Computing the Maslov index for large systems
Computing the Maslov index for large systems
复制标题
计算大型系统的马斯洛夫指数
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
S. Malham
中科院分区：
文献类型：
--
作者：
M. Beck;S. Malham
我们解决了在实线上计算大型线性辛系统的马斯洛夫指数的问题。马斯洛夫指数测量拉格朗日平面路径的有符号交点（与给定的参考平面）。拉格朗日平面的格拉斯曼方程的自然图参数化是实对称矩阵的空间。线性系统演化在图表中引发 Riccati 演化。对于大阶系统，这是一种实用的方法，因为计算复杂度是阶数的二次方。然而，Riccati 解也表现出奇点（通过改变图表来遍历）。我们的新结果涉及表征这些 Riccati 奇点和马斯洛夫指数的两个迹公式，如下所示。首先，我们证明对称图表表示的奇异特征值的数量等于与参考平面相交的维数。其次，凯莱映射是从实对称矩阵空间到酉对称矩阵流形的微分同胚。我们证明凯莱映射的对数等于反正切映射（模 2i），其迹测量朗格朗日平面与参考平面的角度。第三，凯莱映射下的 Riccati 流引起酉对称矩阵流形中的流。利用该流形上的自然酉作用，我们将流拉回酉李代数并监视其迹。这避免了奇点，并且是一个自然的稳健过程。我们通过将这些方法应用于大型特征值问题来证明它们的有效性。我们还讨论了马斯洛夫指数到无限维情况的扩展。
We address the problem of computing the Maslov index for large linear symplectic systems on the real line. The Maslov index measures the signed intersections (with a given reference plane) of a path of Lagrangian planes. The natural chart parameterization for the Grassmannian of La- grangian planes is the space of real symmetric matrices. Linear system evolu- tion induces a Riccati evolution in the chart. For large order systems this is a practical approach as the computational complexity is quadratic in the order. The Riccati solutions, however, also exhibit singularites (which are traversed by changing charts). Our new results involve characterizing these Riccati sin- gularities and two trace formulae for the Maslov index as follows. First, we show that the number of singular eigenvalues of the symmetric chart represen- tation equals the dimension of intersection with the reference plane. Second, the Cayley map is a diffeomorphism from the space of real symmetric matrices to the manifold of unitary symmetric matrices. We show the logarithm of the Cayley map equals the arctan map (modulo 2i) and its trace measures the angle of the Langrangian plane to the reference plane. Third, the Riccati flow under the Cayley map induces a flow in the manifold of unitary symmetric matrices. Using the natural unitary action on this manifold, we pullback the flow to the unitary Lie algebra and monitor its trace. This avoids singularities, and is a natural robust procedure. We demonstrate the effectiveness of these approaches by applying them to a large eigenvalue problem. We also discuss the extension of the Maslov index to the infinite dimensional case.