Componentwise perturbation bounds for the \(LU\), \(LDU\) and \(LDL^T\) decompositions (Q2717019)

Assume that the \(n\times n\) matrix \(A\) has the \(LU\) decomposition \(A=L_1U,\) where \(L_1\) is unit lower triangular and \(U\) is upper triangular. Also assume that the perturbed matrix \(A+\delta_A\) has the \(LU\) decomposition \(A+\delta_A=(L_1+\delta_{L_1})(U+\delta_U),\) where \(L+\delta_{L_1}\) is unit lower triangular and \(U+\delta_U\) is upper triangular. Finally assume that the spectral radius of \(|L_1\delta_AU^{-1}|\) is less than \(1.\) NEWLINENEWLINENEWLINEThe author gives bounds for \(|\delta_{L_1}|\) and \(|\delta_U|\). Similarly he derives a new perturbation bound for the \(LDU\) factorization.