need to use double quad function for four integration with four unknowns

I need to write the correct command that enables me to use the "dblquad" function to solve four integrations with four different variables.

for example: yy = dblquad(....dblquad(@myfun,0,2*pi,0,2*pi))

now the funtion myfun contains phi1,phi2,phi3,phi4 four different variables.

The command line I wrote is not yet complete and I am not sure if it is correct or not.

The myfun should it be then written as follow y = myfun(phi1,phi2,phi3,phi4)

Please I need an answer ASAP

Thank you

[Mohamed]

I really appreciate it Mike, things are working great now. Thank you so much. I was able to get it to work exactly as you mentioned. I was using this to simulate a 2 dimension problem. I am upgrading my code to a 3D one. I have 8 nested integrals and are all function of phi. Ecatly as the one I described earlier but this time I have 8 integrals instead of 4. I asked the University here to purchase Matlab R 2012b. I need to know the command line to use to run the 8 integrals using "integral2" funtion.

Thank you

[Mike Hosea]

I rather doubt 8 nested integrals is practical. You might want to look into sparse grid methods, such as can be found at http://sparse-grids.de/ . Your integrand will need to be well-behaved.

Answer 1

[Mike Hosea]

You should be using INTEGRAL2, but the solution is almost the same.

Using INTEGRAL2:

integral2(@(u,v)arrayfun(@(z,w)integral2(@(x,y)myfun(x,y,z,w),0,2*pi,0,2*pi),u,v),0,2*pi,0,2*pi)

Using DBLQUAD

dblquad(@(u,v)arrayfun(@(z,w)dblquad(@(x,y)myfun(x,y,z,w),0,2*pi,0,2*pi),u,v*ones(size(u))),0,2*pi,0,2*pi)

The INTEGRAL2 way finished in a couple of seconds with the example I gave it, DBLQUAD took 2852 seconds.

You can use QUAD2D instead of INTEGRAL2 if you don't have R2012a. If you do have R2012a, you can also do this

integral(@(x)integral3(@(y,z,w)myfun(x,y,z,w),0,2*pi,0,2*pi,0,2*pi),0,2*pi,'ArrayValued',true)

-- Mike

[Mohamed]

I am using the dblquad function because I have R2007B version. When I am trying to enable a breakpoint on this command line it does not accept it, it keeps on saying that it cannot be enabled because of the syntex error and that there is no anaonymes function.

[Mike Hosea]

Not sure what the error is since I can't see the line the way it appears in your editor. I copy-pasted, however, into R2007b and got

    >> format compact
    >> version
    ans =
    7.5.0.342 (R2007b)
    >> myfun = @(x,y,z,w)x+y+z+w
    myfun = 
        @(x,y,z,w)x+y+z+w
    >> dblquad(@(u,v)arrayfun(@(z,w)dblquad(@(x,y)myfun(x,y,z,w),0,2*pi,0,2*pi),u,v*ones(size(u))),0,2*pi,0,2*pi)
    ans =
      1.9585e+004

[tida]

I am also have the same problem to solve 4 integrals using this dblquad. Could you please explain more about this arrayfun. I mean what is in this function thanx Tida

[Mike Hosea]

The integration functions in MATLAB expect to be able to make "vectorized" calls. An example would be

>> sin(0:pi/12:pi/2)
ans =
         0    0.2588    0.5000    0.7071    0.8660    0.9659    1.0000

A "vectorized" function will accept arrays as input and do its thing on each element of the input array to produce the output array. This is called an "elementwise" operation. Usually all you need to do in MATLAB to vectorize a function is use .* instead of *, ./ instead of /, and .^ instead of ^, but for some functions this is not enough. The ARRAYFUN function applies a function elementwise on input arrays. The line

arrayfun(@sin,0:pi/12:pi/2)

does the same thing as the example I gave before, but this time the sin() function is only ever called with scalar inputs. First it is called with the input 0, then with pi/12, then with pi/6, and so forth. This extends naturally to functions of two variables. For example

arrayfun(@atan2,0:0.1:1,-1:0.1:0)

builds an output array like so

    [atan2(0,-1),atan2(0.1,-0.9),atan2(0.2,-0.8), ..., atan2(1,0)]

Since the output of DBLQUAD is only ever a scalar, we need to do something like this. We could do it by hand, but by using ARRAYFUN with anonymous functions (functions defined "on-the-fly" using @), we don't have to write an m-file with a loop to get it done. And ARRAYFUN is usually pretty fast in terms of the overhead it adds because it is a built-in function in MATLAB.