I have $\frac{n(n-1)}{2}$ equations of the form:
$$2(x_j-x_i)X+2(y_j-y_i)Y+2(z_j-z_i)Z+2v^2\tau(t_i-t_j)=2v^2(t_i^2-t_j^2)+(x_j^2+y_j^2+z_j^2)-(x_i^2+y_i^2+z_i^2)$$
With $n\geq 4$, and where everything except $X,Y,Z$ and $\tau$ are known
I understand that this equation (actually, $\frac{n(n-1)}{2}$ equations/datapoints) can be solved for $X,Y,Z,\tau$ (values which I'll use as initial estimates for a more ``precise" solver) with a linear least-squares fit. How, in practice, is this done? I know not too much about the topic. Would I want a multivariate least-squares fit? A multiple least-squares fit? "Standard" least-squares?
My recollection is that I'd wan't a multiple least-squares fit, however I'm not sure.
Note that I specifically want to use least-squares here, and the goal is to obtain estimates for $X,Y,Z,\tau$.
Thanks in advance.
Your equations are of the form
$Xv_1+Yv_2+Zv_3+\tau v_4 = b$
where $v_1,v_2,v_3, v_4$ and $b$ are vectors. Write A = {the matrix whose columns are $v_1,v_2,v_3,v_4$} then the 4 vector $w$ of numbers you want is the solution of the normal equation
$A^TAw = A^Tb$
which is your least squares solution.