The self-validated method for polynomial zeros of high efficiency
An improved iterative method of Newton type for the simultaneous approximation of all complex zeros of a polynomial having only single roots is proposed. If \(P(z)=\prod_{j=1}^{n}(z-\zeta_{j})\) and \(z_{1},\dots,z_{n}\) are distinct approximations to the zeros, the method reads \[ \widehat{z}_{i}=z_{i}-\frac{1}{\frac{1}{u(z_{i})}-\sum^{n}_{{j=1 j\neq i}}\frac{1}{z_{i}-z_{j}} }\qquad (i\in \mathbf{I}_{n}). \] Here \(\widehat{z}_{i}\) is a new approximation to the zero \(\zeta_{i}\) and \[ u(z)=(\sum_{j=1}^{n}\frac{1}{z-\zeta_{j}})^{-1}. \] This leads to the 4th order method \[ \widehat{z}_{i}=z_{i}-\frac{1}{\frac{1}{u(z_{i})}-\sum^{n}_{{j=1 j\neq i}}\frac{1}{z_{i}-z_{j}+u(z_{j})}} \qquad (i\in \mathbf{I}_{n}). \] If we start instead of \(u(z)\) from Ostrowski's correction \[ \psi(z)=u(z)\frac{P(z-u(z))-P(z)}{2} \] we get the 6th order iterative method: \[ \widehat{z}_{i}=z_{i}-\frac{1}{\frac{1}{u(z_{i})}-\sum^{n}_{{j=1 j\neq i}}\frac{1}{z_{i}-z_{j}+\psi(z_{j})}} \qquad (i\in \mathbf{I}_{n}). \] As before \(\mathbf{I}_{n}=\{1,2,\dots,n\}.\)
scientific article; zbMATH DE number 5627012
scientific article