A complex form containing constant, linear, quadratic, etc., forms

Helow,

I am struggling to find a symbolic representation which contains, constant, linear, quadratic, cubic, etc., forms. I believe the best is to explain via example. Consider the following:

N = 3;
syms x X1 a [N 1] real
syms A X2 [N N] real
syms C X3 [N N N] real
syms E [N N N N] real
for i=1:N  % Linear interactions
    X1(i) = x(i);
end
for i=1:N   %Quadratic interactions
    for j=1:N
        X2(i,j) = x(i)*x(j);
    end
end
for i=1:N   %Cubic interactions
    for j=1:N
        for k=1:N
            X3(i,j,k) = x(i)*x(j)*x(k);
        end
    end
end
F = a+A*X1+squeeze(sum(permute(C,[3 2 1]).*repmat(X2,[[1 1 N]]), [1 2]))

F = 

F =

a1 + A1_1*x1 + A1_2*x2 + A1_3*x3 + C1_1_1*x1^2 + C1_2_2*x2^2 + C1_3_3*x3^2 + C1_1_2*x1*x2 + C1_1_3*x1*x3 + C1_2_1*x1*x2 + C1_2_3*x2*x3 + C1_3_1*x1*x3 + C1_3_2*x2*x3

a2 + A2_1*x1 + A2_2*x2 + A2_3*x3 + C2_1_1*x1^2 + C2_2_2*x2^2 + C2_3_3*x3^2 + C2_1_2*x1*x2 + C2_1_3*x1*x3 + C2_2_1*x1*x2 + C2_2_3*x2*x3 + C2_3_1*x1*x3 + C2_3_2*x2*x3

a3 + A3_1*x1 + A3_2*x2 + A3_3*x3 + C3_1_1*x1^2 + C3_2_2*x2^2 + C3_3_3*x3^2 + C3_1_2*x1*x2 + C3_1_3*x1*x3 + C3_2_1*x1*x2 + C3_2_3*x2*x3 + C3_3_1*x1*x3 + C3_3_2*x2*x3

In above, F is a [3 1] multivariate function in which each of its elements is written interms of constant, linear and quadratic interactions. I believe, it should be clear which 'pattern' I mean now. I had a tough time to generate the quadratic terms. I need to expand F in terms of higher-order interactions (cubic, qurtic, etc.). I believe there should be a more elegant way to do what I am seeking.

I appreciate your help in advance!

Babak

[John D'Errico]

Do you seriously want to have BOTH of these terms:

C1_1_2*x1*x2

C1_2_1*x1*x2

That seems probably wrong. But then, I could be wrong myself.

[Mohammad Shojaei Arani]

Yes John

The reason is that here I have a hyper graph. C1_1_2 means the impact of node 1 on node 2. C1_2_1

is about the impact of node 2 on node 1. Of course, from algebraic points of view we can

write (C1_1_2+C1_2_1)*x1*x2. But, if I do so in my code at the end I cannot find which node

has a bigger impact on the other one. So, I need to find the causal relations and this is only

possible if I do not mix things.

[Catalytic]

@Mohammad Shojaei Arani you have been posting in this forum for two years. It is high time you learn to format your code.

Answer 1

[Walter Roberson]

N = 3;
syms x X1 a [N 1] real
syms A X2 [N N] real
syms C X3 [N N N] real
syms E [N N N N] real
for i=1:N  % Linear interactions
    X1(i) = x(i);
end
for i=1:N   %Quadratic interactions
    for j=1:N
        X2(i,j) = x(i)*x(j);
    end
end
for i=1:N   %Cubic interactions
    for j=1:N
        for k=1:N
            X3(i,j,k) = x(i)*x(j)*x(k);
        end
    end
end
F = a+A*X1+squeeze(sum(permute(C,[3 2 1]).*repmat(X2,[[1 1 N]]), [1 2]))

F = 

collect(F, [x1 x2 x3])

ans =