矩阵多项式的牛顿迭代法

Newton Iterative Algorithm for Matrix Polynomials

研究求解如下矩阵多项式的牛顿迭代算法:P(x)=xm+A1xm-1+…+A m-1x+A m(A i为n×n的复矩阵).首先,在Pereira算法基础上,提出改进算法,以数值示例,比较各自在迭代步骤、计算速度及适用范围上的优缺点.其次,结合初始矩阵的选取方法,研究了二次矩阵多项式的完全解集,给出了求完全解集的主要步骤.

In this paper,the Newton iterative method for solving the following matrix polynomials is studied： P（ x ） =x^m ＋ A1x^m-1 ＋ … ＋Am-1x＋Am（where Ai,i = 1,…,m are n × n complex matrices）. First, based on Pereira＇ s method, the general iterative formula of Newton＇ s method is given, and then some numerical examples for the different algorithms are illustrated, and the calculation speed, scope of application, the advantages and disadvantages of iterative steps are compared. Second, combined with the given method of selecting initial ma-trix, the complete set of the quadratic matrix polynomials is studied, and the main steps for computing it are giv-en.