A high order iterative formula for simultaneous determination of zeros of a polynomial

The authors propose a hybrid method to determine all the zeros of a polynomial simultaneously. First, they give a class of simultaneous iterative formulae by using the Padé approximation. The convergence order of this formula is m+2 for simple zeros and only one for multiple zeros, where m is the order of the highest derivative of a polynomial used for the Padé approximation. Next, the authors combine is simultaneous method and the single-root method which has the high convergence order even for a multiple zero. The convergence order of the hybrid method is 2m+1 for simple zeros and m for multiple zeros. From the numerical results, it can be found that the present hybrid method in the case of m=3 is more efficient for the polynomial of higher degree even if it contains multiple zeros. [In Japanese.]