Analysis of Algorithms for Orthogonalizing Products of Unitary Matrices
Roy Mathias · Numerical Linear Algebra with Applications · 1996
We consider the problem of computing Uk = QkUk−1(where U0 is given) in finite precision (ϵM = machine precision) where U0 and theQi are known to be unitary. The problem is that Ûk, the computed product may not be unitary, so one applies an O(n2) orthogonalizing step after each multiplication to (a) prevent Ûk from drifing too far from the set of untary matrices (b) prevent Ûk from drifting too far from Uk the true product. Our main results are 1. Scaling the rows to have unit length after each multiplication (the cheaptest 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 the distance of Ûk (k = 1, 2, …) to the set of unitary matrices is bounded by n3.5ϵM for any choice of Qi.