An algorithm for numerically computing an interval matrix containing the geometric mean of two Hermitian positive definite (HPD) matrices is proposed. We consider a special continuous-time algebraic Riccati equation (CARE) where the geometric mean is the unique HPD solution, and compute an interval matrix containing a solution to the equation. We invent a change of variables designed specifically for the special CARE. By the aid of this special change of variables, the proposed algorithm gives smaller radii, and is more successful than previous approaches. Solutions to the equation are not necessarily Hermitian. We thus establish a theory for verifying that the contained solution is Hermitian. Finally, the positive definiteness of the solution is verified. Numerical results show effectiveness, efficiency, and robustness of the algorithm.

中文翻译：

---

两个矩阵的几何平均值的验证计算

提出了一种用于数值计算包含两个 Hermitian 正定 (HPD) 矩阵几何平均值的区间矩阵的算法。我们考虑一个特殊的连续时间代数 Riccati 方程 (CARE)，其中几何平均值是唯一的 HPD 解，并计算包含方程解的区间矩阵。我们发明了专为特殊护理设计的变量变化。借助这种特殊的变量变化，所提出的算法给出了更小的半径，并且比以前的方法更成功。方程的解不一定是厄米式的。因此，我们建立了一个理论来验证所包含的解是 Hermitian 的。最后验证解的正定性。数值结果显示了算法的有效性、效率和鲁棒性。