Weierstrass formula and zero-finding methods
For a monic polynomial \(P(z) = \prod_{j=1}^ n (z - \zeta_ j)\) of degree \(n \geq 3\), the identity \[ \zeta_ i = z - {P(z) \over \prod_{j=1,j \neq i}^ n(z - \zeta_ j)}, \quad i = 1, \dots, n, \] gives rise to the iterative process \[ z_ i^{(m+1)} = z_ i^{(m)} - W(z_ i^{(m)}),\quad W(z_ i^{(m)}) = {P(z_ i^{(m)}) \over \prod_{j=1,j \neq i}^ n (z_ i^{(m)} - z_ j^{(m)})},\;i = 1, \dots, n;\;m = 0,1, \dots \] known as Weierstrass' formula. The points \((z_ 1^{(m)}, \dots, z_ n^{(m)})\) simultaneously approximate the zeros of \(P(z)\), and the process presents quadratic convergence. This formula has been subject of several investigations (see the references given in the paper). The authors present some further applications of this method. A new result concerning localization of polynomial zeros, based on Weierstrass' correction \(W(z)\) is proved; this is used for the construction of inclusion disks which are necessary for the application of inclusion methods. Conditions for the convergence of Weierstrass' process, depending only on initial approximations and the degree of \(P(z)\) are given, and two hybrid methods (that combine both ordinary and circular complex arithmetic) for the inclusion of polynomial zeros are constructed. For the interval version of this method, a procedure for finding circular enclosures of the sets containing all zeros of \(P(z)\) is obtained. Finally, an iterative method of Weierstrass' type for the simultaneous finding of the zeros of a class of analytic functions is studied, along with the analysis of its convergence and stability. The algorithms are illustrated with several numerical examples.
scientific article; zbMATH DE number 739416