论文标题
增量SVD中的一个开放问题的答案
An answer to an open question in the incremental SVD
论文作者
论文摘要
品牌提出了增量的奇异值分解（SVD），以有效计算矩阵的SVD。该算法需要计算数千个或数百万个正交矩阵，并将它们倍增。但是，乘法可能会腐蚀正交性。因此，实践中需要许多重新定位。在[线性代数及其应用程序415（2006）20-30]中，Brand问：“这是一个悬而未决的问题，这是保证一定的总体数值精度所必需的；它不会改变整体复杂性。”在本文中，我们回答了这个问题，答案是我们可以避免计算大量正交矩阵，因此通过修改其算法是不需要重新定位的。我们证明修改不会改变算法的结果。提出了数值实验，以说明我们的修改的性能。
Incremental singular value decomposition (SVD) was proposed by Brand to efficiently compute the SVD of a matrix. The algorithm needs to compute thousands or millions of orthogonal matrices and to multiply them together. However, the multiplications may corrode the orthogonality. Hence many reorthogonalizations are needed in practice. In [Linear Algebra and its Applications 415 (2006) 20-30], Brand asked "It is an open question how often this is necessary to guarantee a certain overall level of numerical precision; it does not change the overall complexity." In this paper, we answer this question and the answer is we can avoid computing the large amount of those orthogonal matrices and hence the reorthogonalizations are not necessary by modifying his algorithm. We prove that the modification does not change the outcomes of the algorithm. Numerical experiments are presented to illustrate the performance of our modification.
