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 = Q p?1 k=0 E ?1 k A k down to the product of a single pair ^ E ?1 ^ A. 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 diier in op count, work space, and inherent parallelism. They have in common that they are immune to overrows and use no matrix inversions. A rounding error analysis shows that the special case of collapsing two pairs is numerically backward stable.