The geometry of algorithms with orthogonality constraints
The geometry of algorithms with orthogonality constraints
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DOI：
10.1137/s0895479895290954
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发表时间：
1998-10-20
影响因子：
1.5
通讯作者：
Smith, ST
中科院分区：
文献类型：
--
作者：
Edelman, A;Arias, TA;Smith, ST
在本文中，我们在格拉斯曼（Grassmann）和斯蒂芬（Stiefel）歧管上开发了新的牛顿和共轭梯度算法。这些歧管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产生的约束。除了新算法外，我们还展示了几何框架如何提供深入的新见解，从而使我们能够创建，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分类学，该算法提供了先前无关算法的最高数学观点。我们希望新算法和扰动理论的开发人员将从本文中的理论，方法和示例中受益。
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.