] }, Open ]], Cell[TextData[{ "The attribute ", StyleBox["HoldAll", "Input", FontWeight->"Plain"], " is essential: without it we get" }], "Text"], Cell[BoxData[ \(\(Tim2\ = \ Function[x, First[Timing[x]]]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Tim2[Integrate[Sum[Cos[n\ x], {n, 0, 10}], x]]\)], "Input"], Cell[BoxData[ \(0.`\ Second\)], "Output"] }, Open ]], Cell["\<\ We cannot set attributes for a pure function in the \"usual\" way: \ \ \>", "Text"], Cell[BoxData[ \(SetAttributes[Tim3, HoldAll]; \n Tim3\ = \ Function[x, First[Timing[x]]]; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Tim3[Integrate[Sum[Cos[n\ x], {n, 0, 10}], x]]\)], "Input"], Cell[BoxData[ \(0.`\ Second\)], "Output"] }, Open ]], Cell["And the attribute for Tim is not displayed by Attributes:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Attributes[Tim]\)], "Input"], Cell[BoxData[ \({}\)], "Output"] }, Open ]], Cell["But we can use", "Text"], Cell[BoxData[ \(SetAttributes[Tim4, HoldAll]; \nTim4[x_]\ := \ First[Timing[x]]; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Tim4[Integrate[Sum[Cos[n\ x], {n, 0, 10}], x]]\)], "Input"], Cell[BoxData[ \(1.26000000000021827`\ Second\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Patterns", "Section"], Cell[TextData[{ "In the previous issue I dealt with how positions are used to target \ activity. 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" and to use some mathematics to reduces the calculation needed:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\((w1 = Together[y - z, \ Trig -> True])\); \) // Tim\)], "Input"], Cell[BoxData[ \(0.82999999999992724`\ Second\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\((w2\ = \ Numerator[w1])\); \) // Tim\)], "Input"], Cell[BoxData[ \(0.15999999999985448`\ Second\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\((w3\ = First/@TrigFactorList\ [w2])\); \) // Tim\)], "Input"], Cell[BoxData[ \(0.270000000000209183`\ Second\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify/@w3 // Timing\)], "Input"], Cell[BoxData[ \({2.57999999999992724`\ Second, {1, 0, \(-1\) + x, x, 1 + x, 1 + \((\(-1\) + x)\)\ x, 1 + x + x\^2}}\)], "Output"] }, Open ]], Cell["\<\ So, we have a zero factor in the numerator, whence y - z == 0. In fact the zero factor comes from\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(w3[\([2]\)]\)], "Input"], Cell[BoxData[ \(1 - \((\(-1\))\)\^\(1/7\) + \((\(-1\))\)\^\(2/7\) - \((\(-1\))\)\^\(3/7\) + \((\(-1\))\)\^\(4/7\) - \((\(-1\))\)\^\(5/7\) + \((\(-1\))\)\^\(6/7\)\)], "Output"] }, Open ]], Cell[TextData[{ "Try ", StyleBox["RootReduce", "Input", FontWeight->"Plain"], " on this:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RootReduce[%]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["This suggests that we might introduce another general step:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify/@\(RootReduce/@w3\) // Tim\)], "Input"], Cell[BoxData[ \({1.92999999999994997`\ Second, {1, 0, \(-1\) + x, x, 1 + x, 1 + \((\(-1\) + x)\)\ x, 1 + x + x\^2}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["FullSimplify", "Input", FontWeight->"Plain"], " is often slow, but it is needed here: ", StyleBox["Simplify", "Input", FontWeight->"Plain"], " is not enough." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[w3[\([2]\)]]\)], "Input"], Cell[BoxData[ \(1 - \((\(-1\))\)\^\(1/7\) + \((\(-1\))\)\^\(2/7\) - \((\(-1\))\)\^\(3/7\) + \((\(-1\))\)\^\(4/7\) - \((\(-1\))\)\^\(5/7\) + \((\(-1\))\)\^\(6/7\)\)], "Output"] }, Open ]], Cell[TextData[{ "In this example ", StyleBox["FactorList", "Input", FontWeight->"Plain"], " would have been quicker than ", StyleBox["TrigFactorList", "Input", FontWeight->"Plain"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(First/@FactorList[w2] // Tim\)], "Input"], Cell[BoxData[ \(0.15999999999985448`\ Second\)], "Output"] }, Open ]], Cell[TextData[{ "If we are sure that trigonometric identies are not needed then we will not \ need the Trig variants like ", StyleBox["TrigFactorList", "Input", FontWeight->"Plain"], " or the option ", StyleBox["Trig -> True", "Input", FontWeight->"Plain"], ". " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Integrating a sum", "Section"], Cell["\<\ You may run into the following kind of situation, for example in \ dealing with Fourier series.\ \>", "Text"], Cell[BoxData[ \(\(sm\ = \ Sum[Sin[n\ x], {n, 0, 25}]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[sm, x] // Tim\)], "Input"], Cell[BoxData[ \({8.12999999999999545`\ Second, Null}\)], "Output"] }, Open ]], Cell["\<\ We can speed this up by distributing the integration over the \ summands\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Integrate[#, x]&\)/@sm // Timing\)], "Input"], Cell[BoxData[ \({0.990000000000009094`\ Second, Null}\)], "Output"] }, Open ]], Cell["\<\ This may suggest that we should always expand the integrand int a \ sum and then sum the integrals of the summands, but this is not a universal \ way of speeding up computation:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[\((1 + Sin[x])\)^10, x] // Tim\)], "Input"], 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required high precision computation to \ get a correct value. The difficulty was that ", StyleBox["Plot", "Input", FontWeight->"Plain"], " inputs machine precision numbers. Here is an alternative appoach that \ modifies an expression so that its internal computations can be done to high \ precision, even when given low-precision input (the \"WorkingPercision\" is \ set) " }], "Text"], Cell["\<\ PreciseExpression[expr_, x_, p_:$MachinePrecision] := Function[x, #1][##2]&[ \texpr /. r:(_Real|Complex[_,_Real]|Complex[_Real,_]):> \tSetPrecision[r, p], \tSequence@@ (Switch[#1, \t_?InexactNumberQ, SetPrecision[#1, p], _?NumericQ, #1 \t ]&)/@ Flatten[{x}] ]\ \>", "Input"], Cell[TextData[{ "This sets the precision of evey inexact number in ", StyleBox["expr", "Input", FontWeight->"Plain"], " to ", StyleBox["p", "Input", FontWeight->"Plain"], " and if x is a symbol then it replaces every ", StyleBox["free", FontSlant->"Italic"], " (term to be explained in a later column) occurence of ", StyleBox["x", "Input", FontWeight->"Plain"], " in ", StyleBox["expr", "Input", FontWeight->"Plain"], " with" }], "Text"], Cell[TextData[StyleBox[ "\tSwitch[x,\n\t\t_?InexactNumberQ, 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