Thursday, April 24, 2014

Generalization of Ax = b

Generalization of Ax = b

The linear system \begin{align*} Ax = b \end{align*} is not always solvable, but intesting things can happen if we see it from the numerical point of view.

We never really solve for x anyway, we just get an $x_1 \approx x$ so that $Ax_1 - b \approx 0$. If we write \begin{align*} Ax_1 - b = \min (|Ax - b|^2) \end{align*} then $x_1$ is the more general solution for the system. (By the way, $x_1$ is the least squares estimate of $x$)

This is actually exactly the syntax of the armadillo library: solve(A, b)