Jacobi Correction Equation, Line Search, and Conjugate Gradients in Hermitian Eigenvalue Computation II: Computing Several Extreme Eigenvalues
Jacobi Correction Equation, Line Search, and Conjugate Gradients in Hermitian Eigenvalue Computation II: Computing Several Extreme Eigenvalues
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DOI：
10.1137/070688754
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
E. Ovtchinnikov
中科院分区：
文献类型：
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作者：
E. Ovtchinnikov
本文研究如何有效地将共轭梯度法应用于埃尔米特问题的最左或最右特征值及相应特征向量的计算。一个通用的块CG算法实例化的一些可用的块CG算法被认为是新的近似特征对计算通过应用瑞利-里兹过程中的试验子空间跨越当前的近似特征向量和搜索方向向量，后者中的每一个是一个线性组合的瑞利商和所有的搜索方向从以前的迭代的相应梯度。在CG算法的局部收敛性分析中，利用了一种与Jacobi正交补修正方程有关的方法。基于理论上的考虑，提出了一种新的块共轭方案（一种计算搜索方向的方法），该方案具有一定的最优性，并已被证明在实际特征值计算中具有竞争力。
This paper addresses the question of how to efficiently adapt the conjugate gradient (CG) method to the computation of several leftmost or rightmost eigenvalues and corresponding eigenvectors of Hermitian problems. A generic block CG algorithm instantiated by some available block CG algorithms is considered whereby the new approximate eigenpairs are computed by applying the Rayleigh–Ritz procedure in the trial subspace spanning current approximate eigenvectors and the search direction vectors, each of the latter being a linear combination of the respective gradient of the Rayleigh quotient and all search directions from the previous iteration. An approach related to the so-calledJacobi orthogonal complement correctionequation is exploited in the local convergence analysis of this CG algorithm. Based on theoretical considerations, a new block conjugation scheme (a way to compute search directions) is suggested that enjoys a certain kind of optimality and has proved to be competitive in practical eigenvalue computation.