A Fast Randomized Algorithm for computing the Null Space

06/02/2022

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by   Yuji Nakatsukasa, et al.

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Randomized algorithms in numerical linear algebra can be fast, scalable and robust. This paper examines the effect of sketching on the right singular vectors corresponding to the smallest singular values of a tall-skinny matrix. We devise a fast algorithm for finding the trailing right singular vectors using randomization and examine the quality of the solution using multiplicative perturbation theory. For an m× n (m≥ n) matrix, the algorithm runs with complexity O(mnlog n +n^3) which is faster than the standard O(mn^2) methods. In applications, numerical experiments show great speedups including a 30× speedup for the AAA algorithm and 10× speedup for the total least squares problem.