Two ideas are combined to construct a hybrid symbolicnumeric differential-elimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagonal homotopies to incrementally process new constraints, one at a time. The method is illustrated on several examples, combining symbolic differential elimination (using rifsimp) with numerical homotopy continuation (using phc).

The computational difficulty of completing nonlinear pde to involutive form by differential elimination algorithms is a significant obstacle in applications. We apply numerical methods to this problem which, unlike existing symbolic methods for exact systems, can be applied to approximate systems arising in applications. We use Numerical Algebraic Geometry to process the lower order leading nonlinear parts of such pde systems. The irreducible components of such systems are represented by certain generic points lying on each component and are computed by numerically following paths from exactly given points on components of a related system. To check the conditions for involutivity Numerical Linear Algebra techniques are applied to constant matrices which are the leading linear parts of such systems evaluated at the generic points. Representations for the constraints result from applying a method based on Polynomial Matrix Theory. Examples to illustrate the new approach are given. The scope of the method, which applies to complexified problems, is discussed. Approximate ideal and differential ideal membership testing are also discussed.

Homotopy continuation methods find approximate solutions of a given system by a continuous deformation of the solutions of a related exactly solvable system. There has been much recent progress in the theory and implementation of such path following methods for polynomial systems. In particular, exactly solvable related systems can be given which enable the computation of all isolated roots of a given polynomial system. Extension of such methods to determine manifolds of solutions has also been recently achieved. This progress, and our own research on extending continuation methods to identifying missing constraints for systems of differential equations, motivated us to implement higher order continuation methods in the computer algebra language Maple. By higher order, we refer to the iterative scheme used to solve for the roots of the homotopy equation at each step. We provide examples for which the higher order iterative scheme achieves a speed up when compared with the standard second order scheme. We also demonstrate how existing Maple numerical ODE solvers can be used to give a predictor only continuation method for solving polynomial systems. We apply homotopy continuation to determine the missing constraints in a system of nonlinear PDE, which is to our knowledge, the first published instance of such a calculation. 1.

Riquier Bases for systems of analytic pde are, loosely speaking, a differential analogue of Gröbner Bases for polynomial equations. They are determined in the exact case by applying a sequence of prolongations (differentiations) and eliminations to an input system of pde. We present a symbolic-numeric method to determine Riquier Bases in implicit form for systems which are dominated by pure derivatives in one of the independent variables and have the same number of pde and unknowns. The method is successful provided the prolongations with respect to the dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are non-singular when evaluated at points on the zero sets defined by the functions of the pde. For polynomially nonlinear pde, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. We give a differential algebraic interpretation of Pryce’s method for ode, which generalizes to the pde case. A major aspect of the method’s efficiency is that only prolongations with respect to a single (dominant) independent variable are made, possibly after a random change of coordinates. Potentially expensive and numerically unstable eliminations are not made. Examples are given to illustrate theoretical features of the method, including a curtain of Pendula and the control of a crane.

Abstract. Camera pose estimation is the problem of determining the position and orientation of an internally calibrated camera from known 3D reference points and their images. We introduce a new polynomial equation system for 4-point pose estimation and apply our symbolicnumeric method to solve it stably and efficiently. In particular, our algorithm can also recognize the points near critical configurations and deal with these near critical cases carefully. Numerical experiments are given to show the performance of the hybrid algorithm. 1