Making code faster?

Question

I am dealing with a very simple code, which turns to be slow when I try to plot the final Piecewise function. Is the code computing back again the integral iteratively rather than just substituting d=2 and easily plotting?

AbsoluteTiming[g[t_, d_] = Integrate[d*Sin[t - t1]*f[t1], {t1, 0, t}, 
Assumptions -> t \[Element] Reals && d \[Element] Reals]] 
AbsoluteTiming[f[t1_] = Piecewise[{{-t1^2, 0 <= t1 <= 3}, {t1, t1 > 3}}]]
AbsoluteTiming[g[t, d]]
Plot[g[t, 2], {t, 0, 5}]

Even if I use /.d->2 inside Plot it is very slow.

So what I did is the following, trying to avoid the Evaluate and understand why I had problems (which seem to come from that 4th raw g[t,d]. Two ways worked well: 1) changing the order of the rows

AbsoluteTiming[
f[t1_] = Piecewise[{{-t1^2, 0 <= t1 <= 3}, {t1, t1 > 3}}]]
AbsoluteTiming[
g[t_, d_] = 
Integrate[d*Sin[t - t1]*f[t1], {t1, 0, t}, 
Assumptions -> t \[Element] Reals && d \[Element] Reals]]
AbsoluteTiming[g[t, d]]
Plot[g[t, 2], {t, 0, 5}]

or 2) changing the initial definition of g[t_,d_] into g[t_]deleting g[t,d] and defining a new function:

AbsoluteTiming[
g[t_] = Integrate[d*Sin[t - t1]*f[t1], {t1, 0, t}, 
Assumptions -> t \[Element] Reals && d \[Element] Reals]]
AbsoluteTiming[
f[t1_] = Piecewise[{{-t1^2, 0 <= t1 <= 3}, {t1, t1 > 3}}]]
h[t_, d_] = g[t]
Plot[h[t, 2], {t, 0, 5}]

I think Mathematica gets confused because of that g[t,d] definition after g[t_,d_]? Anyone can explain why?

If I define
g[t_]

rather than g[t_,d_] and then a new function, such as h[t_,d_]=g[t_] and then plot h[t,2], it works. Why? — Andrea G, Mar 29 '17 at 09:10
Plot has the attribute HoldAll and as the documentation will tell you in the section Detail & Options sometimes an Evaluate will help, so this runs fast: Plot[ Evaluate@g[ t, 2 ], {t,0,5}]. — gwr, Mar 29 '17 at 09:32
You could also evaluate at the beginning SetOptions[Plot, Evaluated -> True];. Or, the shortest way: Plot @@ {g[t, 2], {t, 0, 5}}, which is fast because of what @gwr told you. — Rolf Mertig, Mar 29 '17 at 09:37
@Rolf Still learning something new every day: Where is the option Evaluated to be found in the documentation? It is neither listed by itself, nor under Plotor Graphics nor to be found in the tutorials about evaluation... — gwr, Mar 29 '17 at 09:51
@Rolf One should note, that there seem to be deeper issues with that undocumented option. See here. So using Evaluate to me seems saver. — gwr, Mar 29 '17 at 10:57
@gwr, Evaluated is an undocumented option, but has been there for quite a while. — J. M.'s missing motivation, Mar 29 '17 at 11:03
@J.M. So I learn, but there seem to be issues, cf. the linked post? — gwr, Mar 29 '17 at 11:05
@gwr, correct, and yet another example of caveat emptor when using undocumented functionality. — J. M.'s missing motivation, Mar 29 '17 at 11:06
@gwr can you explain me that Evaluate@? I am looking for syntax and i cant find anything about the use of @ symbol — Andrea G, Mar 29 '17 at 11:06
For reference: f[x], f @ x, and x // f are equivalent expressions. — J. M.'s missing motivation, Mar 29 '17 at 11:14
You should also closely look at Set and SetDelayed. See this. — gwr, Mar 29 '17 at 12:11

Making code faster?

0 Answers0