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SymbolicNumeric Completion of Differential Systems by Homotopy Continuation
 Proc. ISSAC 2005. ACM
, 2005
"... Two ideas are combined to construct a hybrid symbolicnumeric differentialelimination 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 diagona ..."
Abstract

Cited by 4 (1 self)
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Two ideas are combined to construct a hybrid symbolicnumeric differentialelimination 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).
Application of Numerical Algebraic Geometry and Numerical Linear Algebra to PDE
 ISSAC'06
, 2006
"... 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 ar ..."
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Cited by 4 (2 self)
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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.