Hankel Matrices and Their Applications to the Numerical Factorization of Polynomials

AbstractGlobally convergent algorithms for the numerical factorization of polynomials are presented. When the zeros of a polynomial are all simple and of different modulus, these procedures are useful in the simultaneous determination of all zeros. These methods are derived based on the algebraic properties of sums of powers of complex numbers and Hankel matrices. The remainder and quotient polynomials which arise from applying the Euclidean and a version of Householder's algorithms are investigated in terms of their convergence properties which turn out to be useful in the splitting of a polynomial into a product of two factors

A family of root-finding methods with accelerated convergence

AbstractA parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given

Solvable Nonlinear Evolution PDEs in Multidimensional Space

A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schr\"odinger type and on a relativistically-invariant system of PDEs of Klein-Gordon type. Isochronous variants of these evolution PDEs are also considered.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA