标题： 正交性约束算法的几何学 显示英文标题

标题： The Geometry of Algorithms with Orthogonality Constraints

摘要： 在本文中，我们开发了新的牛顿和共轭梯度算法在Grassmann和Stiefel流形上。这些流形代表了在对称特征值问题、非线性特征值问题、电子结构计算和信号处理等领域中出现的约束条件。除了新的算法外，我们还展示了几何框架如何提供深入的新见解，使我们能够创建、理解和比较算法。这里提出的理论为数值线性代数算法提供了一个分类，提供了对之前互不相关的算法的高层数学视角。我们希望新算法和扰动理论的开发者将从本文的理论、方法和示例中受益。 显示英文摘要

摘要： In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.