Can You Configure Mathematica to Not Abort?

Question

So I am playing around with an interpolation polynomial. The code for what I am trying to do is computationally heavy:

p[x_, n_]:=Sum[f[Table[15*(k/n),{k,0,n}][[i]]]*Product[Piecewise[{{(x-Table[15*(k/n),{k, 0, n}][[j]])/(Table[15*(k/n),{k,0,n}][[i]]-Table[15*(k/n),{k,0,n}][[j]]),Not[i==j]},{1,i == j}}],{j,1,Length[Table[15*(k/n),{k,0,n}]]}],{i,1,Length[Table[15*(k/n),{k,0,n}]]}];

Or in math because that code looks unreadable:

$$p(x,n):=\left\{ \sum^{n}_{i=0}\left( f(x_i)\left(\prod_{j\neq i}\frac{x-x_j}{x_i-x_j} \right) \right) : i,j\in \mathbb{Z}^{+}_{n} \ \ and\ \ \left\{x_i \right\}_{i=0}^{n}= \frac{15*i}{n} \right\}$$

I want to export this as a gif up to $n=20$. Unfortunately, I get $Aborted around $n=16$.

So now my question: Is there any way I can get Mathematica to not abort?

As always, I appreciate any and all help.

EDIT: I received a comment to include all my code so below is the code I excluded initially:

f[x_]=x*Sin[x^(1.5)]; Export[“coolthing.gif”,Manipulate[Plot[{f[x],p[x,n]},{x,0,15}],{n,1,20}],”GIF”];

Answer 1

Try this

Using the following code below. Btw, there are better ways to make animated gif files from Manipulate, by saving into a table each frame, then export these frames one by one at the end. You can also have more control on time between frames. If you search this site, you will see examples.

It now also runs faster, since the functions are defined to take in numeric arguments. On my PC this finished in about 3 minutes.

I also added SetOptions[$FrontEnd,DynamicEvaluationTimeout->60] just in case you need more than 6 seconds, but it probably not needed now. Also it is good to add TrackedSymbols :> {n}, SynchronousUpdating -> False, ContinuousAction -> False so I added these also.

SetDirectory[NotebookDirectory[]]
SetOptions[$FrontEnd, DynamicEvaluationTimeout -> 60]
p[x_?NumericQ, n_?NumericQ] := 
  Sum[f[Table[15*(k/n), {k, 0, n}][[i]]]*
    Product[Piecewise[{{(x - 
           Table[15*(k/n), {k, 0, n}][[j]])/(Table[
             15*(k/n), {k, 0, n}][[i]] - 
           Table[15*(k/n), {k, 0, n}][[j]]), Not[i == j]}, {1, 
        i == j}}], {j, 1, Length[Table[15*(k/n), {k, 0, n}]]}], {i, 1,
     Length[Table[15*(k/n), {k, 0, n}]]}];

f[x_?NumericQ] := x*Sin[x^(15/10)];

Export["coolthing.gif", Manipulate[
  Grid[{{Row["n=", n]},
    {Plot[{f[x], p[x, n]}, {x, 0, 15}]}}
   ],
  {n, 1, 20}, TrackedSymbols :> {n}, SynchronousUpdating -> False, 
  ContinuousAction -> False
  ],
 "GIF"]

Update

Here is a version for $n=40$. It took 30 minutes on my PC and I had to increase SetOptions[$FrontEnd,DynamicEvaluationTimeout->200] also

Export["coolthing.gif", Manipulate[
  Grid[{{Row[{"n=", n}]},
    {Plot[{f[x], p[x, n]}, {x, 0, 15}]}}
   ],
  {n, 1, 40}, TrackedSymbols :> {n}, SynchronousUpdating -> False, 
  ContinuousAction -> False
  ],
 "GIF"]

Update

per comments, made $n$ integer values and fixed the vertical scale to some values. OP can decide better what these should be.

Export["coolthing.gif",
 Manipulate[
  Grid[{{Row[{"n=", n}]},
    {Plot[{f[x], p[x, n]},
      {x, 0, 15}, PlotRange -> {Automatic, {-40, 40}}]}}
   ],
  {{n, 1, "n"}, 1, 40, 1, Appearance -> "Labeled"},
  TrackedSymbols :> {n},
  SynchronousUpdating -> False,
  ContinuousAction -> False
  ],
 "GIF"]