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I've only had few previous experiences on Mathematica before. So basically an entry-level question that harasses a rookie like me.

I made the following attempt using bisection method to calculate the real root for the function $f(x)=-x^3+2x^2-2$ with basic while loop and if:

a = -5; b = 5; m = (a + b)/2; While [ Abs[f[m]] > 0.000001, If[f[m]*f[a] < 0, b = m, a = m]; m = (a + b)/2]

The answer turns out to be $-(14080895/16777216)$ that is correct.

Then, I wanted to create a animation that shows the change of position of a dot along the function curve for each iteration of the bisection method. But I tried several times with different codes that all fails to do so. One confusion is I am not sure how to set up a stepsize here which involves not a constant number but "While" and "If".

Would you please help me animate the iteration with while and if?

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Two suggestions to make your bisection routine more robust: 1.) compute m as m = a + (b - a)/2, and 2.) change the sign change criterion to Sign[f[m]] != Sign[f[a]]. – J. M. May 28 '12 at 0:56

Perhaps a more Mathematicaish way of doing more or less the same is with NestWhile A possible code could be

 With[{m = Mean[#]},
    Negative[f[First@#] f[m]], {First@#, m}, {m, Last@#}
    ]] &, {-5, 5},
 Abs@f@Mean@# > 0.000001 &

If you have your code written with NestWhile, what you want is as simple as changing NestWhile to NestWhileList.

steps = NestWhileList[
   With[{m = Mean[#]},
      Negative[f[First@#] f[m]], {First@#, m}, {m, Last@#}
      ]] &, {-5, 5},
   Abs@f@Mean@# > 0.000001 &

ListLinePlot[Mean /@ steps, PlotRange -> Full]

Mathematica graphics

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Adding a Sow to your loop let's you store the intermediate values of m. You can grab them later with Reap and animate them with Animate:

a = -5; b = 5; m = (a + b)/2; 
steps = 
 Reap[While[Abs[f[m]] > 0.000001, 
    Sow[m] If[f[m]*f[a] < 0, b = m, a = m]; m = (a + b)/2]][[2, 1]];

 Plot[f[x], {x, -2.5, -0.5}, 
  Epilog -> {Red, PointSize[0.03], 
    Point@{steps[[i]], f@steps[[i]]}}], {i, 1, Length[steps], 1}

enter image description here

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First, what follows is an improvement of your bisection routine, which avoids needless overflow and repeated function evaluations. As with Sjoerd, we use the Sow[]/Reap[] pair for collecting intermediate results:

f[x_] := -2 + 2 x^2 - x^3;
acc = 10.^-6;
a = -5; b = 5;
u = f[a]; v = f[b]; m = $Failed;
pts = Reap[While[NonNegative[u] =!= NonNegative[v],
         m = a + (b - a)/2; h = f[m]; Sow[{m, h}];
         If[NonNegative[u] =!= NonNegative[h],
            (* then *) b = m; v = h,
            (* else *) a = m; u = h];
         If[Abs[h] <= acc, Break[]]; 
         ]][[2, 1]];

I used OP's absolute termination criterion in the code above, but a relative criterion like Abs[b - a] <= (Abs[a] + Abs[b])*tol is usually more preferable in practice.

Having gathered up all required intermediate results in pts, we can elect to either show progress all at once:

Plot[f[x], {x, -5/2, 0}, 
 Epilog -> {Arrow /@ Partition[pts, 2, 1], {AbsolutePointSize[7], Red,
     Point[First[pts]]}, {AbsolutePointSize[7], Blue, 
    Point[Last[pts]]}}, Frame -> True]

bisection plot

or show it gradually:

  Plot[f[x], {x, -5/2, 0}, 
    Epilog -> {Arrow[#1], {AbsolutePointSize[7], Red, 
       Point[First[pts]]}, {AbsolutePointSize[7], Blue, 
       Point[Last[pts]]}}, Frame -> True, 
    PlotLabel -> StringForm["Iteration `1`", First[#2]]] &, 
  Partition[pts, 2, 1]]]

bisection animation

In both cases, we use Arrow[] objects to indicate the direction of progress, a red point to denote the midpoint of the initial interval, and a blue point to denote the final result.

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