Greedy and Random Broyden's Methods with Explicit Superlinear Convergence Rates in Nonlinear Equations

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.