Abstract
A method to calculate the zeros of a polynomial that has several multiple zeros is proposed. The method constructs a companion matrix whose eigenvalues are simple and the zeros of the polynomial, then finds the zeros by a floating-point eigenvalue computation of the matrix by QR algorithm. A three-term recurrence algorithm based on Euclid's method is used to calculate the values of the polynomial with arithmetic operations of O(n^2), where n is the degree of the polynomial. This method can calculate the multiple zeros and their multiplicities accurately, and it is also efficient in calculating the center of the cluster of zeros and the number of zeros in the cluster. Numerical examples are presented to illustrate the efficiency of the method.
