Gaussian elimination

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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.)

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