How to write this function in Matlab having inner product

Here,

inner product is defined as <u,v>=

Is this is correct?

function F_val = F(wi, ci, Li, K, e, sigma, zi)
    % Compute the first term: K <e sum from i to N ci Li, e sum from i to N ci Li>
    term1 = K * e^2 * sum(exp(sum(ci .* Li)) .* exp(sum(ci .* Li)));

    % Compute the second term: K <e sum from i to N ci Li, -2 sigma>
    term2 = K * e * sum(exp(sum(ci .* Li)) * (-2 * sigma));

    % Compute the third term: summation i to N e ci <e^wi, wi>
    term3 = e * sum(zi .* ci .* exp(wi));

    % Compute the final result: F(w) = term1 + term2 + term3
    F_val = term1 + term2 + term3;
end

Answer 1

[William Rose]

@SHAHID,

This is a standard definition of inner product for continuous functions. The limits of integration, 0 to 1 in your example, are chosen to be useful in your situation. For a different problem, different limits might be more appropriate.

[SHAHID]

Thanks fpr your response. In my problem limit is from 0 to 1 also. But I want to ask we will write this notation by integral(f,0,1) or something else. I will wait your response. Thanks

[Torsten]

What is w ? A vector of functions (w_1,...,w_N) ?

How is L_i(e^w_i) defined ?

What are c = (c_1,...,c_N) and h = (h_1,...,h_N) ? Vectors of numbers of length N ? Or also vectors of functions ?

[SHAHID]

dsdz. This is defined for N=2.

[William Rose]

@SHAHID,

Please provide an example script that calls the function, to demonstrate its use.

Your initial post included a definition of inner product for continuous functions. Is that applicable here? If so, what is the variable of integration, x? Why does it not appear in the equation for F(w)? Your Matab code does not integrate anything from 0 to 1.

Perhaps, as @Torsten suggests, the inner product in this case is the inner product of two vectors. Please clarify.

The equation

uses "e" in multiple places. Are they all the same "e", specifically e=2.718...? If they are different e's, then write the equation differently, for clarity.

The first function inside the first inner product is

How do you interpret that? It could mean

which may be written in Matlab

f1=exp(sum(c.*L.*w));

where c, L, and w are vectors of length N. In this case, f1 will be a scalar, not a vector. And it will be a function of its arguments (ci, Li, wi). The fact that you pass the value e to the function suggests that the first "e" is not e=2.718. An alternative interpretation of the first function inside the inner product is

where I have used e0 for the first "e", for clarity. This may be written in Matlab

f1=e0*sum(c.*L.*exp(w));

In this case, as in th previous interpretation, f1 is a scalar, and is a function of its arguments (ci, Li, wi). The second function inside the first inner product is similar, but also includes a term "hi". The same questions about interpretation apply to that function.

[Torsten]

@SHAHID

We are again at the point where we were last time you asked the question in a broader context. So I'll quit here because it seems you haven't been able to clarify matters in the meantime.