Weird numerical integration behavior

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Hussein Ammar
Hussein Ammar 2019 年 9 月 24 日
コメント済み: John D'Errico 2019 年 9 月 24 日
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,

採用された回答

John D'Errico
John D'Errico 2019 年 9 月 24 日
編集済み: John D'Errico 2019 年 9 月 24 日
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.
  3 件のコメント
John D'Errico
John D'Errico 2019 年 9 月 24 日
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.

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