The QLP approximation to the singular value decomposition
The QLP approximation to the singular value decomposition
复制标题
DOI：
10.1137/s1064827597319519
复制
发表时间：
1999-01-01
影响因子：
3.1
通讯作者：
Stewart, GW
中科院分区：
文献类型：
--
作者：
Stewart, GW
在本文中，我们介绍了一种新的分解称为枢轴QLP分解。它是通过对矩阵X的列应用旋转正交三角化来计算的，以得到上三角因子R，然后对R的行应用相同的过程来得到下三角矩阵L。R的对角元素称为X的R值; L的对角元素称为L值。数值算例表明，L值跟踪奇异值的X相当的清晰度，远远优于R值。在L值的间隙处，分解提供X提供的行、列和零空间的类似物的标准正交基。分解所需的工作不超过旋转QR分解所需工作的两倍。R和L的计算可以交错进行，使得计算可以在任何合适的点处终止，这使得分解特别适合于低秩确定问题。交错算法还提出了一个新的，有效的2-范数估计。
In this paper we introduce a new decomposition called the pivoted QLP decomposition. It is computed by applying pivoted orthogonal triangularization to the columns of the matrix X in question to get an upper triangular factor R and then applying the same procedure to the rows of R to get a lower triangular matrix L. The diagonal elements of R are called the R-values of X; those of L are called the L-values. Numerical examples show that the L-values track the singular values of X with considerable fidelity-far better than the R-values. At a gap in the L-values the decomposition provides orthonormal bases of analogues of row, column, and null spaces provided of X. The decomposition requires no more than twice the work required for a pivoted QR decomposition. The computation of R and L can be interleaved, so that the computation can be terminated at any suitable point, which makes the decomposition especially suitable for low-rank determination problems. The interleaved algorithm also suggests a new, efficient 2-norm estimator.