Speed of vpaintegral vs. int+vpa

Dear all,

I am currently testing the possibilities of integrating symbolic functions in Matlab. In the documentation of "int" (https://de.mathworks.com/help/symbolic/sym.int.html), I found the following statement which is not in agreement with my personal observations:

"To approximate integrals directly, use vpaintegral instead of vpa. The vpaintegral function is faster and provides control over integration tolerances."

I run the following example and observe that a combination of "int" and "vpa" needs 0.35 secs vs. 47.69 secs in case of "vpaintegral" both having the same precision of 32 valid digits.

syms x y

% define integration boundary
f1 = 0;
eqn = 0.980580675690920 * (0.4-x) - 0.196116135138184 * (1-y) == 0;
f2 = solve(eqn,y);
f3 = 1;
y_max = 0.5;
f = piecewise(x<0.2, min(f1,y_max), 0.2<=x<0.4, min(f2,y_max), ...
    x>=0.4, min(f3,y_max)); %#ok<CHAIN>

% define integrand
mon = x*y;

% integration over x-y-domain limited by the boundary, y=0, x=0 and x=0.5
tic
F = int(int(mon,y,0,f),x,0,0.5);
vpa(F)

ans = 

0.011145833333333333333333333333333

toc

Elapsed time is 1.432269 seconds.

tic
vpaintegral(vpaintegral(mon,y,0,f),x,0,0.5,'RelTol', 1e-32, 'AbsTol', 0)

ans = 

0.011145833333333333333333333333333

toc

Elapsed time is 16.967404 seconds.

I am curious whether I miss something here.

[Walter Roberson]

Faster (a little) is

F = int(int(mon,y,0,f,'hold',true),x,0,0.5,'hold',true);
vpa(release(F))

[John D'Errico]

Remember that using int followed by vpa requires the integral be computable in analytical form. And even when that is possible, there will be many cases where int will generate a complex mess of terms (often an issue with symbolic computations), whereas vpaintegral directly will not need to do anything of the sort. And then you will be much happier with vpaintegral.

The example you show involves a piecewise function, which will often be problematic with a numerical method, as it then needs to locate where the function changes character. And since vpaintegral is a numerical tool, it does not look at the function itself, instead treating it as a black box. As such, this is the worst kind of case for a numerical tool.

So, are you missing something? I think only that understanding the tools you will use and how they will relate to the problems you will throw at them is the most important part of any such process. Very generic advice, as you found about the difference between these tools is often reasonable. But it is also easily possible to find examples where that advice proves wrong. And sometimes, it may even be best to just try each tool on your specific problem, and see which one performs best in that specific case, at least when your experience is not sufficient.

I might tell you that you should use a screwdriver to drive in a screw, reserving a hammer to drive in nails. And that would generally be wise, sage advice. But SOMETIMES, in SOME circumstances, a hammer may indeed be the better choice to drive in a screw. That would not invalidate the previous, generic advice I have given you.

[Michael Loibl]

Hi,

Thank you for your explanation. Actually I was not sure whether "int" is integrating analytically here because I get the following solution to my integral:

F =
 
int((x*min([1/2, 5*x - 1], [], 2, 'omitnan', false)^2)/2, x, 1/5, 2/5) + 9/1600

Does it mean that it partly computes it analytically (resolving one of my "int") and partly is computing it then numerically using the "vpa" command?

[Walter Roberson]

@Michael Loibl

Yes, one of your int() is being resolved analytically, and then vpa() computes the rest numerically.

Answer 1

[Rahul]

Hi @Michael Loibl,

I understood your query and was able to reproduce the same with the help of the code provided.

You are trying to evaluate the performance benefits of using "vpaintegral" function for the computation of integral as compared to using "int" and "vpa" functions separately. The two methods use different techniques to compute results.

The "int" + "vpa" functions use symbolic methods to evaluate the results and is efficient in cases where integral is computable in closed/analytical form which are less computationally intensive.

The "vpaintegral" function uses numerical methods to solve the integral and is much more efficient in cases which are computationally intensive.

In your case, you are using "piecewise" function which returns symbolic expression or functions. As correctly mentioned by @John D'Errico, "vpaintegral" is not the ideal function to be used in this case as it is designed for numerical cases and looks at the piecewise function as a black box.

Hence in this case using the combination of "int" + "vpa" is much more efficient. If you require to use "vpaintegral" for your case, you can try adjusting the tolerance values which can make the computation more efficient.

tic
vpaintegral(vpaintegral(mon, y, 0, f), x, 0, 0.5, 'RelTol', 1e-6, 'AbsTol', 1e-10)
toc

% You can adjust the 'RelTol' and 'AbsTol' values according to your usecase

You can refer to the following documentations to know more about these functions:

"int": https://www.mathworks.com/help/symbolic/sym.int.html

"vpa": https://www.mathworks.com/help/releases/R2024a/symbolic/vpa.html?searchHighlight=vpa&s_tid=doc_srchtitle

"vpaintegral": https://www.mathworks.com/help/symbolic/sym.vpaintegral.html

"piecewise": https://www.mathworks.com/help/releases/R2024a/symbolic/sym.piecewise.html?searchHighlight=piecewise&s_tid=doc_srchtitle

Hope this helps! Thanks.