solution of a linear underdetermined system of equations

Question

Is there any numerical method to find vector that satisfies an underdetermined system of linear equations?

Example:

• $A \mathbf{x} = \mathbf{b}$
• Let $A$ be $ n \times m$
• Let $\mathbf{x}$ be $ m \times 1$
• $m>n$

This system has infinitude of solutions, need any one of them

Answer 1

Consider a splitting of $A=[\ \underbrace{A_1}_{n \times n} \ \ | \ \underbrace{A_2}_{n \times (m-n)}]$ and a vertical splitting for the unknown vector $x=\binom{x_1}{x_2}$, $x_1$ being $n \times 1$ and $x_2$ being $(m-n) \times 1$.

Thus, initial system $Ax=b$ becomes:

$$A_1x_1+A_2x_2=b \ \ \iff $$

$$\tag{1}A_1x_1=\underbrace{b-A_2x_2}_{b_3}.$$

Giving any value to vector $x_2$, the square system (1) $A_1x_1=b_3$ has in general, a unique solution.

It remains to "pile up" $x_1$ and $x_2$ to generate a solution $x$.