Generalized Davidson and multidirectional-type methods for the generalized singular value decomposition
Generalized Davidson and multidirectional-type methods for the generalized singular value decomposition
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发表时间：
2017-05
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通讯作者：
I. Zwaan;M. Hochstenbach
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作者：
I. Zwaan;M. Hochstenbach
我们提出了计算非平凡极值广义奇异值和向量的新迭代方法。第一种方法是广义davidson型算法，第二种方法采用多向子空间展开技术。后者的关键是一个快速截断步骤，旨在删除低质量的搜索方向，并确保搜索空间的适度增长。这两种方法都依赖于厚重启，并可能与两种不同的放气方法相结合。我们论证了该方法具有单调性和（渐近）线性收敛性，推导并讨论了局部最优展开向量，并解释了为什么快速截断步骤理想地消除了与期望的广义奇异向量正交的搜索方向。进一步，我们确定了广义Davidson-type算法与广义奇异值分解的Jacobi- Davidson算法之间的关系。最后，我们将一些已知的厄米特特征值问题的收敛性结果推广到厄米特正定广义特征值问题。数值实验表明，两种方法都具有一定的竞争力。
We propose new iterative methods for computing nontrivial extremal generalized singular values and vectors. The first method is a generalized Davidson-type algorithm and the second method employs a multidirectional subspace expansion technique. Essential to the latter is a fast truncation step designed to remove a low quality search direction and to ensure moderate growth of the search space. Both methods rely on thick restarts and may be combined with two different deflation approaches. We argue that the methods have monotonic and (asymptotic) linear convergence, derive and discuss locally optimal expansion vectors, and explain why the fast truncation step ideally removes search directions orthogonal to the desired generalized singular vector. Furthermore, we identify the relation between our generalized Davidson-type algorithm and the Jacobi--Davidson algorithm for the generalized singular value decomposition. Finally, we generalize several known convergence results for the Hermitian eigenvalue problem to the Hermitian positive definite generalized eigenvalue problem. Numerical experiments indicate that both methods are competitive.