An improved Šiljakʹs algorithm for solving polynomial equations converges quadratically to multiple zeros

Šiljakʹs method provides a globally convergent algorithm for inclusion of polynomial zeros. The solution procedure is formulated as a minimization process of a positive definite function involving the real and imaginary parts of the polynomial. The main objective of this paper is to propose an improved version of Šiljakʹs algorithm, which exploits the minimizing function to ensure a quadratic convergence to multiple zeros and, at the same time, determine their multiplicity. Time comparisons with other standard zero inclusion methods are provided to demonstrate the efficiency of the proposed improvement of the original algorithm.