We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit viewpoint. More precisely, we change the Newton update equation in a way that the next iterate does not only appear linearly in the update equation. Although the modifications of the update equation make the methods implicit, we show how corresponding iterates can be computed explicitly. Therefore, we can carry out steps of the implicit method using explicit procedures. In several cases, these procedures involve a solution of standard eigenvalue problems. We propose two modifications, one of the modifications leads directly to a well-established method (the self-consistent field iteration) whereas the other method is to our knowledge new and has several attractive properties. Convergence theory is provided along with several simulations which illustrate the properties of the algorithms.

我们研究并推导出非线性特征值问题的算法，其中系统矩阵取决于特征向量，或几个特征向量（或它们相应的不变子空间）。这些算法是从一个隐含的观点派生出来的。更准确地说，我们改变牛顿更新方程，使下一次迭代不仅在更新方程中线性出现。尽管更新方程的修改使方法隐含，但我们展示了如何显式计算相应的迭代。因此，我们可以使用显式过程执行隐式方法的步骤。在一些情况下，这些过程涉及标准特征值问题的解决方案。我们提出两个修改，其中一个修改直接导致了一种完善的方法（自洽场迭代），而另一种方法是我们所知的新方法，并具有几个吸引人的特性。收敛理论与说明算法特性的几个模拟一起提供。