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How to approximate the answers of a nonlinear system of equations where number of unknowns is greater than number of equations?

Let's suppose for simplicity all $a_i \ge 0$ (otherwise replace $a_i$ by $-a_i$, $x_i$ by $-x_i$ and $y_i$ by $-y_i$ for all those where $a_i < 0$), and $a_1$ is largest. Let $v_i = a_i (x_i, y_i)$....

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