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An equation that describes the relationship \ between a term and its predecessor is called a recurrence relation.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "\nMajor curriculum groups have suggested for many years that iteration be \ included in the mathematics curriculum. Experience with mathematical \ iteration in a variety of situations will encourage students to look for \ relationships between quantities, as well as to generate and test patterns. \ The study of iteration may also help students develop an understanding of \ various methods of mathematical reasoning, in particular mathematical \ induction, which moves from the n\[CapitalEHat]\[CapitalOAcute]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"Shadow"->True}], StyleBox[ "case to the (n+1)\[CapitalIAcute]\[CapitalEHat]. Computing technology has \ made mathematical iteration more important.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "When teaching mathematical iteration, the teacher should emphasize the \ iterative procedures have three parts: the initial values, the recurrence \ relation that describes how to get subsequent terms of the sequence, and a \ rule for stoping the procedure. The stopping rule is governed by which term \ or terms of the sequence are needed to solve the problem.\n\nIn the process \ of iteration, we are working with sequence of numbers. When we iterate a \ particular function f[x], then the numbers that appear are x, f[x], f[f[x]], \ f[f[f[x]]],. . . If we can write an equation in the form x=f[x], our goal is \ a sequence that converges - at least in a machine sense - to a fixed point of \ the function. Geometrically, a fixed point of a function is a point where \ the graph of the function y=f[x] meets the line y=x.\n\nIn the upcoming \ example, the iterative procedure will be used to show the convergence of the \ functions y=x\[Currency] and y=x with an intial guess of x=.5. The table of \ values listed first ceases when the difference of the iteration is less than \ .001. This step was required because when the procedure was ran for the \ first time it did not stop iterating itself. The screen showed ordered pairs \ that were identical to six degrees of accuracy but the computer did not stop \ it because it had computed to fifteen degrees of accuracy.", PageWidth->PaperWidth, AspectRatioFixed->True, FontFamily->"CalcMath", FontWeight->"Plain"]], "Input", PageWidth->PaperWidth, AspectRatioFixed->True, FontFamily->"CalcMath"], Cell[CellGroupData[{Cell[TextData[ "Clear[f,t,line,k,h,g,xtemp,newxo,oldx]\nx0 = .5;\ng[x_]=x^2;\nh[x_]=x;\n\n\ newxo[oldx_] := x/.FindRoot[g[x] == oldx,{x,oldx}]\n\npointlist = {};\n\n\ xtemp = x0;\n\nWhile[\n\nN[Abs[g[xtemp]-xtemp]] > .001,\n\n\ AppendTo[pointlist,{xtemp,g[xtemp]}];\n\nxtemp=N[newxo[xtemp]]]\n\n\n\ pointlist"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{0.5, 0.25}, {0.7071067811865475244, 0.5}, {0.8408964152537145594, 0.7071067811865475518}, {0.9170040432046712407, 0.8408964152537145594}, {0.9576032806986663509, 0.917004043204848779}, {0.9785720620877478718, 0.9576032806986670757}, {0.9892280131939996142, 0.9785720620877478748}, {0.9945994234836453062, 0.9892280131939996141}, {0.997296056085476208, 0.9945994234836453062}, {0.9986471128909732186, 0.997296056085476208}}\ \>", "\<\ {{0.5, 0.25}, {0.707107, 0.5}, {0.840896, 0.707107}, {0.917004, 0.840896}, {0.957603, 0.917004}, {0.978572, 0.957603}, {0.989228, 0.978572}, {0.994599, 0.989228}, {0.997296, 0.994599}, {0.998647, 0.997296}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "In the graphics to follow, I stopped the iterator sooner at .01 because you \ could not tell the difference between the last few plotted points. You will \ see the curve y=x\[Currency], the line y=x, and a series of horizontal and \ vertical lines that show the iterative process taking you from the curve to \ the line."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Clear[xmax,xmin,ymax,ymin]\nxmax=Max[(Transpose[pointlist])[[1]]]+1;\n\ xmin=Min[(Transpose[pointlist])[[1]]]-1;\n\ ymax=Max[(Transpose[pointlist])[[2]]]+1;\n\ ymin=Min[(Transpose[pointlist])[[2]]]-1;\nClear[x,pts,xtemp]\nxtemp = x0;\n\n\ pts = {};\nhlines = {};\nvlines = {};\n\nClear[vert,horiz]\nWhile[\n\ N[Abs[g[xtemp]-xtemp]] > .01,\n\nAppendTo[vlines,Line[{{xtemp,xtemp},\n\ {xtemp,g[xtemp]}}]];\n\nAppendTo[hlines,Line[{{xtemp,xtemp},\n\ {newxo[xtemp],g[newxo[xtemp]]}}]];\n\n\ AppendTo[pts,{PointSize[.02],Point[{xtemp,g[xtemp]}]}];\n\ngr = \ Join[pts,hlines,vlines];\n\nPlot[{g[x],h[x]},{x,xmin,xmax},\nEpilog->gr,\n\ DisplayFunction->$DisplayFunction,\nAspectRatio->Automatic,\n\ 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Many applications of calculus involve solving equations that are \ in the form of f[x]=0 or that can be put in this form. Algebraic techniques \ such as factoring can be used to solve only a handful of the possibilities. \ In most situations exact solutions cannot be determined but approximate \ solutions can be found. Here we will look at Newton's Methods, which \ involves approximating the graph of f[x] by its tangent lines.\n\nWe start by \ picking a value x\.ba that we expect to be fairly close to a solution and \ calculate f[x\.ba ]. If f[x\.ba ]=0, we have our desired solution. \ Otherwise, we consider the tangent line to the graph y=f[x] at x=x\.ba . If \ f'[x]\[CapitalAE]0, the tangent line is not horizontal and we let x\ \[DownExclamation] denote the value of x where the tangent line intersects \ the x-axis. 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In this process x\[DownExclamation] is computed from x\ \.ba , x\[Trademark] is computed from x\[DownExclamation] , and in general x\ \[ARing] \[NotEqual] \[DownExclamation] is computed from x\[ARing] . The \ following table is a list of ordered pairs of the iteration process and the \ graphics of Newton's method for approximate solution of a function. The code \ was rigged to round to the fifth decimal place in order to minimize needless \ calculations, since we are dealing with approximate solutions anyhow."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Clear[f,t,line,k]\n\nx0 = 1.5;\n\nf[x_] = x^3-x;\n\nline[k_] := \ f'[k](x-k)+f[k];\n\nt[k_]:= (x/.Solve[line[k] == 0,x])[[1]]\npointlist = {};\n\ \nxtemp = x0;\n\nWhile[Chop[N[f[t[xtemp]]]] != 0,\n \ AppendTo[pointlist,{xtemp,f[xtemp]}];\n xtemp = t[xtemp]];\n\ AppendTo[pointlist,{Chop[xtemp,10^-5],0}]\n"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{1.5, 1.875}, {1.17391304347826087, 0.4438234568915920109}, {1.032307127497771018, 0.06777922703707818216}, {1.00145595377899462, 0.002918270048541813733}, {1.000003168936889527, 0}}\ \>", "\<\ {{1.5, 1.875}, {1.17391, 0.443823}, {1.03231, 0.0677792}, {1.00146, 0.00291827}, {1., 0}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "Now you will see the complete graphics iteration that yielded the table from \ above. 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We are going \ in the wrong direction to find a root. With a better initial guess the \ iteration process will work. We will use the same function, but start with \ an x\.ba =.8 and an approximate root appears after only six iterations. I \ stopped the graphics after five, because you can not tell the difference \ between the last few anyhow. 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