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- W3046650850 abstract "In theory the minimum diagonal element d in a $QR$ factorization of an $m times n$ matrix A can be much larger than the smallest singular value $s_n $ of A. However, in practice d is often the same magnitude as $s_n $ and some packages rely on this to test for approximate singularity. We develop sufficient conditions and separate necessary conditions, involving A's nth singular vector, for $d gg s_n $. For a natural class of random matrices these conditions are used to obtain bounds on the probability that $d/s_n geq B$. In certain cases for large B this probability is not insignificant. This suggests that tests for approximate singularity which are based on the size of d should be used with caution. In estimating some of our probabilities we introduce techniques that lead to an enormous reduction in the required sample size." @default.
- W3046650850 created "2020-08-07" @default.
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- W3046650850 date "1990-01-01" @default.
- W3046650850 modified "2023-10-16" @default.
- W3046650850 title "The Probability of Large Diagonal Elements in the QR Factorization." @default.
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