RANDOM POLYNOMIAL SPACE AND COMPUTATIONAL COMPLEXITY THEORY

Under the assumption on the coefficient distribution of the random polynomial space made by Smale in[1], we discuss the computational complexity for finding zeros of a polynomial, and come to the following conclusions: (ⅰ) It is inadequate in the result given by Wang Zeke and Xu Senlin to cIaim that Kuhn's algorithm is better than Newton's in solving a polynomial equation. (ⅱ) By applying the concept of a quadratic approximate zero rather than an approximate zero and estimating the volume of the set in a multi-complex space, we make certain of the fact that a combined method connecting Kuhn's algorithm and Newton's algorithm is quite efficient. (ⅲ) Further analysis shows that the estimated computational cost for an improved algorithm of Lehmer's is less than that for Kuhn's algorithm. (ⅳ) The estimated computational cost for the combined method connecting Lehmer's algorithm and Weierstrass's parallel algorithm(L-W method) is less than that for the combined method connecting Kuhn's algorithm and Newton's. Moreover, the L-W method involves no difficulty with some processes mentioned in Wilkinson's monograph on rounding errors.