In this paper, we propose the greedy and random Broyden's method for solving nonlinear equations. Specifically, the greedy method greedily selects the direction to maximize a certain measure of progress for approximating the current Jacobian matrix, while the random method randomly chooses a direction. We establish explicit (local) superlinear convergence rates of both methods if the initial point and approximate Jacobian are close enough to a solution and corresponding Jacobian. Our two novel variants of Broyden's method enjoy two important advantages that the approximate Jacobians of our algorithms will converge to the exact ones and the convergence rates of our algorithms are asymptotically faster than the original Broyden's method. Our work is the first time to achieve such two advantages theoretically. Our experiments also empirically validate the advantages of our algorithms.

翻译：在本文中,我们提出了贪婪和随机的Broyden解决非线性方程式的方法。 具体地说,贪婪方法贪婪地选择了最大限度地提高当前Jacobian矩阵进展度的方向,而随机方法则随机地选择了方向。 我们建立了两种方法的明确( 本地) 超线趋同率, 如果初始点和近似Jacobian 足够接近于解决方案, 和相应的Jacobian 。 我们的两种新式方法具有两个重要优势: 我们的算法近似Jacobian 将集中到准确的算法, 而我们算法的趋同率比原始的Broyden方法要快。 我们的工作是首次在理论上实现这两种优势。 我们的实验还以经验验证了我们算法的优势。