Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
Polynomials orthogonal on the semicircle
Journal of Approximation Theory
On some improvements of square root iteration for polynomial complex zeros
Journal of Computational and Applied Mathematics
A high-order iterative formula for simultaneous determination of zeros of a polynomial
ISCM '90 Proceedings of the International Symposium on Computation mathematics
A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
The theory of Smale's point estimation and its applications
Proceedings of the international meeting on Linear/nonlinear iterative methods and verification of solution
On some simultaneous methods based on Weierstrass' correction
Journal of Computational and Applied Mathematics
Point estimation of simultaneous methods for solving polynomial equations: a survey
Journal of Computational and Applied Mathematics
Enclosing clusters of zeros of polynomials
Journal of Computational and Applied Mathematics
Ten methods to bound multiple roots of polynomials
Journal of Computational and Applied Mathematics
Family of simultaneous methods of Hansen-Patrick's type
Applied Numerical Mathematics
A family of root-finding methods with accelerated convergence
Computers & Mathematics with Applications
The convergence of a family of parallelzero-finding methods
Computers & Mathematics with Applications
Computers & Mathematics with Applications
Hi-index | 7.29 |
A one parameter family of iterative methods for the simultaneous approximation of simple complex zeros of a polynomial, based on a cubically convergent Hansen-Patrick's family, is studied. We show that the convergence of the basic family of the fourth order can be increased to five and six using Newton's and Halley's corrections, respectively. Since these corrections use the already calculated values, the computational efficiency of the accelerated methods is significantly increased. Further acceleration is achieved by applying the Gauss-Seidel approach (single-step mode). One of the most important problems in solving nonlinear equations, the construction of initial conditions which provide both the guaranteed and fast convergence, is considered for the proposed accelerated family. These conditions are computationally verifiable; they depend only on the polynomial coefficients, its degree and initial approximations, which is of practical importance. Some modifications of the considered family, providing the computation of multiple zeros of polynomials and simple zeros of a wide class of analytic functions, are also studied. Numerical examples demonstrate the convergence properties of the presented family of root-finding methods.