An effective criterion for a stable factorisation of strictly non-singular 2 × 2 matrix functions: use of the ExactMPF package

In this paper, we propose a method to factorise arbitrary strictly non-singular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:semantics> </mml:math> matrix functions, allowing for stable factorisation. For this purpose, we use the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mstyle displaystyle="true" scriptlevel="0"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="sans-serif">E</mml:mi> <mml:mi mathvariant="sans-serif">x</mml:mi> <mml:mi mathvariant="sans-serif">a</mml:mi> <mml:mi mathvariant="sans-serif">c</mml:mi> <mml:mi mathvariant="sans-serif">t</mml:mi> <mml:mi mathvariant="sans-serif">M</mml:mi> <mml:mi mathvariant="sans-serif">P</mml:mi> <mml:mi mathvariant="sans-serif">F</mml:mi> </mml:mrow> </mml:mrow> </mml:mstyle> </mml:math> package working within the Maple environment previously developed by the authors and perform an exact factorisation of a non-singular polynomial matrix function. A crucial point in the present analysis is the evaluation of a stability region of the canonical factorisation of the polynomial matrix functions. This, in turn, allows us to propose a sufficient condition for the given matrix function admitting stable factorisation.