Abstract: An elegant property of the maximal solution to the matrix equation X+A^TX-1A=I is presented. The property shows that the maximal solution is well-conditioned. Two new iteration methods for finding the maximal solution are proposed. Of these two methods, one is a linearly convergent iteration without matrix inversion, and one is related to Newton's method and quadratically convergent. The convergence analysis is also given. Comparisons have been made with other known methods. In all test problems the new quadratically convergent method appeared to be far superior to the other methods.

Key words: matrix equation, maximal solution, numerical methods

摘要： 首先证明了矩阵方程X+A^TX-1A=I的最大解是十分良态的，然后给出了2种求解最大解的迭代方法，并且讨论了这些方法的收敛性。这2种方法，一种是线性收敛的，其优点是迭代过程不需要求矩阵的逆；另一种是二次收敛的，数值试验的结果表明该方法在计算速度和精度方面都明显地优于现有的其他几种迭代方法。

关键词: 矩阵方程, 最大解, 数值方法