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Computing GCRDs of Approximate Differential Polynomials
Differential (Ore) type polynomials with approximate polynomial coefficients
are introduced. These provide a useful representation of approximate
differential operators with a strong algebraic structure, which has been used
successfully in the exact, symbolic, setting. We then present an algorithm for
the approximate Greatest Common Right Divisor (GCRD) of two approximate
differential polynomials, which intuitively is the differential operator whose
solutions are those common to the two inputs operators. More formally, given
approximate differential polynomials and , we show how to find "nearby"
polynomials and which have a non-trivial GCRD.
Here "nearby" is under a suitably defined norm. The algorithm is a
generalization of the SVD-based method of Corless et al. (1995) for the
approximate GCD of regular polynomials. We work on an appropriately
"linearized" differential Sylvester matrix, to which we apply a block SVD. The
algorithm has been implemented in Maple and a demonstration of its robustness
is presented.Comment: To appear, Workshop on Symbolic-Numeric Computing (SNC'14) July 201
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