# FTRAN

This applies to the factored system,

$LaTeX: E_1 E_2 ... E_n x = b$,

where each $LaTeX: E_i$ is an elementary matrix. The recursion is:

$LaTeX: \begin{array}{rcl} E_1 x_1 & = & b \\ E_2 x_2 & = & x_1 \\ & \vdots & \\ E_n x_n & = & x_{n-1} \end{array}$

In the end, $LaTeX: x = x_n$ is the solution. Each elementary system is solved as follows. For notational convenience, suppose the system is $LaTeX: Ex = y$, and $LaTeX: v$ is the distinguished $LaTeX: p$-th column of $LaTeX: E$. Then,

$LaTeX: x(p) = y(p)/v(p)$, and $LaTeX: x(i) = y(i) - v(i)x(p)$ for $LaTeX: i \ne p$.

This is what is done in the (revised) simplex method, and each elementary solution is a pivot operation.