Bounds for relative errors of complex matrix factorizations Original Research Article

The goal of this paper is to obtain optimal first order bounds for absolute and relative errors of unitary and Hermitian factors of some commonly used matrix factorizations. We have chosen the strong derivative calculus approach and we have expressed the factors as a differentiable function of the data but since these expressions define the functions implicitly, the inverse function theorem plays a central role in finding the Jacobian matrix. Then, first order bounds are deduced by means of the mean value theorem for the derivatives. We either improve or generalize some of the bounds proposed by Bhatia [1], Stewart [2], and Sun [3].