We treat pseudo-factorization of a multivariate polynomial p over C as an overdetermined bilinear system for the coefficients of the factors. If the specified data (coefficients of p) are sufficiently close to the manifold of consistent data we project onto it by solving a well-chosen subsystem (otherwise we quit). On the manifold, we reduce the backward error by a linearized minimization procedure. The resulting algorithm appears to extend the range of treatable cases - in terms of number of variables and total degree - considerably.

Article Singular systems of polynomials Share on Authors: Hans J. Stetter Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, Austria Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, AustriaView Profile , Günther H. Thallinger Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, Austria Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, AustriaView Profile Authors Info & Claims ISSAC '98: Proceedings of the 1998 international symposium on Symbolic and algebraic computationAugust 1998 Pages 9–16https://doi.org/10.1145/281508.281525Online:01 August 1998Publication History 5citation281DownloadsMetricsTotal Citations5Total Downloads281Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

We consider a cluster of m zeros of a multivariate polynomial system which we interpret as a perturbation of a system with an m-fold zero. By algebraic techniques, we find a first order correct representation of the primary ideal of the cluster zeros from which we obtain approximations for the individual zeros in the cluster.

Efficient and asymptotically (h --, 0) correct global error estimation procedures for initial value problem ODE-codes may be based on a suitable version of defect correction.An implementation of such a procedure for a variable order, variable step Adams PECE-code (typically STEP of Shampme) is described; the global error vector is produced simultaneously with the solution vector at each output point which need not be a grid point.The results of extensive experimentation are presented and d~scussed; m these tests, the qualitative and quantitative behavior of the true error was correctly reflected by the numerical estnnates throughout.

The order p which is obtainable with a stable k -step method in the numerical solution of y′ = f ( x , y ) is limited to p = k + 1 by the theorems of Dahlquist. In the present paper the customary schemes are modified by including the value of the derivative at one “nonstep point;” as usual, this value is gained from an explicit predictor. It is shown that the order of these generalized predictor-corrector methods is not subject to the above restrictions; stable k -step schemes with p = 2 k + 2 have been constructed for k ≤ 4. Furthermore it is proved that methods of order p actually converge like h p uniformly in a given interval of integration. Numerical examples give some first evidence of the power of the new methods.