Fast Approximate Determinants Using Rational Functions
<|reference_start|>Fast Approximate Determinants Using Rational Functions: We show how rational function approximations to the logarithm, such as $\log z \approx (z^2 - 1)/(z^2 + 6z + 1)$, can be turned into fast algorithms for approximating the determinant of a very large matrix. We empirically demonstrate that when combined with a good preconditioner, the third order rational function approximation offers a very good trade-off between speed and accuracy when measured on matrices coming from Mat\'ern-$5/2$ and radial basis function Gaussian process kernels. In particular, it is significantly more accurate on those matrices than the state-of-the-art stochastic Lanczos quadrature method for approximating determinants while running at about the same speed.<|reference_end|>
@article{colthurst2024fast,
    title={Fast Approximate Determinants Using Rational Functions},
    author={Thomas Colthurst, Srinivas Vasudevan, James Lottes, and Brian Patton},
    journal={arXiv preprint arXiv:2405.03474},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.03474},
    primaryClass={cs.DS cs.NA math.NA}
}