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Computing nearby non-trivial Smith forms. (English) Zbl 1452.65084
Summary: 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.

MSC:
 65F60 Numerical computation of matrix exponential and similar matrix functions 68W30 Symbolic computation and algebraic computation
Software:
Maple; MultRoot; mctoolbox
Full Text:
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