Triple intergral for the array valued function

Hello,

I would like to perform triple intergral for my array valued function. The example of kind is below :

Generally, the command quadv works for the single variable integral. However, the integral3 or triplequad could not handle this. It throws the error. Example code below which throwing an Error. From my understanding, the integral3 can not handle array valued function. What would be the alternative way to handle this kind of situation

N1 =@(x,y,z) x.^2+y.^2+z.^2;
N2 =@(x,y,z) x.^2+y.^2+z.^2;
N3 =@(x,y,z) x.^2+y.^2+z.^2;
N4 =@(x,y,z) x.^2+y.^2+z.^2;

A =@(x,y,z)[N1(x,y,z), N2(x,y,z); N3(x,y,z), N4(x,y,z)];
    
integral3(A,0,2,0,2,0,2)

Error using integral2Calc>tensor (line 253)
Integrand output size does not match the input size.

Error in integral2Calc>integral2t (line 55)
[Qsub,esub,FIRSTFUNEVAL,NFE] = tensor(thetaL,thetaR,phiB,phiT,[],[], ...

Error in integral2Calc (line 9)
[q,errbnd] = integral2t(fun,xmin,xmax,ymin,ymax,optionstruct);

Error in integral3>innerintegral (line 128)
Q1 = integral2Calc( ...

Error in integral3>@(x)innerintegral(x,fun,yminx,ymaxx,zminxy,zmaxxy,integral2options) (line 111)
f = @(x)innerintegral(x, fun, yminx, ymaxx, ...

Error in integralCalc>iterateScalarValued (line 334)
fx = FUN(t);

Error in integralCalc>vadapt (line 148)
[q,errbnd] = iterateScalarValued(u,tinterval,pathlen, ...

Error in integralCalc (line 77)
[q,errbnd] = vadapt(vfunAB,interval, ...

Error in integral3 (line 113)
Q = integralCalc(f,xmin,xmax,integralOptions);

Answer 1

[Ayush Anand]

Hi,

You need to integrate each component of your array-valued function separately and then combine the results. You can call "integral3" multiple times to do this, once for each element in your array:

% Define N1,N2,N3,N4
N1 =@(x,y,z) x.^2+y.^2+z.^2;
N2 =@(x,y,z) x.^2+y.^2+z.^2;
N3 =@(x,y,z) x.^2+y.^2+z.^2;
N4 =@(x,y,z) x.^2+y.^2+z.^2;

% Integrate each component separately
I1 = integral3(N1, 0, 2, 0, 2, 0, 2);
I2 = integral3(N2, 0, 2, 0, 2, 0, 2);
I3 = integral3(N3, 0, 2, 0, 2, 0, 2);
I4 = integral3(N4, 0, 2, 0, 2, 0, 2);

% Combine the results into an array
A = [I1, I2; I3, I4];

Hope this helps!

[Rohan]

Hi, Thanks for the answer. Howeverm this is what i want to avoid.Because here above is just example. However, the real one looks more complicated and bigger. i would like to solve in one shot, with intrinsic function or in worst case looping.

Answer 2

[Torsten]

N = cell(2,3);
N{1,1} =@(x,y,z) x.^2+y.^2+z.^2;
N{1,2} =@(x,y,z) x.^2+y.^2+z.^2;
N{1,3} =@(x,y,z) x.^2; 
N{2,1} =@(x,y,z) y.^2+z.^2;
N{2,2} =@(x,y,z) x.^2+y.^2+z.^2;
N{2,3} =@(x,y,z) z.^2;
[I,J] = meshgrid(1:2,1:3);
A = arrayfun(@(i,j)integral3(N{i,j},0,2,0,2,0,2),I,J).'

A = 2x3

32.0000 32.0000 10.6667 21.3333 32.0000 10.6667

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[Rohan]

Hi @Torsten

Really thanks for the answer. Does that also work with array valued function in form of matrices not cell array ? I tried some combination but failed.

Would you please provide variant for the matrices instead of cell array ? or it is not possible. I mean function handle in following kind.

N1 =@(x,y,z) x.^2+y.^2+z.^2;

N2 =@(x,y,z) x.^2+y.^2+z.^2;

N3 =@(x,y,z) x.^2+y.^2+z.^2;

N4 =@(x,y,z) x.^2+y.^2+z.^2;

A =@(x,y,z)[N1(x,y,z), N2(x,y,z); N3(x,y,z), N4(x,y,z)];

Kind regards,

Rohan

[Torsten]

Does that also work with array valued function in form of matrices not cell array ?

No. Your A from above is not a matrix of 4 function handles, but 1 matrix-valued function handle. Thus the elements of the matrix are not function handles on their own and cannot be extracted as such for the 3d-integration. And integrating all matrix functions in one go is not possible with integral3.