# Computing Nearby Non-trivial Smith Forms

## TL;DR

This paper introduces an optimization-based numerical method for computing the nearest matrix polynomial with a non-trivial Smith Normal Form, enabling error analysis and spectral property control.

## Contribution

It presents a novel optimization approach to find nearby matrix polynomials with specified Smith form properties, including maximum McCoy rank, with practical implementation details.

## Key findings

- Effective optimization technique for Smith form computation.
- Demonstrated robustness of the method with Maple examples.
- Generalization to maximum McCoy rank cases.

## Abstract

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 then 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 be used for an 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 an implementation of our algorithms and demonstrate the robustness with examples in Maple.