Analysis of Algorithms for Orthogonalizing Products of Unitary Matrices

We consider the problem of computing U k+1 = Q k U k+1 ; U 0 given in nite precision (M = machine precision) where U 0 and the Q i are known to be unitary. The problem is that ^ U k , the computed product may not be unitary, so one applies an O(n 2) orthogonalizing step after each multiplication to (a) prevent ^ U k from drifting too far from the set of unitary matrices (b) prevent ^ U k from drifting too far from U k the true product. Our main results are 1. Scaling the rows to have unit length after each multiplication (the cheapest of the algorithms considered) is usually as good as any other method with respect to either of the criteria (a) or (b). 2. A new orthogonalization algorithm that guarantees that the distance of ^ U k (k = 1; 2; : : :) to the set of unitary matrices is bounded by n 3:5 M for any choice of Q i .