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On explicit and exact solutions of the Wiener-Hopf factorization problem for some matrix functions

Presented by: 
Victor Adukov
Tuesday 13th August 2019 - 16:00 to 16:30
INI Seminar Room 1

By an explicit solution of the factorization problem we mean the solution that can be found by finite number of some steps which we call "explicit". When we solve a specific factorization problem we must rigorously define these steps. In this talk we will do this for matrix polynomials, rational matrix functions, analytic matrix functions, meromorphic matrix functions, triangular matrix functions and others. For these classes we describe the data and procedures that are necessary for the explicit solution of the factorization problem. Since the factorization problem is unstable, the explicit solvability of the problem does not mean that we can get its numerical solution. This is the principal obstacle to use the Wiener-Hopf techniques in applied problems. For the above mentioned classes the main reason of the instability is the instability of the rank of a matrix. Numerical experiments show that the use of SVD for computation of the ranks often allows us to correctly find the partial indices for matrix polynomials. To create a test case set for numerical experiments we have to solve the problem exactly. By the exact solutions of the factorization problem we mean those solutions that can be found by symbolic computation. In the talk we obtain necessary and sufficient conditions for the existence of the exact solution to the problem for matrix polynomials and propose an algorithm for constructing of the exact solution. The solver modules in SymPy and in Maple that implement this algorithm are designed.

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons