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SUMMARY:The ExactMPF: the exact matrix polynomial factorisation - Natalia 
 Adukova (Aberystwyth University)
DTSTART:20240705T130000Z
DTEND:20240705T133000Z
UID:TALK215023@talks.cam.ac.uk
DESCRIPTION:\nIn this talk\, we discuss a notion of exact solution of the 
 Wiener&ndash\;Hopf factorisation problem for matrix polynomials. By the ex
 act solution\, we understand the fulfilment of the following two condition
 s: 1) the input data belongs to the Gaussian field Q(i) of complex rationa
 l numbers and 2) all (finite) steps of the explicit algorithm can be perfo
 rmed in the rational arithmetic. Since the factorisation is generally spea
 king unstable with respect to small perturbation\, those requirements are 
 crucial to guarantee that the instability issue does not arise. Unfortunat
 ely\, even the conditions 1) &ndash\; 2) are not sufficient for the exact 
 solution to exist. We have proven the following necessary and sufficient c
 ondition: a matrix polynomial over the field of Gaussian rational numbers 
 admits the exact Wiener&ndash\;Hopf factorisation if and only if its deter
 minant is exactly factorable. For the factorisation\, we use the explicit 
 algorithm based on the method of essential polynomials. It has been proven
  already its efficiency (it provides both left and right factorisation sim
 ultaneously) but is rather technical. To help possible users\, we develop 
 its realisation within an ExactMPF package in Maple Software. We illustrat
 e its performance presenting several examples. \n
LOCATION:Seminar Room 1\, Newton Institute
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