# Gaussian elimination

A method to solve $LaTeX: Ax=b$ that performs elementary row operations on $LaTeX: A$ to annihilate successive elements of $LaTeX: A$ in order to reduce $LaTeX: A$ to an upper triangular matrix, $LaTeX: U$. On paper, the same operations are applied to $LaTeX: b$, then the solution is obtained by solving the resulting upper triangular system. In a computer, the product of the matrices effecting the elementary row operations is a lower triangular matrix, $LaTeX: L$, with unit diagonal. Once this phase is completed, the system $LaTeX: Ax=b$ becomes $LaTeX: LUx=b$. This is then solved in two steps: forward substitution solves $LaTeX: Ly=b$; then backward substitution solves $LaTeX: Ux=y$. (Of course, computer implementations vary.)