I don't really know the correct terminology to describe this, but let's say we have many values of $(x_n, y_n, z_n)$. Also let's say that our description of 'many' means that $i$ ranges from $1$ to some number in the order of millions.
I would like to convert each value of $(x_n, y_n, z_n)$ to $(a_n, b_n)$ such that:
$$a_n \cdot(m,n,o) + b_n \cdot(p,q,r) \approxeq (x_n, y_n, z_n)$$
where $(m,n,o)$ and $(p,q,r)$ are constant (that is, the same for all values of $n$).
All numbers in this system are positive, and range from $0$ to $1$.
As I see it, the problem is to find values of $(m,n,o)$ and $(p,q,r)$ that could be used with suitable $(a_n, b_n)$ to produce the closest values possible to the set of ($x_n$, $y_n$, $z_n$). Some inaccuracy is expected so this should be a 'best fit', with some way of reporting how closely it fitted.
Here is an example of what I'd like to end up with (this was made by going backwards, randomly generating various $a_n$, $b_n$, $m$, $n$, $o$, $p$, $q$, $r$ that produces $x_n$, $y_n$, $z_n$):
Input: $$ \begin{array}{c|lcr} n & x & y & z \\ \hline 1 & 0.310 & 0.259 & 0.776 \\ 2 & 0.222 & 0.189 & 0.71 \\ 3 & 0.231 & 0.197 & 0.774 \\ \ldots \\ 999999 &0.254 & 0.21 & 0.55 \\ 1000000 &0.237 & 0.199 & 0.635 \end{array}$$
Output (along with some description of how suitably this fits the input data): $$(m,n,o) = (0.311,0.252,0.411)\\ (p,q,r) = (0.134,0.122,0.796)\\ \begin{array}{c|lcr} n & a & b \\ \hline 1& 0.758 & 0.555 \\ 2& 0.433 & 0.652 \\ 3& 0.425 & 0.738 \\ \ldots \\ 999999& 0.683 & 0.313\\ 1000000& 0.550 & 0.493 \end{array}$$
How should I approach this problem generally? Is there a name for this sort of problem? Are there any algorithms that are designed to solve this sort of thing?
Thank you!
If I do understand correctly then answer to your question is: No it is not always possible to express $\left(x, y, z\right)^t$in terms of fixed $\left(p, q, r\right)^t$ and $\left(m, n, o\right)^t$. For ex. $\left(1, 0, 0\right)^t$ and $\left(0, 1,0\right)^t$ fixed you cannot express $\left(0, 0, 1\right)^t =a \left(1, 0, 0\right)^t+b \left(0, 1, 0\right)^t$