Some new theorems generalizing a result of Oettli and Prager are applied to the a posteriori analysis of the compatibility of a computed solution to the uncertain data of a linear system (or of a polynomial equation).

In 1999 Amodio and Mazzia presented a new backward error analysis for $LU$ factorization and introduced a new growth factor $?_n$. Their very interesting approach allowed them to obtain sharp error bounds. In particular, they derive nice results assuming that partial pivoting is used. However, the forward error bound for the solution of a linear system whose coefficient matrix $A$ is an $M$-matrix given in Theorem 4.1 of that paper is not correct. They first obtain a bound for the condition number $?(U)$ assuming that one has the $LU$ factorization of an $M$-matrix and then they apply the bounds obtained when partial pivoting is used. But if $P$ is the permutation associated with partial pivoting then $PA = LU$ can fail to be an $M$-matrix and the bound for $?(U)$ can be false, as shown in our Example 1.1. We also prove that, for a pivoting strategy presented in the paper, the growth factor of an $M$-matrix $A$ is $?_n(A) = 1$ and $?(U) \leq ?(A)$, where $U$ is the upper triangular matrix obtained after applying such a pivoting strategy.

In the spirit of Wilkinson?s [1, 2] backward error analysis, conditions are established under which a given approximate solution of a system of $n$ linear equations with $n$ unknowns is the exact solution of a modified system whose coefficients and right-hand sides are within a given neighborhood of those of the original system.

An equilibration is introduced that minimizes the maximum ratio of the upper bounds for the backward rounding error and the inherent error in the given data. This can be applied to the solution of systems of linear equations and to the linear eigenproblem.

A new backward error analysis of $LU$ factorization is presented. It allows to obtain a sharper upper bound for the forward error and a new definition of the growth factor that we compare with the well known Wilkinson growth factor for some classes of matrices. Numerical experiments show that the new growth factor is often of order approximately $\log_2 n$ whereas Wilkinson?s growth factor is of order n or $\sqrt n$.