标题： 求解具有特征向量非线性的特征值问题的逆迭代方法 显示英文标题

标题： An inverse iteration method for eigenvalue problems with eigenvector nonlinearities

摘要： 考虑一个对称矩阵$A(v)\in\RR^{n\times n}$，它依赖于一个向量$v\in\RR^n$，并满足对于任何$\alpha\in\RR\backslash{0}$的性质$A(\alpha v)=A(v)$。我们将在此研究寻找$(\lambda,v)\in\RR\times \RR^n\backslash\{0\}$的问题，使得$(\lambda,v)$是矩阵$A(v)$的一个特征对，并我们提出了针对这种类型的特征向量非线性的特征值问题的逆迭代方法的推广。所提出方法的收敛性得到研究，并显示若干收敛性质与标准特征值问题的逆迭代类似，包括局部收敛性质。如果以特定方式选择位移，该算法也被证明等价于相关常微分方程的一个特定离散化。该算法被适应到一种称为Gross-Pitaevskii方程的Schrödinger方程变体。我们使用数值模拟来说明收敛性质，以及算法和适应性的效率。 显示英文摘要

摘要： Consider a symmetric matrix $A(v)\in\RR^{n\times n}$ depending on a vector $v\in\RR^n$ and satisfying the property $A(\alpha v)=A(v)$ for any $\alpha\in\RR\backslash{0}$. We will here study the problem of finding $(\lambda,v)\in\RR\times \RR^n\backslash\{0\}$ such that $(\lambda,v)$ is an eigenpair of the matrix $A(v)$ and we propose a generalization of inverse iteration for eigenvalue problems with this type of eigenvector nonlinearity. The convergence of the proposed method is studied and several convergence properties are shown to be analogous to inverse iteration for standard eigenvalue problems, including local convergence properties. The algorithm is also shown to be equivalent to a particular discretization of an associated ordinary differential equation, if the shift is chosen in a particular way. The algorithm is adapted to a variant of the Schr\"odinger equation known as the Gross-Pitaevskii equation. We use numerical simulations toillustrate the convergence properties, as well as the efficiency of the algorithm and the adaption.