Computing Nearby Non-trivial Smith Forms

We consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem. Furthermore, we describe an effective optimization technique to find a nearby matrix polynomial with a non-trivial Smith form. The results are later generalized to include the computation of a matrix polynomial having a maximum specified number of ones in the Smith Form (i.e., with a maximum specified McCoy rank). We discuss the geometry and existence of solutions and how our results can used for a backwards error analysis. We develop an optimization-based approach and demonstrate an iterative numerical method for computing a nearby matrix polynomial with the desired spectral properties. We also describe the implementation of our algorithms and demonstrate the robustness with examples in Maple.