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=∏ₖ₌₀ᵖ⁻¹ Eₖ⁻¹Aₖ 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.