Optimizing parameters for a system of linear equations

This is rather an easy problem but I cannot get my head around with it. Please help. The question is followign:

I have two data sets in the form of F(fi, xi) and G(gi, xi) with i = 1..65. These two curves supposed to be the same but different in raw format. I want to find the parameters that make these two curves equal such that

aF + bG = 0, or ai * fi + bi * gi = 0 subject to .

In matrix form,

constraints:

I want to find a's and b's.

How do I do it in matlab?

[LS_SL]

I mistyped the functional form of F and G. It should be corrected as F = F(xi, fi) and G = G(xi, gi).

Answer 1

[Torsten]

One possible choice:

a_i = g_i / 2 * sum_{j=1}^{j=65} (f_j - g_j) / sum_{j=1}^{j=65} (f_j * g_j)
b_i = -f_i / 2 * sum_{j=1}^{j=65} (f_j - g_j) / sum_{j=1}^{j=65} (f_j * g_j)

Best wishes

Torsten.

Answer 2

[LS_SL]

Hi Torsten,

Thanks for your suggestion. These two equations algibircally solve for the coefficients ai, bi.

One qeustion remains: How do I incorporate the contraint?

That's where I thought optimization methods would help. Any idea?

[Torsten]

But they satisfy the constraint ...

Or do you want to prescribe a fixed value for "constant" in

sum_{j=1}^{j=65} (a(j)-1)*f(j) = sum_{j=1}^{j=65} (b(j)-1)*g(j) = constant

?