Higher Order Convergence Modified Chebyshev Method

This thesis consists of four chapters. In chapter one, we offer a general survey of the simple historical process of solving polynomial equations, and summarize several main methods of simultaneous iteration.In chapter two, by use of Durand-Kerner approximation, we conclude a method of five order based on the accelerated Chebyshev method (?)_i=(?)_i-(W_i((1+G_1,i)²-W_iG_2,i))/((1+G_1,i)³+W_i²G_3,i), i=1,…, n. and give the prover for its convergence order.In chapter three, under the general analysis for iteration methods, we give its initial convergence condition concretely, which depends only on the polynomial coefficients, its degree and initial approximations.In chapter four, we extend the method to the Hansen-Patrick family, and give the prover for its convergence order. In the end, we give two numerical examples, which validate the method with high order and computing efficiency.