Accurate computation of the product-induced singular value decomposition with applications

We present a new algorithm for floating-point computation of the singular value decomposition (SVD) of the product B-tau C, where B and C are full row rank matrices. The algorithm replaces the pair (B, C) with an equivalent pair (B', C') and then it uses the Jacobi SVD algorithm to compute the SVD of the explicitly computed matrix B'(tau) C'. In this way, each nonzero singular value sigma is approximated with some sigma + delta sigma, where the relative error \delta sigma\/sigma is, up to a factor of the dimensions, of order epsilon{min(Delta is an element of D) kappa(2) (Delta B) + min(Delta is an element of D) kappa(2)(Delta C)}, where D denotes the set of diagonal nonsingular matrices, kappa(2) (.) denotes the spectral condition number, and epsilon is the roundoff unit of floating-point arithmetic. The new algorithm is applied to the eigenvalue problem HMx = lambda x with symmetric positive definite H and M. It is shown that each eigenvalue lambda is computed with high relative accuracy and that the relative error \delta lambda\/lambda of the computed approximation lambda + delta lambda is, up to a factor of the dimension, of order epsilon{min(Delta is an element of D) kappa(2)(Delta H Delta) + min(Delta is an element of D) kappa(2)(Delta M Delta)}. The new algorithm can also be used for accurate SVD computation of a single matrix G that admits an accurate factorization G = B-tau C.