Broyden’s method is a quasi-Newton method which is used to solve a system of nonlinear equations. Almost all convergence theories in the literature assume existence of a root and bounds on the nonlinear function and its derivative in some neighbourhood of the root. All these conditions cannot be checked in practice. The motivation of this work is to derive a local convergence theory where all assumptions can be verified, and the existence of a root and its superlinear rate of convergence are consequences of the theory. The BFGS algorithm is a quasi-Newton method for unconstrained minimization. Also, all known convergence theories assume existence of a solution and bounds of the function in a neighbourhood of the minimizer. The second main result of this paper is a local convergence theory where all assumptions are verifiable and existence of a minimizer and superlinear convergence of the iteration are conclusions. In addition, both theories are simple in the sense that they contain as few constants as possible.

Broyden方法是一种拟牛顿法，用于求解非线性方程组。文献中几乎所有收敛理论都假设存在一个根，并且非线性函数及其在根的某些邻域中的导数有界。在实践中无法检查所有这些条件。这项工作的目的是要推导出一个可以证明所有假设都成立的局部收敛理论，并且根的存在及其收敛的超线性是该理论的结果。BFGS算法是用于无约束最小化的准牛顿法。同样，所有已知的收敛理论都假定在最小化器附近存在函数的解和界。本文的第二个主要结果是一个局部收敛理论，其中所有假设都可以验证，并且存在最小化子和迭代的超线性收敛是结论。此外，从包含尽可能少的常量的意义上讲，这两种理论都很简单。