Why doesn't the following code for recursive convolution work?

Question

I'm trying to play with central limit theorem using Mathematica, so I wrote the following code,

f[x_] := UnitBox[x];  
For[i = 1, i < 4, i++,  
   f[x_] := Convolve[UnitBox[z], f[z], z, x]; 
    ]
Plot[f[x],{x,-3,3}]

hoping to produce a 3-fold convolution of the original function, but only to produce the following error notifications:

$RecursionLimit::reclim2: Recursion depth of 1024 exceeded during evaluation of UnitBox[z].
$RecursionLimit::reclim2: Recursion depth of 1024 exceeded during evaluation of UnitBox[z].
$RecursionLimit::reclim2: Recursion depth of 1024 exceeded during evaluation of UnitBox[z].
General::stop: Further output of $RecursionLimit::reclim2 will be suppressed during this calculation.

but I don't know how to comprehend this. What exactly is wrong with the code?

@J.M., I saw that, but I'm too green to mathematica to understand the details of that post. — Jia Yiyang, Apr 10 '17 at 17:52
I think that because you use 'SetDelayed' rather than 'Set', only the last assignment to 'f (x)' has any effect. Consequently, when you evaluate it, it calls itself recursively, without limit. — mikado, Apr 10 '17 at 18:33
At the very least, did you see that you can use B-splines for this? — J. M.'s missing motivation, Apr 10 '17 at 23:11
@J.M., I saw your answer, but I don't understand how B-spline works even after reading the Wiki introduction to B-splines. — Jia Yiyang, Apr 12 '17 at 02:19
@J.M., I see, then it's of limited utility for my interest, because I'm playing with central limit theorem, which needs to be true for many other initial functions. — Jia Yiyang, Apr 12 '17 at 02:33
Ah, then yes, the B-splines are specific to the convolutions of the box function. Nevertheless, the general solution in my other answer should be usable here. — J. M.'s missing motivation, Apr 12 '17 at 02:37

David G. Stork · Answer 1 · 2017-04-08T02:34:26.360

3

g[z_] := Nest[Convolve[#, UnitBox[x], x, z] &, UnitBox[x], 3];
Plot[g[z], {z, -2, 2}]

And try this, which is more instructive:

g[z_] := Nest[Convolve[#, UnitTriangle[x], x, z] &, UnitTriangle[x], 3];
Plot[g[z], {z, -2, 2}]

edited Apr 08 '17 at 02:34

answered Apr 08 '17 at 02:12

David G. Stork

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Better to define g with Set than with SetDelayed for this function; particularly so for the UnitTriangle version. – Edmund Apr 08 '17 at 02:57
Better to include option, Exclusions -> None for the plot. It joins the skip zones. – Carl Apr 08 '17 at 03:05
1

Thanks for the answer. But does the first code block really produce a 3-fold convolution of UnitBox? I would expect the result of 3-fold convolution to be a (piece-wise) quadratic function. – Jia Yiyang Apr 08 '17 at 03:11

score 1 · Accepted Answer · answered Apr 08 '17 at 11:43

1

The following is a more recursive solution

g[0] = UnitBox;
g[n_] := g[n] = Function[{x}, 
   Evaluate[Module[{z}, Convolve[UnitBox[z], g[n - 1][z], z, x]]]]

Plot[g[3][x], {x, -3, 3}]

answered Apr 08 '17 at 11:43

mikado

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This works very well, thanks and +1. But exactly what's wrong with my original code? – Jia Yiyang Apr 10 '17 at 17:52
After some testing it seems what made the difference is "Evaluate", i.e. my code would also work if I enclose the "Convolve" with an "Evaluate". But I don't really understand the operating principle here. – Jia Yiyang Apr 10 '17 at 17:59
When learning Mathematica, I found making it evaluate my functions in the order I intended was one of the biggest challenges. I admit that I gave you something that worked rather than trying very hard to understand what you had done. – mikado Apr 10 '17 at 18:27

Why doesn't the following code for recursive convolution work?

2 Answers2