On the \(R\)-order of some accelerated methods for the simultaneous finding of polynomial zeros (Q1207050)

Starting from fixed-point relations derived in previous work, generalized iterative methods for the simultaneous approximation of the roots of a polynomial are investigated. Assuming fairly good initial guesses, the single step methods considered are shown to have an \(R\)-order of convergence not less than four. After a tedious proof of convergence using initial knowledge about the separation of the roots, the \(R\)-order is derived by appealing to a lower bound estimate through the spectral radius of a matrix. Furthermore the computational efficiency on a serial machine is derived, considering the processor time for different operations. Numerical examples demonstrating these claims are provided.