This is my first MathOverflow post: I apologize if my message is lacking of something. I also posted this question in Math StackExchange, but as I haven't seen an answer I post it here. 

Suppose you have system of equations:
$$Ax = b  \rightarrow [A_{1}, A_{2},...,A_{M}]^{T}x = [b_{1},...,b_{M}]^{T} \: \: \: (1)$$ 

where each and all matrices $A_{1},...,A_{M}$  $\in \mathbb{R}^{S \times L}$. We call this system of equations the *whole system*.

In the same manner, each and all $b_{1},..., b_{M} \in \mathbb{R}^{S \times 1}$ and $x \in \mathbb{R}^{L \times 1}$.

As you see in (1), $A$ is composed by matrices $A_{1},...,A_{M}$ as well as $b$ by vectors $b_{1},...,b_{M}$. For each of these $A_{i}$ and $b_{i}$ you can write down equations:

$$A_{i}x_{i} = b_{i}, \: \; i = 1,...,M \: \: \: (2)$$

We call each of these equations a *partial system*.

By Least Squares you know the solutions $x_{1},...,x_{M}$ to each of the *partial systems* in (2). However, by restrictions **you CAN'T solve directly** the *whole system* of equations in (1) through Least Squares. 

Let's suppose we know the error of each of the partial solutions $x_{i}$ with respect to the whole solution $x$. That is to say that we know: 

$$ e(x_{i}) = || x_{i} - x||_{p} \: \: \: (3)$$

where in (3) $||\cdot||_{p}$ is some norm to be defined.

We want to build $x$ from $x_{1},..,x_{M}$ in such a way that the more $x_{i}$ that we include into the solution we build, the smaller the error with respect to $x$, and all the $x_{i}$ have the same importance. 

> In other words, we want to produce a sequence of functions $y_{1},...y_{M}$ as follows:

$$y_{1} = f(x_{k_{1}}), \: \: \: f: \mathbb{R}^{L} \rightarrow \mathbb{R}^{L} $$
$$y_{2} = f(x_{k_{1}},x_{k_{2}}), \: \: \: f: \mathbb{R}^{L} \times \mathbb{R}^{L} \rightarrow \mathbb{R}^{L}$$
$$y_{3} = f(x_{k_{1}},x_{k_{2}},x_{k_{3}}), \: \: \: f: \mathbb{R}^{L} \times \mathbb{R}^{L} \times \mathbb{R}^{L} \rightarrow \mathbb{R}^{L}$$
$$.$$
$$.$$
$$.$$
$$y_{M} = f(x_{k_{1}},...,x_{k_{M}}) = x, \: \: \: f: \mathbb{R}^{L} \times ... \times \mathbb{R}^{L} \rightarrow \mathbb{R}^{L}$$

such that $e(y_{M}) < e(y_{M-1}) < ... < e(y_{2}) < e(y_{1})$. 

The indexes $k_{1},...,k_{M}$ represent which of the solutions $x_{i}$ are taking into account to build each $y$. We use those indexes to say that any of the partial solutions $x_{i}$ **is NOT more important** than any other.