We briefly survey several existing methods for solving polynomial system with inexact coefficients, then introduce our new symbolic-numeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method.
There has been considerable progress in the theory and computer implementation of symbolic computation algorithms to automatically determine and exploit exact symmetries of exact differential equations. Such programs usually apply a finite number of exact differentiations and eliminations to the overdetermined linearized equations for the unknown symmetries (the symmetry defining equations), to complete them to certain involutive or standard forms. The symmetry properties can be determined from these involutive forms. In many applications, however, the differential equations describing a model are only known approximately. For example they may contain parameters that are only known approximately. Symbolic methods are unstable if applied to the symmetry defining equations directly, and indirect techniques (e.g. replacing approximate parameters by symbolic ones) might not be practical in cases with many parameters. Discussion and examples are given of such difficulties. A new generation of symbolic-numeric methods is described and applied to the problem of determining symmetries of differential equations. We introduce a class of differential-elimination methods which uses Numerical Linear Algebra, and in particular the Singular Value Decomposition, to perform the elimination process on the symmetry defining equations. Our approach uses symbolic differentiations but not symbolic eliminations. Substitution of an appropriate random point in the independent variables, followed by numerical projection is used to test for the conditions of completion to a projective involutive form. We prove that this form is equivalent to the involutive form of the Cartan-Kuranishi theory of partial differential equations. Our method is applied to determining symmetry properties of 50 ode from the collection in Kamke's book.
In this paper we extend complex homotopy methods to finding witness points on the irreducible components of real varieties. In particular we construct such witness points as the isolated real solutions of a constrained optimization problem.First a random hyperplane characterized by its random normal vector is chosen. Witness points are computed by a polyhedral homotopy method. Some of them are at the intersection of this hyperplane with the components. Other witness points are the local critical points of the distance from the plane to components. A method is also given for constructing regular witness points on components, when the critical points are singular.The method is applicable to systems satisfying certain regularity conditions. Illustrative examples are given. We show that the method can be used in the consistent initialization phase of a popular method due to Pryce and Pantelides for preprocessing differential algebraic equations for numerical solution.
Complicated nonlinear systems of pde with constraints (called pdae) arise frequently in applications. Missing constraints arising by prolongation (differentiation) of the pdae need to be determined to consistently initialize and stabilize their numerical solution. In this article we review a fast prolongation method, a development of (explicit) symbolic Riquier Bases, suitable for such numerical applications. Our symbolic-numeric method to determine Riquier Bases in implicit form, without the unstable eliminations of the exact approaches, applies to square systems which are dominated by pure derivatives in one of the independent variables. The method is successful provided the prolongations with respect to a single dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are nonsingular when evaluated at points on the zero sets defined by the functions of the pdae. For polynomially nonlinear pdae, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. Our method generalizes Pryce's method for dae to pdae. Given a dominant independent time variable, for an initial value problem for a system of pdae we show that its semi-discretization is also naturally amenable to our symbolic-numeric approach. In particular, if our method can be successfully applied to such a system of pdae, yielding an implicit Riquier Basis, then under modest conditions, the semi-discretized system of dae is also an implicit Riquier Basis.
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