Pivoting strategies leading to small bounds of the errors for certain linear systems

Abstract

For the solution of a linear system Ax=b using Gaussian elimination, some new properties of scaled partial pivoting strategies for a strictly monotone vector norm ∥·∥ are obtained. Given a nonsingular matrix A, we have proved that, if there exists a permutation matrix P such that PA=LU and |PA|=|L| |U| (where |B| denotes the matrix of absolute values of the entries of B), then P is associated with the scaled partial pivoting strategy for ∥·∥. This result is applied to extend a well-known small componentwise relative backward error for the Gaussian elimination of certain matrices to larger classes of matrices. On the other hand, given a matrix A=LU it is shown that, if there exists an optimal pivoting strategy in order to diminish the Skeel condition number Cond(U) of the resulting upper triangular matrix U, then it coincides with the scaled partial pivoting for ∥·∥. Furthermore, the class of matrices for which no row exchanges is an optimal pivoting strategy to diminish Cond(U) is characterized.