Abstract
We introduce a pair of dual concepts: pivoted blocks and reverse pivoted blocks. These blocks are the outcome of a special column pivoting strategy in QR factorization. Our main result is that under such a column pivoting strategy, the QR factorization of a given matrix can give tight estimates of any two a priori-chosen consecutive singular values of that matrix. In particular, a rank-revealing QR factorization is guaranteed when the two chosen consecutive singular values straddle a gap in the singular value spectrum that gives rise to the rank degeneracy of the given matrix. The pivoting strategy, called cyclic pivoting, can be viewed as a generalization of Golub's column pivoting and Stewart's reverse column pivoting. Numerical experiments confirm the tight estimates that our theory asserts.
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Pan, CT., Tang, P.T.P. Bounds on Singular Values Revealed by QR Factorizations. BIT Numerical Mathematics 39, 740–756 (1999). https://doi.org/10.1023/A:1022395308695
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DOI: https://doi.org/10.1023/A:1022395308695