The problem of computing the distance of two real coprime polynomials to the set of polynomials with a nontrivial greatest common divisor (GCD) appears in computer algebra, signal processing, and control theory. It has been studied in the literature under the names approximate common divisor, $\varepsilon$-GCD, and distance to uncontrollability. Existing solution methods use different types of local optimization methods and require a user-defined initial approximation. In this paper, we propose a new method that allows us to include constraints on the coefficients of the polynomials. Moreover, the method proposed in the paper is more robust to the initial approximation than the Newton-type optimization methods available in the literature. Our approach consists of two steps: (1) reformulate the problem as a problem of determining the structured distance to singularity of an associated Sylvester matrix, and (2) integrate a system of ODEs, which describes the gradient associated to the functional to be minimized.

### An ODE-based method for computing the distance of coprime polynomials to common divisibility

#### Abstract

The problem of computing the distance of two real coprime polynomials to the set of polynomials with a nontrivial greatest common divisor (GCD) appears in computer algebra, signal processing, and control theory. It has been studied in the literature under the names approximate common divisor, $\varepsilon$-GCD, and distance to uncontrollability. Existing solution methods use different types of local optimization methods and require a user-defined initial approximation. In this paper, we propose a new method that allows us to include constraints on the coefficients of the polynomials. Moreover, the method proposed in the paper is more robust to the initial approximation than the Newton-type optimization methods available in the literature. Our approach consists of two steps: (1) reformulate the problem as a problem of determining the structured distance to singularity of an associated Sylvester matrix, and (2) integrate a system of ODEs, which describes the gradient associated to the functional to be minimized.
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