Plot inside Manipulate with indexed coefficients fails

Question

I am trying to plot a polynomial inside Manipulate with indexed coefficients.

Surprisingly, only an empty plot is generated.

Any help welcome.

Block[{n = 3},
 Manipulate @@ {
 Column[{
  Sum[a[i] x^i, {i, 0, n}],
  Plot[Evaluate@Sum[a[i] x^i, {i, 0, n}], {x, -5, 5}]
 }],
 Sequence @@ Table[{{a[i], 0}, -2, 2}, {i, 0, n}]
}]

Regards Robert

Answer 1

I couldn't do it with Manipulate[], so I used pure Dynamic[] constructs instead:

With[{nmax = 12},
     DynamicModule[{n = 3, cofs = ConstantArray[0, 4]},
                   Panel[Row[{Column[{Style["Degree:", Bold], 
                                      Slider[Dynamic[n, (n = #;
                                                     cofs = PadRight[cofs, n + 1, 0];) &],
                                             {1, nmax, 1}],
                                      Style["Coefficients:", Bold], 
                                      Dynamic[Multicolumn[
                                              Array[Row[{Subscript["c", # - 1],
                                              Spacer[5],
                                              Slider[Dynamic[cofs[[#]]], {-2, 2}, 
                                                     ImageSize -> Small]}] &, n + 1],
                                              {9, Automatic}]]}], 
                   Spacer[20], 
                   Dynamic[Column[{Pane[
                   Row[{"f(x) = ", TraditionalForm[Expand[
                                                   FromDigits[Reverse[cofs], x]]]}]], 
                   Panel[Plot[FromDigits[Reverse[cofs], x], {x, -5, 5}, 
                              Axes -> None, Frame -> True]]}]]}]]]]

For instance:

Answer 2

I think what you are trying to accomplish is not easy. Both evaluation control and symbol localization can be tricky in Mathematica, and your code has subtle errors related to both.

First I will give a possible solution, then I will discuss the problems with your approach.

manip[n_] :=
 Block[{a,x}, (* prevent conflict with any global a, x *)
   With[{
     vars = Sequence @@ Table[{{a[i], 0}, -2, 2}, {i, 0, n}],
     poly = Sum[a[i] x^i, {i, 0, n}]
     },
    Manipulate[
     Column[{
       poly,
       Plot[poly, {x, -5, 5}]
       }],
     vars
    ]
   ]
 ]

Problems with your code:

• Using Manipulate @@ {...} instead of Manipulate[...] prevents the HoldAll attribute of Manipulate from taking effect. You probably did this on purpose to allow the second argument to evaluate. But the first one evaluates too, including the Plot part. Manipulate will never have a chance to see the Plot command (only its empty result).

• Block sets n only temporarily. This means that after evaluating the content of Block, the result must not contain any unevaluated n symbols. If it does, these will never get the value 3 again. This is in contradiction with the requirement not to evaluate Plot ...

• When compound expressions are used as Manipulate variables, i.e. a[1], a[2], ..., they will internally be replaced by single symbols. In practice this means that each of a[1], a[2], ... must appear explicitly in the body of Manipulate. a[i], with i being substituted only during dynamic evaluation, will not work.

These difficulties are overcome by pre-generating both the variable list and the polynomial, and injecting them into a Manipulate.

Another note: Usually people recommend against using Subscript[a,1] instead of a[1]. I think in this case it's quite safe to use the subscript, as the computation will be done with replacement symbols anyway. The Manipulate interface will look better.