Evaluating products of matrix pencils and collapsing matrix products

This paper describes three numerical methods to collapse a formal product of p pairs of matrices $$P=\mathop{\prod}\limits_{k=0}^{p-1} E_{k}^{-1}A_{k}$$ down to the product of a single pair Ê −1 Â . In the setting of linear relations, the product formally extends to the case in which some of the E k 's are singular and it is impossible to explicitly form P as a single matrix. The methods differ in flop count, work space, and inherent parallelism. They have in common that they are immune to overflows and use no matrix inversions. A rounding error analysis shows that the special case of collapsing two pairs is numerically backward stable. Copyright © 2001 John Wiley & Sons, Ltd.