大系统中Evans函数计算的高效打靶算法
An efficient shooting algorithm for Evans function calculations in large systems
AI总结：
针对大系统Evans函数计算中高阶外积空间维数爆炸问题，提出一种极坐标算法，通过正交基标量倍数表示单项式积，角方程采用数值优化的连续正交化方法，径向方程通过求积计算，并保持参数解析性。
摘要：
在渐近常系数线性算子谱的Evans函数计算中，一个基本问题是根据相关特征值ODE高效且数值稳定地计算子空间的演化。对于小系统，可以通过将子空间表示为单个外积来获得快速打靶算法\cite{AS,Br.1,Br.2,BrZ,BDG}。然而对于大系统，外积空间的维数迅速变得不可行，增长为$inom{n}{k}$，其中$n$是写成一阶ODE的系统维数，$k$（通常$\sim n/2$）是子空间的维数。我们通过引入一种简单的极坐标算法解决了这一困难，该算法将“纯”（单项式）积表示为正交基的标量倍数，其中角方程是Drury--Davey连续正交化方法\cite{Da,Dr}的数值优化版本，径向方程可通过求积计算。值得注意的是，极坐标方法保留了关于参数解析性的重要性质。
英文摘要：
In Evans function computations of the spectra of asymptotically constant-coefficient linear operators, a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be obtained by representing subspaces as single exterior products \cite{AS,Br.1,Br.2,BrZ,BDG}. For large systems, however, the dimension of the exterior-product space quickly becomes prohibitive, growing as $\binom{n}{k}$, where $n$ is the dimension of the system written as a first-order ODE and $k$ (typically $\sim n/2$) is the dimension of the subspace. We resolve this difficulty by the introduction of a simple polar coordinate algorithm representing ``pure'' (monomial) products as scalar multiples of orthonormal bases, for which the angular equation is a numerically optimized version of the continuous orthogonalization method of Drury--Davey \cite{Da,Dr} and the radial equation is evaluable by quadrature. Notably, the polar-coordinate method preserves the important property of analyticity with respect to parameters.
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