Using Riemannian SVD for Problems in Approx- imate Algebra

Many fundamental problems for approximate polynomials can be reformulated as structured linear approximation problems. Problems such as finding polynomials with a GCD which are nearest to given relatively prime polynomials, finding the nearest Ore polynomials with a non-trivial GCRD, finding the nearest polynomial which functionally decomposes, and finding the nearest multivariate polynomial which factors or divides another polynomial, can all be cast as so-called Structured Total Least Squares (STLS) problems. More generally, given a basis of “structure” matrices for a vector space V of structured matrices, and a matrix A in V , the STLS problem seeks to find the “nearest” B to A in V such that B is singular. We present an implementation of one heuristic (the Riemannian SVD) to solve the STLS problem, and demonstrate its effectiveness on the problems discussed above. For the approximate GCD problem, we compare the solutions this approach finds to solutions found by other methods.