Why does Mathematica constantly reevaluate an expression?

Question

Manipulate[
  f[n_] := 1/n^2; 
  pt = Accumulate[Table[Table[f[i], {i, max}][[n]], 
          {n, PermutationReplace[Table[i, {i, max}], RandomPermutation[max]]}]]; 

  DiscretePlot[pt[[n]], {n, 1, max}],

  {{max, 5}, 2, 15, 1, Appearance -> "Labeled"},

  Deployed -> True]

The problem is that Mathematica constantly reevaluates the randomization. Even if the mouse is completely outside the Manipulate, the plot constantly flashes and changes.

But now here goes the really weird part. If the whole Manipulate gets a bit less complex... this behaviour ceases.

Manipulate[
  pt = Accumulate[PermutationReplace[Table[i, {i, 1, max}], 
          RandomPermutation[max]]]; 

  DiscretePlot[pt[[n]], {n, 1, max}], 

  {{max, 5}, 2, 15, 1, Appearance -> "Labeled"},
  Deployed -> True]

Everything looks nicely here. Yes, it does seem Mathematica re-randomizes this two or three Times too often, so it takes a second for the plot to calm down and stop changing; but after that time, everything looks well.

How to fix this? And why does this happen?

PS. Sorry for spamming the site... I really need to have a better knowledge of Mathematica, I know. But, for my defence, this really seems weird and unandersandable.

Answer 1

Manipulate updates each time its expression changes. The expression is a function of the control variables. Or if the expression contains a symbol or expression that appears only inside the Manipulate expression (i.e. not global before), even though it is not control variable, and this symbol/expression gets updated during evaluation of the manipulate expression.

In your example below, we see these symbols f[],n,i,max,pt

The control variable is max, so when max changes, pt changes. The symbols i,n are local to Table by definition, hence they do not affect the update.

So what is left is f[]. Notice that when pt changes, inside the Table you are calling f[i], then Table will make repeated evaluation for each Table index See

http://reference.wolfram.com/language/tutorial/EvaluationInIterationFunctions.html

To solve this, you make make a module inside Manipulate, and define any symbols used in the expression, which are not control variable, in the module, like this:

Manipulate[
 Module[{f, n, i},
  f[n_] := 1/n^2;
  pt = Accumulate[
    Table[
     Table[Evaluate@f[i], {i, max}][[n]], {n, PermutationReplace[Table[i, 
        {i, max}], RandomPermutation[max]]}]
    ];
  DiscretePlot[pt[[n]], {n, 1, max}]
  ],
 {{max, 5}, 2, 15, 1, Appearance -> "Labeled"},
 Deployed -> True]

Another way to solve this, which is what I prefer to do, is to use TrackedSymbols.

 Manipulate[
 f[n_] := 1/n^2;
 pt = Accumulate[
   Table[
    Table[f[i], {i, max}][[n]], {n, PermutationReplace[Table[i, {i, max}],
       RandomPermutation[max]]}]
   ];
 DiscretePlot[pt[[n]], {n, 1, max}],
 {{max, 5}, 2, 15, 1, Appearance -> "Labeled"},
 TrackedSymbols :> {max},
 Deployed -> True]

Here is a simple example that shows the problem you were having

Manipulate[
 g[n_] := n;
 Table[g[i], {i, 1, max}],
 {{max, 10, "max"}, 1, 10, 1}
 ]

This will now re-evaluate repeatedly.