On some simultaneous methods based on Weierstrass' correction
The authors consider iteration methods for the simultaneous approximation of all zeros of a polynomial \(f(z)\) with complex coefficients. All methods are based on the Weierstrass correction \(W_i(z)=f(z)/\prod^n_{\substack{ k=1\\ k\neq i}} (z-z_k)\), which appears in the iteration formula \(\widehat z_i=z_i- W_i(z_i)\), \(i=1,2,\dots,n\). The \(z_i\) are approximations to the zeros of \(f(z)\). Simultaneous methods of order \(q+1\) are derived from a one point method of order \(q\) by replacing \(f(z)\) by \(W_i(z)\), \(f'(z)\) by \(W_i'(z)\), etc. For example, for Newton's method \((q=2)\), the corresponding simultaneous method is the third-order Maehly-Ehrlich-Aberth method. Simultaneous multistep methods, based on rational interpolation, are then discussed, and the paper concludes by presenting a new combined interval method based on the secant method and the \(W\)-correction. Convergence analysis is included and some numerical examples presented.
scientific article; zbMATH DE number 932485
scientific article