迭代法求解实对称矩阵绝对值方程

An Iterative Method for Absolute Value Equations Associated with Real Symmetric Matrices

给出了实对称矩阵绝对值方程的一个求解方法.当假设矩阵A的特征值的绝对值大于1时,绝对值方程存在唯一解,进而把绝对值方程问题转化为线性互补问题,利用不动点原理,给出了求解此类绝对值方程问题的迭代算法,并证明该算法经过有限次迭代之后收敛到原问题的一个最优解.数值实验表明此方法是有效的.

Absolute value equations （AVE） are an NP-hard problem in its general form. A new method for solving absolute value equation problems with real symmetric matrices is proposed in this paper. Firstly, the existence and uniqueness theorem of the solution to the absolute value equation is presented under the condition that the absolute value of eigenvalue of A exceeds one. Next, based on the above, the absolute value equation is transformed into a linear complementarity problem. Then, using the fixed-point princi- ple, an iterative method is obtained for the absolute value equation. It is proved that this method conver- ges to an optimal solution of the original problem after finite iterations. Finally, some numerical examples are given to indicate that the method is feasible and effective.