The stochastic Lanczos quadrature method has garnered significant attention recently. Upon examination of the error analyses given by Ubaru, Chen and Saad and Cortinovis and Kressner, certain notable inconsistencies arise. It turns out that the former's results are valid for cases with symmetric quadrature nodes and may not be adequate for many practical cases such as estimating log determinant of matrices. This paper analyzes probabilistic error bound of the stochastic Lanczos quadrature method for cases with asymmetric quadrature nodes. Besides, an optimized error allocation technique is employed to minimize the overall number of matrix vector multiplications required by the stochastic Lanczos quadrature method.

翻译：随机Lanczos求积方法近年来受到了广泛关注。在回顾Ubaru、Chen与Saad以及Cortinovis与Kressner给出的误差分析时，发现其中存在某些显著的不一致之处。结果表明，前者的结论仅适用于对称求积节点的情况，而对于许多实际场景（例如矩阵对数行列式的估计）可能并不适用。本文分析了非对称求积节点下随机Lanczos求积方法的概率误差界。此外，采用了一种优化的误差分配技术，以最小化随机Lanczos求积方法所需的矩阵向量乘法总次数。