RANK-REVEALING QR FACTORIZATIONS AND THE SINGULAR VALUE DECOMPOSITION
RANK-REVEALING QR FACTORIZATIONS AND THE SINGULAR VALUE DECOMPOSITION
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DOI：
10.2307/2153029
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发表时间：
1992-01-01
影响因子：
2
通讯作者：
PAN, CT
中科院分区：
文献类型：
--
作者：
HONG, YP;PAN, CT
T.Chan指出，即使当矩阵A的奇异值分解已知时，如果A存在数值秩亏，如何找到A的揭示秩QR分解(RRQR)仍然不明显。本文构造性地证明了具有数值秩为r的m×n矩阵A的RRQR分解的存在性.与Chan的O(2n-r)相比，本文得到的保证RRQR存在的界都是平方根nr阶的.一段时间以来，已知如果A仅在数值上秩一亏，则通过考察A的右奇异向量对应于A的最小奇异值的元素的大小，可以得到保证A-PI的QR分解中r(Nn)小的A的列置换-PI。本文在某种程度上推广了这一众所周知的结果。
T. Chan has noted that, even when the singular value decomposition of a matrix A is known, it is still not obvious how to find a rank-revealing QR factorization (RRQR) of A if A has numerical rank deficiency. This paper offers a constructive proof of the existence of the RRQR factorization of any matrix A of size m x n with numerical rank r. The bounds derived in this paper that guarantee the existence of RRQR are all of order square-root nr, in comparison with Chan's O(2n-r). It has been known for some time that if A is only numerically rank-one deficient, then the column permutation-PI of A that guarantees a small r(nn) in the QR factorization of A-PI can be obtained by inspecting the size of the elements of the right singular vector of A corresponding to the smallest singular value of A. To some extent, our paper generalizes this well-known result.