We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special case. By exploiting the quasiseparable structure of the projected matrices, we show that the basis vectors can be updated using a short recurrence, which can be seen as a generalization to the rational case of the Golub-Kahan bidiagonalization. We also prove error bounds that relate the error of these methods to uniform rational approximation. The effectiveness of the algorithms and the accuracy of the bounds is illustrated with numerical experiments.
翻译:我们提出了一个基于理性 Krylov 方法的算法类别, 以计算向量上通用矩阵函数的动作。 这些算法将基于Golub- Kahan 的比对性化现有方法作为特例。 通过利用预测矩阵的准可分离结构, 我们显示基础矢量可以使用短暂的重复来更新, 这可以被视为对Golub- Kahan 的比对性化合理案例的概括。 我们还证明了这些方法的错误界限, 将这些方法的错误与统一的合理近似联系起来。 算法的有效性和界限的准确性可以用数字实验来说明。