Gaussian elimination
From the numerical linear algebra perspective, Gaussian elimination is a procedure for factoring a matrix "A" into its "LU" factorization, which Gaussian elimination accomplishes by left-multiplying "A" by a succession of matrices formula_20 until "U" is upper triangular and "L" is lower triangular, where formula_21. Naive programs for Gaussian elimination are notoriously highly unstable, and produce huge errors when applied to matrices with many significant digits. The simplest solution is to introduce pivoting, which produces a modified Gaussian elimination algorithm that is stable.