Weird numerical integration behavior

Hello all,

Please check this simple example:

d1 = 8;
d2 = 1.4142;
apPDF = @(x1) (1./sqrt(2*pi)) .* exp(-power((d2.*x1./sqrt(2)) - d1,2) ./ d2^2);

integral(apPDF, 0, Inf)

myVarBoundary = 10^24;
integral(apPDF, 0, myVarBoundary)

My Matlab output:

>> d1 = 8;
d2 = 1.4142;
apPDF = @(x1) (1./sqrt(2*pi)) .* exp(-power((d2.*x1./sqrt(2)) - d1,2) ./ d2^2);

integral(apPDF, 0, Inf)

myVarBoundary = 10^24;
integral(apPDF, 0, myVarBoundary)

ans =

    1.0000


ans =

     0

The correct answer that I should get from the second integration should be one, right?

What is wrong in my integration? Should I replace myVarBoundary with Inf after some value.

Best Regards,

Answer 1

[John D'Errico]

No. You should NOT make it inf. Even 1e24 is incredibly large, wild overkill.

As to why you are using integral to compute the area under what loos like a normal PDF is completely beyond me.

d1 = 8;
d2 = 1.4142;
apPDF = @(x1) (1./sqrt(2*pi)) .* exp(-power((d2.*x1./sqrt(2)) - d1,2) ./ d2^2);

You need to understand what integral does when it sees a function with limits that wide. It evaluates the function at a variety of points in the interval. Lets try a few, just for kicks.

apPDF(0)
ans =
      5.04917109360107e-15
>> apPDF(1e24)
ans =
     0
>> apPDF(1e24 / 2)
ans =
     0
>> apPDF(1e24 / 100000)
ans =
     0
>> apPDF(1e24 / 10000000000000)
ans =
     0

Do you see anything significant? Do you see a function that seems to be everywhere zero on the interval [0,1e24]? And even when not identically zero, it deviats from zero on the order of the convergence tolerance. Should it somehow, magically know that in effectively a tiny corner of that HUGE interval, it is non-zero?

apPDF(5)
ans =
       0.00443082846737379

So now if we do this:

integral(apPDF,0,100)
ans =
                         1

What a surprise! It integrates to 1.

[Steven Lord]

Integrating a function that's non-zero only on a very, very short portion of the interval over which you're integrating is like counting vehicles in a parking lot from the window of an airplane cruising at 30,000 feet. The ground looks pretty flat from an airplane; seeing an individual vehicle is likely to be difficult if not impossible.

As the plane is taking off or landing (when you have a closer view of the parking lot in question) or if you're viewing it from the top of a building, getting an accurate count is much easier.

[Hussein Ammar]

Hello all, thank you for your replies.

I understand what your pointing for, but unfortunately I can't control the value of "myVarBoundary" (depending on a lot of things it can be very small or very large). so, I think I can assume that if "myVarBoundary" crosses some threshold, my answer should tend to 1. right?

Anyway, I think what's wrong is to use myVarBoundary directly as the upper limit of the numerical integration, because when myVarBoundary is very large, the numerical integration won't see the very tiny part that is non-zero at the start of integration domain.

Regards,

[John D'Errico]

Steve makes an excellent point. Here, you might need to be looking at the ground using the Hubble space telscope though, to get the necessary resolution.

As I showed, the function was zero above x1=100. So out of an interval of width 1e24, it is zero on only the fraction (well) below 100. 1 part in 1e22?

1e22 is a number almost as large as Avogadro's number. Big.