Papers › A randomized small-block Lanczos method for large-scale null space computations
A randomized small-block Lanczos method for large-scale null space computations
Daniel Kressner, Nian Shao
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Computing the null space of a large sparse matrix A is a challenging computational problem, especially if the nullity -- the dimension of the null space -- is not small. When applying a block Lanczos method to A^𝖳 A for this purpose, conventional wisdom suggests to use a block size d that is not smaller than the nullity. In this work, we show how randomness can be utilized to allow for smaller d without sacrificing convergence or reliability. Even d = 1, corresponding to the standard single-vector Lanczos method, becomes a safe choice. This is achieved by using a small random diagonal perturbation, which moves the zero eigenvalues of A^𝖳 A away from each other, and a random initial guess. We analyze the effect of the perturbation on the attainable quality of the null space and derive convergence results that establish robust convergence for d=1. As demonstrated by our numerical experiments, a smaller block size combined with restarting and partial reorthogonalization results in reduced memory requirements and computational effort. It also allows for the incremental computation of the null space, without requiring a priori knowledge of the nullity. Our algorithm is best suited for situations when the nullity of A is moderate.
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