TL;DR
This paper introduces an algorithmic method for computing non-commutative derivatives of elements in free fields and applies it to solve matrix rational equations using a non-commutative Newton iteration.
Contribution
It provides a new algorithmic approach to compute non-commutative derivatives via linear representations and demonstrates its application to matrix equation root-finding.
Findings
Algorithm for non-commutative derivatives using linear representations
Application of derivatives to non-commutative Newton iteration
Effective method for solving matrix rational equations
Abstract
By representing elements in free fields (over a commutative field and a finite alphabet) using Cohn and Reutenauer's linear representations, we provide an algorithmic construction for the (partial) non-commutative (or Hausdorff-) derivative and show how it can be applied to the non-commutative version of the Newton iteration to find roots of matrix-valued rational equations.
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