A randomized small-block Lanczos method for large-scale null space computations
<|reference_start|>A randomized small-block Lanczos method for large-scale null space computations: 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 large. When using a block Lanczos method 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 $A^{\mathsf{T}} A$ 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.<|reference_end|>
@article{kressner2024a,
    title={A randomized small-block Lanczos method for large-scale null space
  computations},
    author={Daniel Kressner and Nian Shao},
    journal={arXiv preprint arXiv:2407.04634},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.04634},
    primaryClass={math.NA cs.NA}
}