Danielle A. Brake, Jonathan D. Hauenstein, Frank-Olaf Schreyer et al.
Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two methods to compute them and present several applications.
Daniel J. Bates, Andrew J. Sommese, Charles W. Wampler
A path tracking algorithm that adaptively adjusts precision is presented. By adjusting the level of precision in accordance with the numerical conditioning of the path, the algorithm achieves high reliability with less computational cost than would be incurred by raising precision across the board. We develop simple rules for adjusting precision and show how to integrate these into an algorithm that also adaptively adjusts the step size. The behavior of the method is illustrated on several examples arising as homotopies for solving systems of polynomial equations.
Andrew J. Sommese, Jan Verschelde, Charles W. Wampler
By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets. Of particular interest is the special case where one of the sets is defined by a single polynomial equation. This leads to an algorithm for finding a numerical representation of the solution set of a system of polynomial equations introducing the equations one-by-one. Preliminary computational experiments show this approach can exploit the special structure of a polynomial system, which improves the performance of the path following algorithms.