Scaled Pivots and Scaled Partial Pivoting Strategies

Abstract

Scaled pivots for Gaussian elimination of an n × n matrix are introduced. They are used to obtain bounds for the Skeel condition number of the resulting upper triangular matrix and for a growth factor which has been introduced by Amodio and Mazzia [BIT, 39 (1999), pp. 385--402]. A bound of this growth factor for row scaled partial pivoting strategies is also included. It is proved that the Skeel condition number of an n × n upper triangular matrix which is strictly diagonally dominant by rows is bounded above by a number which is independent of n. It is also shown that the calculation of the n scaled pivots associated with a pivoting strategy presenting nice properties when applied to nonsingular n × n M-matrices adds ${\mathcal{O}}(n)$ elementary operations to the complete cost of this pivoting strategy.