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.

翻译：我们展示了如何将对数的有理函数近似（例如 $\log z \approx (z^2 - 1)/(z^2 + 6z + 1)$）转化为快速近似计算超大型矩阵行列式的算法。通过实证表明，当与良好的预处理器结合使用时，在来自 Matérn-$5/2$ 和径向基函数高斯过程核的矩阵上，三阶有理函数近似在速度与精度之间实现了非常优越的权衡。特别地，与当前最先进的用于近似行列式的随机 Lanczos 求积方法相比，该方法在这些矩阵上显著更准确，同时运行速度大致相同。