How to integrate a functional of integrals numerically?

Greetings, Trying to numerically integrate the following http://www.scribd.com/doc/221721132/Numerical-Integration expression> using MATLAB. The answer should be a table of phi as a function of j (rows) and x (columns) as indicated on the attached file. Any help is appreciated. Thanks Naeem

[Star Strider]

Nothing attached.

[Naeem]

sorry, attached on scribd.com (i edited the post and provided link inline)

Answer 1

[Star Strider]

The integrals in the numerator and denominator of φ(z,j) are actually double integrals of ψ w.r.t. x and y, so I calculated them that way in the loop, substituting in the required values for z, and then computed φ(z,j) from them and beta. (The entire calculation required about 214 seconds on my machine.)

This code:

zv = 0.1:0.1:1;
jv = 0:31;

fpsixyz = @(x,y,z)  exp(-(((x-y).*z).^2)/4)./(1+y.^2);      % Function to integrate
B = @(j) 1E-5 * 2.^j;                                       % Beta

for k1 = 1:size(jv,2)
    for k2 = 1:size(zv,2)
        z = zv(k2);
        psiz = integral2(@(x,y) fpsixyz(x,y,z), 0, Inf, -Inf, Inf); % Integrate over [x,y]
        j = jv(k1);
        phizj(k1,k2) = psiz ./ (B(j) + psiz);       % Compute phi(z,j)
    end
end

figure(1)
surfc(phizj)
xlabel('\itz\rm')
ylabel('\itj\rm')
zlabel('\phi(\itz, j\rm)')
set(gca, 'XTick', 1:2:10, 'XTickLabel', zv(1:2:end))
axis([xlim    0  31    0  1])

produces:

[Naeem]

Star Strider, Not sure that the ntegrals in the numerator and denominator of φ(z,j) are actually double integrals of ψ w.r.t. x and y. The integral of f(x) divided by the integral of f(x) equals 1 while the integral of f(x)/f(x) is the integral of 1.

[Star Strider]

They actually are double integrals in the final outcome, but it’s necessary to do the integration over x and y with the appropriate values of z as I did in the loops. You’re integrating x from zero to +Inf, and y from -Inf to +Inf. The loop over z is necessary to carry out the integration. As far as the fraction is concerned, remember that beta(j) is in the denominator, so the value of phi(z,j) will not uniformly be equal to one.

Also, I don’t understand ‘f(x)/f(x) is the integral of 1’. That is simply not true. The integral of 1 with respect to x is simply x.

[Naeem]

see the attached file file for explaination of my point http://www.scribd.com/doc/221775970/Explaination

[Star Strider]

What I definitely intended to say is that φ(z,j) is the RHS of the top equation. That is what I coded. The LHS of that is clearly not the original equation. They are definitely not equal in that situation because beta never appears in an integral.

I believe the original system was written as it was to emphasize that z had to be incorporated into the integral for ψ.

The last two statements there I agree with, but they seem irrelevant.

Answer 2

[Naeem]

sorry, it's my mistake, the original equation is actually equivalent to RHS (see the corrected top equation) http://www.scribd.com/doc/221721132/Numerical-Integration

[Star Strider]

I can’t get that one to integrate in MATLAB because the integral of a function of two variables integrated with respect to only one will not return the result as a function of the other variable. I tried it with a much simpler function, and when I attempted to integrate it with respect to only x, it gave an undefined variable error for y. It is obviously not a double integral problem in the usual sense.

I suggest you search the literature to see if it is possible to integrate expressions such as this, or perhaps for expressions of the integrated variables themselves. You might also consider a 2- or 3 term Taylor expansion (linearisation) of ψ, since that might make φ(z,j) easier to calculate.

Otherwise, you could write your own multiple-integration routine specifically for this problem. Numerically integrate it with the trapezoid rule or some such.

Answer 3

[Naeem]

Star Strider, Sorry, ment to say the LHS is the correct form. Please see the attached corrected form and let me know if you have any idea on how to approach this. Thanks much.

[Star Strider]

I understand that now. I worked for a while to see if I could integrate it in MATLAB and finally gave up. It’s just not possible.

[Naeem]

thanks much for trying and for your time though.

[Star Strider]

My pleasure! I always learn from Answering Questions here.

I wish we could have integrated your second equation, but it is sufficiently difficult that it would require several days — if not a few weeks — to code the integrals. It is necessary to integrate two integrals over four variables, not a simple task under any circumstances.