Why is building a table of function values so much slower than just plotting the function?

Question

Here is a function that take some time to evaluate:

n = 1000;
coeffs = RandomVariate[NormalDistribution[], n];
f[x_] := Sum[coeffs[[k]] Sin[k x]/k, {k, 1, n}];

If I just plot the function with 1000 sample points, it takes about 0.24 seconds on my machine:

Plot[Evaluate@f[x], {x, 0, 2 Pi}, PlotPoints -> n, MaxRecursion -> 0] // Timing

On the other hand, if I try to collect 1000 samples of the function into a table, it takes five times as long:

Table[Evaluate@f[x], {x, 0, 2 Pi, 2 Pi/n // N}] // Timing

(Without the // N Mathematica tries to evaluate things symbolically, which is even slower.)

What is going on, and how can I make the Table as fast as the Plot?

Answer 1

Try this:

n = 1000;
coeffs = RandomVariate[NormalDistribution[], n];
f[x_] := Sum[coeffs[[k]] Sin[k x]/k, {k, 1, n}];
Plot[Evaluate@f[x], {x, 0, 2. Pi}, PlotPoints -> n, MaxRecursion -> 0,
   Mesh -> All] // Timing

With[{n = 1000},
 First@Timing[Table[Evaluate@f[x], {x, 0, 2. Pi, 2. Pi/n}]]
 ]

2 times as fast as plot. I remembered my own question difference-of-speed-in-making-a-table on tables some time ago and this trick of making Table much faster. The point is : to keep it packed, need to have all limits be numerics ! When you had n in there, the upper limit was not. So the table was not packed. Now it is.

You could just also write

Table[Evaluate@f[x], {x, 0, 2. Pi, 2. Pi/1000}]

and get the speed advantage. The point again, all its limits needs to be numeric. See above question for more information and additional links on this subject.

Another example

2000 points. Table still about 2 times faster: