Stability of the Gaussian elimination algorithm.

Consider the system

Ax = b,

where A, L, E ? Rⁿⁿ , x, b ? Rⁿ , with A = L + E. L is unit lower triangular, where all the subdiagonal elements are ?1, i.e., l_i,j = ?1 for i i,i = 1 for i = 1 . . . n, and l_i,j = 0 otherwise. Additionally, E is a matrix of such that e_i,n = 1 for i = 1 . . . n ? 1 and e_i,j = 0 otherwise. For example, when n = 5, we have

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