How I compute multiple sum in matlab for 4D matrix?

How I compute multiple sum in matlab for 4D matrices as below? U and V are 4D matrices.

[Stephen23]

Use sum.

[Mehdi]

Yes as you see in enclosed photo there are 6 sum.

[Guillaume]

Why do some summations start at 0 and others at 1?

[Mehdi]

it is the format my problem, I cant change it.

[Amir Xz]

Why you don't use "for" loops?

 U = rand(5,4,3,2);
V = rand(5,4,3,2);
I=5;J=4;M=2;N=3;
sumUV=0;
for i=1:I
    for j=1:J
        for k=1:I
            for r=2:M
                for s=2:M
                    for q=2:N
                        sumUV = sumUV + U(i,j,q,r)*V(k,j,q,s);
                    end
                end
            end
        end
    end
end

[Mehdi]

becasuse the speed is very low, I have many similar expressions must be computed in program.

[Matt J]

Mehdi's comment relocated here:

in Maple you can simply do it by add command like below:

A=add(add(add(add(add(add(U[i,j,q,r]*V[k,j,q,s],i=0..I),j=0..J),k=0..I),r=1..M))s=1..M),q=1..N)

is there any similar way to do it in Matlab?

[Matt J]

If you mean, how can you sum over specific dimensions of an array, my answer shows how to do that using the sum() command.

[Mehdi]

you mean I must replace add with sum like:

A=sum(sum(sum(sum(sum(sum(U[i,j,q,r]*V[k,j,q,s],i=0..I),j=0..J),k=0..I),r=1..M)),s=1..M),q=1..N)

[Matt J]

No, I mean you should do as I posted in my answer below. Reading the sum() commmand documentation would also probably be beneficial.

[Mehdi]

As I said I have many similar expressions like that and it seems your method is not applicable for more expressions. I am searching a similar command like add in Maple.

[Matt J]

If they are "similar", my method - which is to use the sum command - is applicable. Have you read the documentation on sum()? It is the best analogue of add that you will find in Matlab.

Answer 1

[Matt J]

Notice that the only indices shared between U and V are j and q. Therefore, you can minimize multiplication operations as follows,

partialU=sum(sum(U,4),1); %sum over i and r
partialV=sum(sum(V,4),1); %sum over k and s

A=sum(partialU(:).*partialV(:)); %sum over j and q

[Matt J]

In fact, it would probably be faster to do this as a vector-vector inner product,

A= partialU(:).'*partialV(:);

[Mehdi]

believe me, your method is very different from add command in Maple.add is very simpler. What about if U and V be symbolic parameters?

[Matt J]

What about if U and V be symbolic parameters?

But they are not. You said that they are numeric arrays, which is a smart choice, since you said that speed is important here. However, sum() should still work with symbolic variables. There is also symsum().

I don't see how add is simpler. Your expressions using add are a lot longer than the ones I've presented using sum.

[Mehdi]

I am amazed. Here I ask my question again: What is the best way to calculate following expression in Matlab? Note U and V are known but q's are unknown (symbolic)

[Matt J]

You should edit your original post with this new equation and information about q_r and q_s. The new expression is rather unclear. You have the symbol q being used for two different things, both as a symbolic variable and as an index of summation.

[Mehdi]

I asked the question in new page as below: https://uk.mathworks.com/matlabcentral/answers/418526-what-is-the-best-way-to-calculate-multiple-summations-expression-in-matlab