A numerical approach to the solution of stable resultant linear systems (Q2726132)

The author presents a purely numerical approach for solving stable \((m + n) \times (m + n)\) resultant linear systems of the type \([T_n[c] |T_m[\widehat{a}]] x = b\), where \(c = [1,c_1, \ldots ,c_m]^T\) and \(\widehat{a} = [a_n, a_{n-1}, \ldots ,1]^T\) are the coefficient vectors of two given Laurent polynomials \(c(z)\) and \(\widehat{a}(z)\). \(T_n[c]\), \(T_m[\widehat{a}]\) are the associated triangular Toeplitz matrices. It is supposed that the roots of the polynomials \(c(z)\) and \(a(z) = z^{-n} \widehat{a}(z^{-1})\) lie inside the unit circle in the complex plane. NEWLINENEWLINENEWLINEThe starting point of the proposed solution methods is an equivalent polynomial formulation of the considered problem. The most expensive step in the presented algorithm for solving this polynomial equation is the evaluation of certain central coefficients of the Laurent expansion of the reciprocal of \(a(z) c(z^{-1})\). Efficient direct and iterative methods for finding these coefficients are developed. Here, ideas from structured numerical linear algebra, computational complex analysis and linear operator theory are used. A discussion of numerical experiments shows the effectivity and robustness of the proposed algorithms.